### Introduction

This writeup is on an option combination, or strategy, known as a butterfly. It is an option strategy that is used to make money when an underlying ends up exactly at a certain level at expiry, while maintaining a modest risk profile.

### How to set up a butterfly?

An ordinary butterfly consists of three option positions on the same underlying with the same expiry, but a different strike. The call butterfly consists of:

• One long call with a strike K - X
• Two short calls with a strike K
• One long call with a strike K + X
So, an example could be a long 40 - short 2 times 50 - long 60. Let's investigate the payoff of this thing, by considering various scenarios that may occur at expiry.
• Below or at the lowest strike: All options expiry worthlessly. Payoff is 0.
• Above the lowest strike, and at or below the middle strike: The long call is in the money. Payoff equal to the amount by which the underlying is above the lowest strike
• Above the middle strike, but above the upper strike: The option with the lowest strike is in the money, but so are the two middle options. Exactly at the middle strike, the payoff is K; this goes down with 1 for each 1 the underlying goes up.
• At the top strike: All options are in the money. Due to the choice of strikes, all payoffs cancel.

As such, a butterfly has the maximum payoff when the underlying ends up exactly at the middle strike. The minimum payoff, is, conveniently, 0. As such, a butterfly is never free, and one always has to pay for it. The payment is essentially equal to the chance the underlying ends up at or close to the middle strike, and depends chiefly on the strikes, the time to expiry and the volatility of the underlying.

It is noted that a similar construction that uses puts exclusively also can be made. The payoff is quite similar. For American options, there can be a difference in value between the two that is a result of the possibility of early exercise.

### Iron Butterfly

This sounds like a psychedelic rock band, and actually, there IS a psychedelic rock band with that name. However, it is also a different way of building a butterfly. Instead of the options above, it consists of:

• A long straddle at strike K
• A short strangle with strikes K - X and K + X.
Here, the payoff is simpler to derive: it is 1 for each 1 the underlying is away from the strike of the straddle. The maximum, however, is X at the strikes of the strangle.

The iron butterfly is a such the "inverse" of a regular butterfly. Buying an iron butterfly indicates one hopes an underlying will move away from the middle strike, whereas with a regular butterfly, one hopes the opposite. Another way of looking at this is stating that a short iron butterfly is the same as a long regular butterfly, only for the iron butterfly, one does not have to pay up front, but at expiry. By selling it, one pockets the premium, with a maximum potential loss of X. In general, large trading companies prefer the iron butterfly over the regular one, as it does not require a usually illiquid in the money leg.

### Butterflies in option theory

The butterfly plays an important role in option theory. As mentioned above, a butterfly is essentially paying for the chance an underlying ends up at a certain strike. Now, imagine it is possible to price options with any possible strike. By setting up a very narrow butterfly, of which so many are bought that the maximum payoff is 1, one can estimate the probability it ends up in that region. By taking mathematical limits, it is as such possible to compute the implied probability density an underlying ends up at a certain strike. Such an infinitesimal butterfly is called an Arrow-Debreu butterfly.

### Use of butterflies in investing

Butterflies are not used extensively in investing. The main reason is that a butterfly has a high cost: one needs to trade either one liquid and two illiquid or 4 liquid options, for a relatively "flat" payoff. This basically means that if one would habitually trade butterflies, the combination of a rather efficient market in options and these high costs would likely mean that one would lose money in the long run.

Professional parties, that can trade at much lower costs, do trade butterflies. One of the reasons they can do this is because they don't trade the individual options, but rather the butterfly as a whole. A butterfly is not very risky - at most as risky as a straddle, but usually a lot less - and as such, they can trade them at sharp prices with market makers. Some exchanges offer the option to make so-called strategies, in which bid or offer in the whole butterfly can be made. This is preferable to having to trade the separate legs, because it is likely other market participants will recognize the trade as less risky and as such will do the trade more cheaply.

### Conclusion

Butterflies are option constructions that can be used to profit from a scenario in which an underlying ends up exactly at a strike. As such, they offer a very finely tailored payoff with limited risk. The disadvantage is the relatively high costs needed to set them up. They can also be used as a tool to compute the probability density an underlying ends up at a certain strike at expiry.

Disclaimer: This writeup is not meant as investment advice. Don't take investment advice from anonymous strangers on the Internet.