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Butterfly Arbitrage: Strike Convexity, Weighted Spreads, and Execution

Test European call-price convexity with equal- and unequal-strike butterflies, then distinguish fitted or midpoint violations from executable static arbitrage after depth, fees, settlement, and account constraints.

Updated

For educational purposes only; not investment advice. Investing may result in loss.

Direct answer

For matched European calls on one underlying and expiration, no-arbitrage call prices must be nonincreasing and convex in strike. At equally spaced strikes K_1 = K − h, K_2 = K, and K_3 = K + h, the long +1:−2:+1 call butterfly costs B = C(K_1) − 2C(K_2) + C(K_3) and has a nonnegative terminal payoff, so B ≥ 0 in the frictionless model.

A negative fitted value or midpoint is only a diagnostic. Executable static arbitrage requires synchronized asks for both purchased wings, bids for the two sold body calls, sufficient size, matched contract specifications, and a net initial receipt that remains positive after commissions, exchange fees, financing, margin, tax and settlement costs. American exercise, physical delivery, partial assignment, legging and broker liquidation can add interim obligations even when the combined expiration payoff is nonnegative.

Seven-step convexity and execution test

  1. Lock one matched claim set. Record underlying, option root, call side, K_1 < K_2 < K_3, expiration, exercise style, settlement, multiplier, deliverable, adjustment status, currency, venue, quote timestamp and official terminal-value definition. Calls with different specifications do not form one static convexity test.
  2. Normalize the strike geometry. For equal spacing require K_2 − K_1 = K_3 − K_2 and use ratio +1:−2:+1. For unequal spacing define w = (K_3 − K_2) ÷ (K_3 − K_1) and test C(K_2) ≤ wC(K_1) + (1 − w)C(K_3). Equivalent integer quantities are proportional to (K_3 − K_2):−(K_3 − K_1):(K_2 − K_1) after reducing by a common divisor where possible.
  3. Separate research marks from executable sides. Use synchronized quotes and conditions. A long butterfly pays the ask for each positive wing and receives the bid for each negative body call. Last trades, independent timestamps, theoretical values and midpoints can diagnose a surface but cannot establish proceeds.
  4. Prove the terminal payoff by region. Evaluate S_T ≤ K_1, K_1 < S_T ≤ K_2, K_2 < S_T < K_3, and S_T ≥ K_3 with the exact ratio and official S_T. For equal spacing the payoff is zero in both tails and peaks at h at K_2; for unequal spacing the weighted construction also has zero tails and nonnegative interior payoff.
  5. Convert the edge to executable account cash. Multiply signed net credit by actual multiplier and complete-package quantity, then subtract all opening and expected lifecycle costs. Limit quantity to simultaneous depth in every leg and respect ratios, position limits, margin, capital, borrow and broker permissions.
  6. Stress lifecycle breakage. Test complex-order rejection, partial fills, legging, quote cancellation, halts, American early or partial assignment, exercise-by-exception, contrary instructions, physical stock and strike cash, official cash settlement, adjustments and broker liquidation. A nonnegative expiration identity does not coordinate the legs operationally.
  7. Reconcile and preserve evidence. Save raw quotes, feed or venue, timestamp, sizes, order ratio, limit price, fills, fees, assignment and settlement files, tax lots and remaining positions. Reprice the completed package and distinguish a fitted shape violation, an executable quote opportunity, an accepted order and a fully settled realized result.

Worked examples

  • Midpoint violation versus executable package. Equal strikes are K_1 = $90, K_2 = $100, and K_3 = $110. Midpoint call values $15.00, $10.00, and $4.00 imply B_mid = $15.00 − 2 × $10.00 + $4.00 = −$1.00, or a theoretical $100 credit at multiplier 100; terminal payoff peaks at $10 × 100 = $1,000 when S_T = $100. But wing asks $15.30 and $4.25 plus body bid $9.75 give B_exec = $15.30 − 2 × $9.75 + $4.25 = $0.05, a $5 debit. The midpoint curve violates convexity, but those executable sides do not offer the arbitrage.
  • Unequal strikes and integer ratio. Let K_1 = $90, K_2 = $100, and K_3 = $115. Then w = ($115 − $100) ÷ ($115 − $90) = 0.60, so a convex curve requires C($100) ≤ 0.60C($90) + 0.40C($115). With synchronized executable values C($90) = $16.00, C($100) = $11.20, and C($115) = $3.00, the right side is $10.80, a $0.40 normalized violation. The tradable ratio reduces from +15:−25:+10 to +3:−5:+2; its cost is 3 × $16.00 − 5 × $11.20 + 2 × $3.00 = −$2.00, a $200 credit at multiplier 100, while its nonnegative payoff peaks at $30 × 100 = $3,000 at S_T = $100.
  • Fully executable European cash-settled example. Assume a synchronized +1:−2:+1 complex order on matched European cash-settled calls fills for a $0.12 credit with multiplier 100. Gross receipt is $0.12 × 100 = $12. If four contract fills cost $1 per contract, net initial receipt is $12 − $4 = $8. At S_T = $103 for 100/105/110 strikes, terminal option payoff is ($103 − $100) × 100 = $300, so total before tax is $8 + $300 = $308; in either tail the payoff is zero and the retained amount is $8. This is an arbitrage only under the stated matched, executable, funded and collectible assumptions.
  • Depth caps the realizable scale. A complex quote shows a $0.15 credit, but synchronized displayed sizes support 4 low wings, 12 middle calls and 10 high wings. The +1:−2:+1 ratio therefore supports only min(4, 12 ÷ 2, 10) = 4 complete butterflies, not 10. Gross credit is 4 × $0.15 × 100 = $60. At $0.65 per contract across 4 × 4 = 16 contracts, fees are $10.40, leaving $60 − $10.40 = $49.60. The remaining six packages are not proven executable; selling body calls first would create unmatched short-call exposure.

Risks and validation controls

  • Match underlying, root, call side, expiration, exercise style, settlement, multiplier, deliverable, currency and adjustment status.
  • Verify strike ordering and exact spacing before choosing equal or weighted ratios.
  • Reduce unequal-spacing ratios to valid integer contracts without changing payoff weights.
  • Use synchronized executable asks for purchased calls and bids for sold calls, not midpoint or last trade.
  • Record quote source, venue, timestamp, session, condition, displayed size and latency.
  • Check complete-package depth; each body consumes two contracts in an equal butterfly.
  • Submit a ratio-correct complex order where available; legging creates delta, gamma, vega and gap exposure.
  • Treat order acceptance as distinct from full execution; partial fills can leave naked short calls.
  • Include commissions, exchange, clearing, routing, financing, settlement and tax costs.
  • Obtain strategy, portfolio and broker house-margin treatment; capital use can exceed the initial edge.
  • Check position and exercise limits, account permissions, concentration and broker liquidation policies.
  • Separate American or European exercise from physical, cash or futures settlement.
  • Model early and partial assignment of short body calls; long wings do not act automatically.
  • Verify exercise-by-exception, contrary instructions, broker cutoffs, pin risk and after-hours moves.
  • Define S_T from the contract; close, live index, ETF and futures prices may not equal official settlement.
  • Recalculate after splits, mergers, distributions or other deliverable adjustments.
  • Treat a negative fitted density as a model or data diagnostic until executable quotes confirm it.
  • Enforce decreasing, convex and calendar-consistent surfaces without hiding raw data or uncertainty.
  • Stress stale quotes, crossed markets, halts, delayed openings, corrections and unavailable exits.
  • Reconcile premiums, shares, strike cash, settlement, margin, collateral, fees and tax lots after final files.

Common misconceptions

  • “A negative midpoint butterfly is free money.” Midpoints are not executable, and synchronized natural sides can turn the apparent credit into a debit.
  • “The +1:−2:+1 formula works for any three strikes.” It requires equal spacing; unequal strikes need weighted ratios that may require many contracts.
  • “A nonnegative terminal payoff removes every risk.” Partial fills, assignment, funding, margin, settlement, broker and tax risks remain.
  • “A negative fitted density proves a trade exists.” Smoothing, interpolation, stale or crossed quotes can create a diagnostic violation without an executable package.
  • “A small edge can be scaled at the displayed price.” The scarcest leg and exact ratio cap synchronized quantity, while fees and impact can consume the edge.

Authoritative sources

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