Constructing a Butterfly Payoff from Call Options

Summary

The document presents a compact payoff-replication example for a butterfly spread. It represents a desired piecewise-linear payoff by its slopes across strike intervals, then expresses call-option payoffs as vectors over those same intervals. The position in each call is chosen so that the weighted combination of option payoff vectors reproduces the target slope pattern. With strikes at 98, 100, and 102 and a desired slope sequence of zero, positive one, negative one, and zero, the example gives call positions of one, negative two, and one.

This is an algebraic way to see how a butterfly can be assembled from options at three strikes, including the short middle-strike position. The document is only a question seeking references for the notation and method; it provides no derivation, general algorithm, pricing discussion, or payoff diagram. It therefore illustrates the replication idea but does not establish how to handle strike spacing, premiums, exercise conventions, or other payoff shapes. Those issues would need to be addressed separately when applying the representation.

Key ideas

• A piecewise-linear option payoff can be represented by its slopes across intervals separated by strikes.
• Call payoff vectors can be combined to reproduce a specified target slope pattern.
• The example constructs a butterfly using long calls at the outer strikes and two short calls at the middle strike.
• The document gives an illustration but does not develop a general method or discuss pricing and implementation details.

Tags

# Wheres is this method/notation of option portfolio payoff design from?


# Wheres is this method/notation of option portfolio payoff design from?












The "desired position" in the image is a set of slopes $(0,1,-1,0)$, and a set of strike prices between these slopes $\mathbf{K}=(98,100,102)$.

The payoff is then designed by finding the positions $n_1,n_2,n_3$ in three call options

$$c_1=(0,1,1,1)\mathbf{K}$$ $$c_2=(0,0,1,1)\mathbf{K}$$ $$c_3=(0,0,0,1)\mathbf{K}$$

So that they total the desired payoff $(0,1,-1,0)$.

In this case of a butterfly spread, the required postions are $n_1=1$ $n_2=-2$ and $n_3=1$.

Any points to litterature where this method is used is appreciated.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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