The broken-wing butterfly

A butterfly with one wing pushed further away than the other.
The asymmetry is the whole point: it can be opened for a credit, and it moves the risk to one side.

What it is

Three strikes, one expiry.
Two options sold at the body, one bought either side.
One wing is further out than the other — that is the “broken” part.
Moving that wing out makes the whole thing cheaper — and moves the risk onto the side you widened.

What you are actually agreeing to

It costs far less than a standard butterfly — the ticket above shows how little.
In the market it often opens for a credit. That comes from skew: far out-of-the-money puts are dearer than a flat model makes them. These pages price every leg at one volatility, so the credit version does not appear here.
There is a wide zone where you keep everything you were paid or nearly all of what you staked.
The side you widened is where the loss lives, and it is larger than the credit.
You are trading a small, frequent win against a rarer, larger loss on one side only.

The trade, written out

Everything above is the idea. This is the position — the lines you would actually send to a broker.

Order ticket

Leg	Action	Qty	Expiry	DTE	Strike	Delta	Price
1	Buy to open	+1	16 Oct 26	45	100 Put	-0.46 (46Δ)	$3.25
2	Sell to open	-2	16 Oct 26	45	95 Put	+0.49 (49Δ)	$1.35
3	Buy to open	+1	16 Oct 26	45	85 Put	-0.03 (3Δ)	$0.09
Net	Debit	1 position	16 Oct 26	45	—	+0.01	$0.65

BUY +1 BUTTERFLY XYZ 16 OCT 26 85/95/100 PUT @ 0.65 DEBIT

Reading it back

Buy one put above the body, sell two at the body.
Buy one further below — further than the first wing is above.
The unequal gap is what makes it so cheap.
The ticket shows what it actually costs at one flat volatility. With real skew it would cost less again, and often nothing at all.

The argument

Bull

makes the claim

Almost free to open, a wide area where I keep everything, and no loss at all if the stock goes the wrong way entirely. Most weeks the market does nothing and this simply pays.

Bear

doubts it

You have moved your protection further away and been paid a small sum for it. The gap you opened is exactly where the loss goes, and it is bigger than everything you collected. This is the shape of a strategy that works until it doesn’t.

Ferret

settles it

The interesting number is the size of the losing zone relative to how often the stock lands in it. Nobody in this argument has counted that, and it is countable.

The shape of it

The broken-wing butterfly on XYZ at $100 · 25% volatility · 4% rate · 45 days. Priced from the model, not written by hand.

	At expiry
Debit paid	$65
Maximum profit	$435
Maximum loss	$-565
Breakevens	$90.65, $99.35

Flat where nothing happens, a peak at the body, and one side that falls further than the other.
Maximum profit is at the body strike.
Maximum loss is on the widened side, and the table computes it — it is not the credit.
There is a whole region where the credit is simply kept, and that region is the reason people trade these.

Where you actually enter

The chart above is the shape at expiry. It puts the bend right next to the money, which makes the trade look like it is already on the edge. It is not — here is the day you open it.

The same structure set up at 25Δ · 45 days · credit $65. The shaded band is the room between the stock and the strike you sold.

The same price, two different days

	On day one	At expiry
Stock at $95 (the strike you sold)	$-36	$435
Stock unchanged at $100	$0	$-65

You enter above the body, in the zone that pays if nothing happens.
The day-one line is far flatter than the expiry shape, so early moves feel like very little.
The wide side does not announce itself early. It only bites near the end.
Which makes this a structure where the exit rule matters more than usual.

Different ways to set it up

The dials here are where the body sits and how far the broken wing is pushed. The panels below move the body; the wing gap stays fixed.

Same delta, four tenors

Days	Short strike	Away from spot	Credit	Max loss	Risk : reward
7	98	+2.0%	$218	$-718	—
21	96	+4.0%	$119	$-619	—
45	95	+5.0%	$65	$-565	—
90	94	+6.0%	$14	$-514	—

Short-dated versions collect little and resolve quickly.
Longer ones collect more and leave the wide side exposed for longer.
The shape barely exists until near expiry — the panels show how flat day one is.

Same tenor, three deltas

Delta	Short strike	Away from spot	Credit	Max loss	Risk : reward
40Δ	99	+1.0%	$32	$-532	—
25Δ	95	+5.0%	$65	$-565	—
15Δ	92	+8.0%	$62	$-562	—

Move the body closer to the money and the credit grows.
So does the chance of finishing on the wrong side of it.
The zone where you keep everything shrinks as the body comes in, which is the trade being made.

A butterfly is a bet on where the stock finishes. Breaking a wing turns it into a bet on where it doesn’t.

What the Greeks are doing

Four questions about the same position, and a fifth that barely applies here.
Every structure answers them differently, which is why this site never writes a general page about the Greeks.

Days left	Delta	Gamma	Theta / day	Vega	Rho
45	+0.8	-1.99	$+1.70	$-6.14	$+0.02
21	-12.0	-1.91	$+1.78	$-2.74	$-0.75
7	-35.6	+4.27	$-3.25	$+2.05	$-0.70

The position at entry — stock $100 · 25% volatility. Per contract, from the model.

Delta — which way you need the stock to go

Small at entry, and it changes sign depending on which side of the body the stock sits.
The asymmetry means it is never as neutral as it looks.

Theta — what time does to you

Positive while the stock is in the profitable zone.
It turns against you near the body, which is the opposite of intuitive.

Vega — what a change in fear does to you

Short volatility overall, and unevenly so across the price range.

Gamma — how fast your delta turns against you

Negative around the body, where the two short options sit.
It is the sharpest part of the structure and it sharpens with time.

Rho — what a change in interest rates does to you

Negligible here.
One row.

Theta and gamma are one thing

This is the part that decides whether the strategy works, and it is almost never put plainly.

You cannot be paid theta without being short gamma. They are not two features of the trade. They are the rent and the risk on one lease.

The same position, three points in its life

Nothing about the position changes.
The stock sits still at $100 throughout.
Only the days left move.

Days left	Delta	Gamma	Theta / day	Vega	Rho
45	+0.8	-1.99	$+1.70	$-6.14	$+0.02
21	-12.0	-1.91	$+1.78	$-2.74	$-0.75
7	-35.6	+4.27	$-3.25	$+2.05	$-0.70

Now move the stock to $95.50

Same three dates, same butterfly.
Now with the stock down at the body — the peak, and the most unstable place on the whole shape.

Days left	Delta	Gamma	Theta / day	Vega	Rho
45	+13.2	-3.33	$+2.47	$-9.36	$+1.51
21	+9.6	-7.39	$+5.68	$-9.69	$+0.46
7	-5.3	-18.41	$+14.46	$-8.05	$-0.15

Gamma goes from -3.33 to -18.41.
Theta goes from $+2.47 a day to $+14.46.
Both climb, and neither is available without the other.
Delta now moves roughly 6 times as far for every dollar the stock travels.

Which is what DTE actually sets

The body is where all the gamma is, and time concentrates it there.
A broken-wing butterfly is almost entirely an expiry-week structure.
Everything interesting about it happens in the last few days, which is precisely when it is most dangerous to be holding it.

The trap

The trap is the phrase “opened for a credit”, which is how this structure is almost always sold.

Being paid to open a position tells you nothing about what it can cost you.

A credit at entry — or a ticket that costs almost nothing — feels like a free position.
That cheapness is payment for widening one wing, and the widened side is where the maximum loss lives.
That loss is many times what you put in — the table in section four computes both.
The zone where you keep everything is wide, so the strategy wins often. That is the shape most likely to be mistaken for an edge.

Why it matters

A high win rate with one large, rare loss is exactly the profile that needs expectancy rather than a win rate.
It is also the profile that survives a short backtest and fails a long one, because the losing side may simply not have occurred yet.

How this shows up in our tests — pending. No result has been published on this structure yet.