Broken-Wing Butterfly: Unequal Wings, Tail Risk, and Assignment

Build a matched broken-wing call butterfly, derive its signed-debit payoff by region, and reconcile execution, assignment, margin, and settlement risk.

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

Direct answer

A call broken-wing butterfly buys one lower-strike call, sells two middle-strike calls, and buys one higher-strike call, but the two strike intervals are unequal. Lock equal spread quantity Q, one underlying and expiration, the same exercise style, settlement, multiplier M, currency, and deliverable before treating the four contracts as one structure.

For K1 < K2 < K3, define the lower width L = K2 - K1, upper width U = K3 - K2, and signed net debit per underlying unit d, where a debit is positive and a credit is negative. Its expiration profit per unit is

Pi(S_T) = max(S_T - K1,0) - 2max(S_T - K2,0) + max(S_T - K3,0) - d.

The low-tail result is -d, the middle-strike result is L - d, and the high-tail result is L - U - d. The unequal widths therefore create directional tail exposure. A cheap debit or net credit does not by itself define maximum loss, and a put broken-wing structure is not automatically interchangeable with the call structure without reconciling its fixed cash component, exercise, and settlement terms.

Mechanism

Lock the exact series and claim. Record every root, call, strike, expiration, ratio +1/-2/+1, exercise style, settlement method, multiplier, deliverable, currency, adjustment memo, and official S_T; reject unmatched legs.
Build the complete timeline. Separate order entry, ex-dividend date, last trading time, customer and broker exercise cutoffs, expiration, official valuation, assignment notice, and cash or physical settlement. American exercise and physical settlement are independent fields.
Price the executable package. Use the complex-book bid, ask, size, ratio, auction state, fees, and partial-fill status. Convert the fill to signed d; four favorable leg midpoints are not one executable price.
Derive every expiration region. For S_T <= K1, use -d; for K1 < S_T <= K2, use S_T - K1 - d; for K2 < S_T < K3, use 2K2 - K1 - S_T - d; and for S_T >= K3, use L - U - d. Multiply by M x Q only after checking units.
Validate extrema and roots. Peak profit per unit is L - d; maximum contractual expiration loss is max(0,d,d + U - L). Under the common debit condition 0 < d < L with U > L, breakevens are K1 + d and K2 + L - d; otherwise solve region by region and keep only roots inside their regions.
Stress the live account. Revalue delta, gamma, theta, vega, skew, gaps, dividends, borrow, halts, zero, partial, or full short-leg assignment, Ex-by-Ex and contrary instructions, stock and strike cash, strategy, portfolio, and house margin, and broker liquidation.
Reconcile exit and lifecycle. Match package fills, closes, each roll as a close plus a new opening, exercise and assignment records, shares, cash, fees, tax lots, settlement, and next-session exposure. A defined expiration graph is not a guarantee of interim funding or realized loss.

Worked examples

Debit payoff by region. Buy the 100 call, sell two 105 calls, and buy the 115 call for d = $1.00, with M = 100 and Q = 1. Here L = $5 and U = $10. At S_T = $95/$103/$105/$109/$110/$120, account P&L is -$100/+$200/+$400/$0/-$100/-$600. Peak profit is $400, maximum low-tail loss is $100, maximum high-tail loss is $600, and the two valid breakevens are $101 and $109.
Credit does not remove the wide-wing loss. Use the same strikes but receive $0.50, so d = -$0.50. Low-tail profit is $50, body profit is $550, and high-tail loss is $450. There is no lower breakeven because the low tail is already profitable; the valid upper breakeven is $105 + $5 - (-$0.50) = $110.50. Calling the credit “income” ignores the remaining fixed wide-wing obligation.
Executable package and fees. For two spreads, suppose the opening complex market is $0.90 bid / $1.10 ask and the buy fills at $1.10. Maximum low-tail loss is $1.10 x 100 x 2 = $220; maximum high-tail loss is ($1.10 + $10 - $5) x 100 x 2 = $1,220; peak profit is ($5 - $1.10) x 100 x 2 = $780. A later executable package bid of $2.40 gives gross realized profit ($2.40 - $1.10) x 100 x 2 = $260. At $1 per contract for all eight contracts on both opening and closing, total fees are $16 and net realized profit is $244.
Partial assignment and settlement contrast. With the 100/105/115 physical American call structure and stock at $112, assignment of one of the two short 105 calls sells 100 shares for $10,500; the other three option positions remain separate. If the long 100 call has an executable bid of $12.40, selling it and buying stock for $11,200 produces management cash $10,500 + $1,240 - $11,200 = $540; exercising it instead produces $10,500 - $10,000 = $500, forfeiting $40 of extrinsic value. Neither number is total butterfly P&L. A matched European cash-settled version at official S_T = $112 instead has spread payoff -$2.00 and, after the original $1.00 debit, P&L -$300, with no shares or early assignment.

Trade and lifecycle checklist

Match exact underlying, option root, call type, strikes, expiration, ratio, multiplier, currency, and deliverable.
Verify K1 < K2 < K3, compute L and U, and state which wing is wider rather than relying on the strategy label.
Treat d as signed: positive debit, negative credit; do not switch sign conventions between formulas and broker screens.
Use the contract’s official S_T, not an unrelated stock close, after-hours print, ETF, futures quote, or live index.
Keep American or European exercise separate from physical, cash, or futures settlement.
Use executable complex-book bid or ask, displayed size, ratio, auction state, fees, and fill timestamps.
Stress partial fills and legging; an incomplete +1/-2/+1 ratio can have materially different or unlimited exposure.
Recalculate both tail results, peak result, and only breakeven roots that lie in their assumed payoff regions.
Do not call d the maximum loss until the d + U - L wide-tail result has also been checked.
Treat delta, gamma, theta, and vega as local, state-dependent sensitivities across all three strikes and the volatility surface.
Compare short-call extrinsic value, dividends, financing, borrow, tax, and broker cutoff before assessing early-assignment risk.
Model OCC-to-clearing-member and firm-to-customer allocation separately; one, both, or neither middle short may be assigned.
Keep each long wing separate after assignment; it is not automatically sold or exercised to repair the account.
Stress stock delivery, strike cash, short-stock borrow, dividend obligations, settlement timing, and overnight gaps.
Treat Ex-by-Ex as an administrative process subject to contrary instructions, broker rules, adjusted contracts, and halts.
Recheck corporate-action adjustments, nonstandard multipliers, deliverables, cash-in-lieu, and series identifiers.
Distinguish regulatory or exchange margin minima from strategy, portfolio, concentration, and broker house requirements.
Include broker liquidation, inability to borrow, trading halts, stale marks, and loss of complex-book liquidity.
Track commissions, exchange and clearing fees, taxes, bid-ask cost, slippage, and tax-lot consequences.
Reconcile every close, roll, exercise, assignment, share, cash, settlement, and next-session position from account records.

Common misconceptions

“Broken wing means one long option is missing.” The standard structure still has four contracts per spread; the strike intervals, not the contract count, are unequal.
“The debit is the maximum loss.” The wider tail can add U - L per unit to the loss, while a credit can coexist with material tail risk.
“Both displayed breakevens always exist.” A credit or an extreme debit can remove a root or move it outside the region used to derive it.
“The long wings automatically handle assignment.” Each short can be assigned independently, and disposition or exercise of a long wing requires a separate valid action.
“Defined expiration risk fixes margin and realized P&L.” Interim shares, cash, borrow, house margin, liquidation, execution, fees, tax, and settlement can differ from the four-region diagram.

Authoritative sources

Complex Spreads II: Long Condors and Broken-Wing Butterflies - The Options Industry Council; supports strategy education, not an executable quote, suitability decision, or broker margin promise.
Long Call Butterfly - The Options Industry Council; supports matched butterfly construction, expiration shape, and assignment risk, while equal-wing formulas must be modified for unequal strikes.
Characteristics and Risks of Standardized Options - The Options Clearing Corporation; covers standardized-option rights, exercise, assignment, settlement, adjustments, and risks, not a strategy recommendation.
OCC By-Laws & Rules - The Options Clearing Corporation; supports clearing-member exercise and assignment procedures, not a broker’s customer cutoff, allocation, or liquidation policy.
Trading Options: Understanding Assignment - FINRA; supports independent multi-leg assignment and resulting account obligations, not the probability of assignment.
4210. Margin Requirements - FINRA; supplies regulatory margin definitions and authority, not exact portfolio or house buying power or protection from liquidation.
Cboe Margin Manual - Cboe Global Markets; illustrates exchange margin treatment, not universal broker requirements or a guarantee that interim risk equals expiration loss.
Cboe Titanium U.S. Options Complex Book Process - Cboe Global Markets; supports Cboe-specific ratios, package bids and offers, auctions, and partial execution, not every venue’s execution logic.