Butterfly Spreads

Most option strategies bet that the stock moves somewhere. A butterfly spread bets the opposite: that the stock lands near a specific price and stays there, and it does this while risking only a small, fixed amount and never requiring you to guess the direction of a move that doesn't happen.

A dart that only pays if it's near the bullseye

Imagine a dartboard game where you get paid the most for landing exactly on the bullseye, a shrinking amount as you land further away, and nothing at all past a certain radius — and crucially, the most you can ever lose for playing is the small price of the dart. A butterfly spread constructs exactly that payoff out of options: buy one option at a low strike, sell two options at a middle strike, buy one option at a high strike, all same expiry, evenly spaced. The middle strike is the bullseye — maximum profit if the stock finishes exactly there — and profit tapers off linearly to zero at the outer strikes, beyond which you simply lose the (small) amount you paid to set the trade up.

The formula

$$\text{Payoff}_T = (S_T - K_1)^+ - 2(S_T - K_2)^+ + (S_T - K_3)^+, \qquad K_2 - K_1 = K_3 - K_2$$

In plain English: buy a call at the low strike $K_1$, sell two calls at the middle strike $K_2$, buy one call at the high strike $K_3$, with the strikes evenly spaced. At expiry, below $K_1$ every term is zero — you lose only what you paid. Between $K_1$ and $K_2$, the payoff rises linearly with the stock, driven by the single long call. Between $K_2$ and $K_3$, the two short calls start eating into that gain twice as fast as the long call adds it, so the payoff falls back down. Above $K_3$, everything cancels and the payoff is flat at zero. The maximum possible payoff, exactly at $K_2$, equals $K_2 - K_1$ (the strike spacing) minus whatever net premium you paid.

Worked example 1: pricing and the payout at the pin

Strikes at $95, $100, $105, all calls, same expiry. Prices: $95 call $8.20, $100 call $4.60, $105 call $2.10. Cost of the butterfly: buy $95 ($-8.20$), sell two $100 ($+2 \times 4.60 = +9.20$), buy $105 ($-2.10$). Net cost: $-8.20 + 9.20 - 2.10 = -1.10$, so you pay $1.10 to enter. If the stock finishes exactly at $100, the payoff is $\text{max}(100-95,0) - 2\max(0,0) + \max(0,0) = 5 - 0 + 0 = 5.00$, i.e. $5.00, for a profit of $5.00 - 1.10 = 3.90$, i.e. $3.90 — a return of over 350% on the $1.10 risked. If the stock finishes at $92 or $112 (outside the wings), the payoff is $0 and you lose the full $1.10, and nothing more.

Worked example 2: the break-even points

Using the same butterfly (cost $1.10, strikes $95/$100/$105), find where the position breaks even. On the way up, the payoff between $K_1$ and $K_2$ is $S_T - 95$; setting this equal to the $1.10 cost gives $S_T = 96.10$. On the way down from the peak, the payoff between $K_2$ and $K_3$ is $105 - S_T$ (by symmetry of the evenly spaced structure); setting this equal to $1.10 gives $S_T = 103.90$. So the trade is profitable only within the narrow $96.10–$103.90 range, a band just under 8% wide around the $100 pin, out of a maximum possible profit window running from $95 to $105. Most of the position's value depends on the stock landing inside a fairly tight zone.

Drag the strikes on this explorer and watch the tent-shaped payoff — widen the wings and the profit zone grows but the maximum payout per dollar risked shrinks; narrow them and it's the reverse.

What this means in practice

Market makers use butterflies to trade the shape of the volatility smile without taking outright directional or volatility-level risk — a butterfly's price is highly sensitive to how curved the implied volatility surface is near a strike, which is exactly the quantity Breeden-Litzenberger Formula connects to the implied probability density. Traders around known pinning events (index rebalancing dates, options expiries with heavy open interest at a strike) also use butterflies to express a view that the stock will gravitate toward a specific level.