Understand iron butterfly — Stock-Market notes

WHAT is it?

WHY build it this way?

Selling the ATM straddle harvests the most premium (ATM options are richest in time value).
Buying the far wings is cheap insurance: they cost little but hard-cap the loss.
Net effect: you receive a credit and your maximum loss is known in advance.

HOW the payoff is derived (from first principles)

Let $S$ = stock price at expiry, $K$ = center, $w$ = wing width, $C$ = net credit received.

Each option's value at expiry (intrinsic only):

Long call payoff: $\max(S-K_c,0)$
Long put payoff: $\max(K_p-S,0)$
Short = negative of the above.

Sum the four legs, then add the credit $C$ you were paid.

P(S) = C \underbrace{-\max(S-K,0)}_{\text{short call}} \underbrace{-\max(K-S,0)}_{\text{short put}} \underbrace{+\max(S-(K+w),0)}_{\text{long call wing}} \underbrace{+\max((K-w)-S,0)}_{\text{long put wing}}

Evaluate at the key point $S=K$ (why: all "distance" terms vanish, so we see the peak):

P(K) = C - 0 - 0 + 0 + 0 = C

So max profit = net credit $C$ , achieved when the stock pins the center strike. Why this step? At $S=K$ every option is at- or out-of-the-money → all intrinsic values are 0 → you keep the whole credit.

Evaluate at a wing, say $S=K+w$ (why: this is where a wing kicks in and stops the bleeding):

P(K+w) = C - \underbrace{w}_{\text{short call loses } w} - 0 + 0 + 0 = C - w

Beyond $K+w$ , the long call gains exactly offset further short-call losses, so loss stays flat at $C-w$ . By symmetry the same happens at $S=K-w$ .

Formula The four numbers you must know

$\boxed{\text{Max Profit} = C}$ $\boxed{\text{Max Loss} = w - C}\quad(\text{the } C-w \text{ above is negative; loss magnitude is } w-C)$ $\boxed{\text{Breakevens} = K \pm (\,\text{Wing gains }= C\,)\ \Rightarrow\ K - C \text{ and } K + C}$ $\boxed{\text{Risk/Reward: risk } (w-C)\text{ to make } C}$ where $C$ = net credit, $w$ = wing width, $K$ = center strike (all per-share; multiply by lot size for money).

WHY are breakevens at $K \pm C$ ? Starting from the peak $C$ at $S=K$ , profit falls at slope 1 as $S$ moves away (one short option going ITM). It reaches 0 after moving a distance equal to $C$ . So breakeven $= K \pm C$ .

Worked Examples

Common Mistakes (Steel-manned)

Active Recall

Recall Forecast-then-Verify (cover the answers!)

Where is max profit? → At $S=K$ , equal to the net credit $C$ .
Max loss formula? → $w-C$ .
Breakevens? → $K-C$ and $K+C$ .
Which vol environment to sell it in? → High IV.
What outcome do you root for? → Stock pins the center strike.

Recall Feynman: explain to a 12-year-old

Imagine you bet that a spinning coin will land right on its edge in the exact middle of a table. If it lands dead-center, you win the whole prize (the credit). If it rolls a little, you win less. If it rolls off toward the table's edge, you lose — but you put fences (the wings) near the edges so you can never lose everything. You want the coin to freeze in the middle.

Flashcards

What 4 legs make an iron butterfly?

Sell ATM call, sell ATM put, buy OTM call (

K+w

), buy OTM put (

K-w

), same expiry.

Is an iron butterfly a debit or credit strategy?

Net credit (you receive premium

C

Iron butterfly max profit?

The net credit

C

, achieved when

S=K

(center strike) at expiry.

Iron butterfly max loss?

Wing width minus credit:

w-C

Iron butterfly breakevens?

K-C

and

K+C

Why buy the wings?

They cap risk, converting the short straddle's unlimited loss into a defined loss.

Iron butterfly vs iron condor?

Butterfly = same ATM short strikes (higher credit, narrow peak); condor = two different OTM short strikes (lower credit, wider plateau).

Best IV environment to sell an iron butterfly?

High implied volatility (fatter credit

C

What outcome do you want as the seller?

The stock to pin the center strike so all options expire near-worthless.

Regroup the iron butterfly as two spreads?

A short call vertical + a short put vertical, each of width

w

Connections

Short Straddle — the "engine"; iron butterfly = short straddle + protective wings.
Iron Condor — the wider-strike cousin; contrast the payoff shapes.
Vertical Spreads — the butterfly is two credit spreads glued together.
Implied Volatility — governs the size of the credit $C$ .
Option Greeks — short Vega & short Gamma, positive Theta near the center.
Payoff Diagrams — how to draw and read the tent shape.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Iron butterfly ek aisi strategy hai jahan aap bet lagate ho ki stock hilega nahi, ek center strike ke aas-paas hi ruka rahega. Aap ATM ki call aur put dono bech dete ho (isse maximum premium milta hai kyunki ATM options me time value sabse zyada hoti hai) — yeh ek short straddle ban gaya. Phir upar aur neeche dono taraf ek-ek OTM option kharid lete ho — yeh "wings" aapki insurance hain, jo unlimited loss ko limited bana dete hain. Net me aapko credit milta hai, aur maximum loss pehle se pata hota hai.

Formulas ek line me: Max profit = credit $C$ (jab stock exactly center strike $K$ par band ho), Max loss = $w - C$ (wing width minus credit), aur breakevens = $K \pm C$ . Payoff ka shape ek tent (▲) jaisa hota hai — beech me sabse ooncha (max profit), donon taraf slope down, aur wings ke baad flat (max loss). Aapko chahiye ki stock beech me hi pin ho jaye.

Ek important baat: iron butterfly high IV me bechna chahiye, taaki credit $C$ mota mile — chhota credit matlab risk zyada aur reward kam. Aur ise iron condor se confuse mat karo: butterfly me dono short strikes same (ATM) hote hain (sharp peak), condor me alag-alag OTM strikes hote hain (chaudi plateau). Common galti: log samajhte hain max loss = wing width, lekin actual me credit ghata do → loss $= w - C$ . Doosri galti: log chahte hain stock wing tak jaye — nahi! Wing tak gaya to loss hota hai; aapko stock center par chahiye.

Test yourself — Options Strategies

Connections