What is a Butterfly Spread?
A butterfly is a three-strike option structure — long the wings, short the body in 1:2:1 ratio. It expresses a view that spot will land at the middle strike, or a view on volga.
Definition
A long call butterfly is: long 1 K1 call, short 2 K2 calls, long 1 K3 call (with K1 < K2 < K3, typically equally spaced). The payoff at expiry is a tent: zero outside the wings, rising linearly toward K2 from both sides, peaking at K2 − K1 (= K3 − K2) and then descending. The structure costs a small debit and has a small maximum loss; the maximum gain happens if spot pins exactly at K2 at expiry. As a vol structure, a butterfly is "short ATM vol, long wing vol" — it benefits if the smile flattens (ATM falls or wings rise) and from spot landing at the middle strike. The "iron butterfly" replaces calls in the wings with same-strike puts, with a similar P&L profile but different margining.
Why it matters & how it's calculated
In terms of Greeks, a long butterfly is short gamma at the body strike (you want spot to stop moving), positive theta (time decay helps), and structurally long volga (the wings get vega-fat fast if vol explodes). The 1:2:1 ratio matters: it makes the structure roughly vega-neutral at trade inception when the surface is flat, so the P&L is dominated by spot pinning + smile shape. Butterflies are the textbook instrument for "betting on the smile shape" — the implied 3-strike P&L at expiry is closely related to the butterfly's arbitrage-free price, which is itself related to the smile curvature. If you fit a vol smile and the model says the 4,900/5,000/5,100 butterfly should cost $8 but the market is pricing it at $12, the market is implying more curvature (more spot-pinning at 5,000) than your model — useful information for both vol arbitrage and for reading positioning.
Worked example
SPX at 5,000. You buy 100 butterflies: long 100 SPX 4,950 calls, short 200 SPX 5,000 calls, long 100 SPX 5,050 calls — 30 days to expiry. Net debit $8 per fly. If SPX pins at 5,000 on expiry: payoff = ($50 − $8) × 100 × 100 = $420,000. If SPX is below 4,950 or above 5,050: loss = $8 × 100 × 100 = $80,000. Risk/reward ≈ 5:1, with the breakeven being modest pinning around the body.
Related concepts
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