Strategies

Defined risk versus undefined risk

A position is defined-risk when maximum loss is a finite number computable at entry. Every long option, every vertical spread, and every structure whose short legs are fully covered by long legs at equal or better strikes is defined-risk.

• Naked short calls carry theoretically unlimited loss because the underlying has no upper bound.
• Naked short puts carry maximum loss of K - P per share, realised at S = 0. This is bounded but often large.
• A ratio spread with more short than long contracts is undefined-risk on the side with the naked excess, regardless of the long leg.

Credit and debit vertical equivalence

A bull call spread and a bull put spread at the same strikes and expiration have identical expiration payoff profiles. The same holds for the bear pair. They differ only in cash flow at entry and in assignment exposure.

Field	Value
Formula	BullCall(K1,K2) payoff == BullPut(K1,K2) payoff, for all S at expiry
Worked	The profiles coincide when C = (K2 - K1) - D, which is the no-arbitrage condition. K1 = 100, K2 = 110. Bull call for D = 2.10: max profit 7.90, max loss 2.10, breakeven 102.10. Bull put at the same strikes for C = 10 - 2.10 = 7.90: max profit 7.90, max loss 10 - 7.90 = 2.10, breakeven 110 - 7.90 = 102.10. Both verified by brute-force evaluation of the payoff expressions at 0.0001 granularity

• The credit version holds short options that are in the money when the trade is losing, creating early-assignment exposure on the short leg.
• The debit version holds long options that are in the money when the trade is winning.
• Choice between them is usually driven by assignment risk, financing, and which strikes carry better bid-ask liquidity - not by expected payoff.

Put-call parity

The no-arbitrage relationship linking a European call and put of the same strike and expiration to the underlying and a zero-coupon bond. Violations imply a riskless arbitrage net of costs.

Field	Value
Formula	C - P = S*exp(-q*T) - K*exp(-r*T)
Rearranged for synthetic long stock	S = C - P + K*exp(-r*T), with dividends adjusted
Conversion	Long stock + long put + short call, locking a rate
Reversal	Short stock + short put + long call
Worked	S = 100, K = 100, T = 0.25, r = 0.04, q = 0, sigma = 0.20. Call = 4.485236 and put = 3.490220, so C - P = 0.995017. Independently S*exp(-q*T) - K*exp(-r*T) = 100 - 100*exp(-0.01) = 0.995017. The identity holds at any sigma, since sigma cancels

• Holds exactly only for European exercise. American options may deviate because early exercise has value.
• Apparent parity violations on listed American equity options are usually explained by borrow cost, hard-to-borrow rates, or pending dividends rather than by genuine arbitrage.

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Long call payoff algebra

Buying one call at strike K for premium P. The only structure with an unbounded profit and a loss capped at the premium.

Field	Value
Formula	Payoff(S) = max(S - K, 0) - P
Max profit	Unbounded. Payoff grows 1:1 with S above K
Max loss	P, realised for all S <= K
Breakeven	K + P
Delta at entry	exp(-q*T)*N(d1), between 0 and 1
Worked	K = 100, P = 3.20. Breakeven 103.20. At S = 110 payoff = 10.00 - 3.20 = 6.80 per share, 680.00 per contract. At S = 98 payoff = -3.20, loss 320.00

• Loss is capped in dollars but is 100 percent of capital at any S at or below K. The capped-loss property does not make it a small loss.
• The breakeven moves with the premium, not with delta. Two calls at the same strike bought at different prices have different breakevens and identical payoff slopes above K.
• Time value is the entire gap between the breakeven and the strike. A long call held to expiry must clear K + P, not merely K, to return anything.

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Short call payoff algebra

Selling one call at strike K for premium P without owning the underlying. Profit is capped at the premium received; loss is unbounded because the underlying has no upper bound.

Field	Value
Formula	Payoff(S) = P - max(S - K, 0)
Max profit	P, realised for all S <= K
Max loss	Unbounded
Breakeven	K + P
Worked	K = 100, P = 3.20. Breakeven 103.20. At S = 110 payoff = 3.20 - 10.00 = -6.80 per share, loss 680.00. At S = 150 loss = 46.80 per share, 4,680.00

• The short call and long call at the same strike and price share one breakeven and are exact mirror images. Their payoffs sum to zero at every S.
• Loss is unbounded in theory and, in a takeover or a gap, unbounded enough in practice to exceed the account. This is the structure margin rules treat most severely.
• Assignment risk concentrates before an ex-dividend date once extrinsic value approaches zero. See the dividend early-exercise test.

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Long put payoff algebra

Buying one put at strike K for premium P. Profit is bounded because the underlying cannot fall below zero.

Field	Value
Formula	Payoff(S) = max(K - S, 0) - P
Max profit	K - P, realised only at S = 0
Max loss	P, realised for all S >= K
Breakeven	K - P
Worked	K = 100, P = 2.80. Breakeven 97.20. Max profit 97.20 per share at S = 0. At S = 90 payoff = 10.00 - 2.80 = 7.20 per share, 720.00

• Max profit is finite, which is why a long put is never a mirror of a long call despite the symmetry of the formulas.
• A long put carries positive rho of the wrong sign for the holder: higher rates reduce put value, all else equal.
• Deep in-the-money long puts can be worth exercising early once the interest on the strike proceeds exceeds remaining time value.

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Short put payoff algebra

Selling one put at strike K for premium P. Loss is bounded but large, realised at S = 0.

Field	Value
Formula	Payoff(S) = P - max(K - S, 0)
Max profit	P, realised for all S >= K
Max loss	K - P, realised at S = 0
Breakeven	K - P
Worked	K = 100, P = 2.80. Breakeven 97.20. Max loss 97.20 per share, 9,720.00 per contract at S = 0

• Bounded is not small. The maximum loss on one uncovered put at a 100 strike is 9,720.00 against a 280.00 credit in the worked case.
• Payoff is identical to a covered call struck at the same strike, net of the difference in carry. The two are the same risk expressed two ways.
• A short put is the leg most often assigned early in a rising-rate environment on a deep in-the-money strike.

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Covered call payoff algebra

Long the underlying at S0 with one call written at strike K for premium P. The written call converts unlimited upside into a fixed cap in exchange for the premium.

Field	Value
Formula	Payoff(S) = min(S, K) - S0 + P
Max profit	K - S0 + P, realised for all S >= K
Max loss	S0 - P, realised at S = 0
Breakeven	S0 - P
Position delta	1 - exp(-q*T)*N(d1) per share held
Worked	S0 = 98, K = 105, P = 2.10. Max profit 105 - 98 + 2.10 = 9.10 per share (910.00). Max loss 95.90 (9,590.00). Breakeven 95.90. At S = 120 payoff is still 9.10, capped

• Downside is reduced by exactly P and by nothing else. The written call is not a hedge; it is a partial rebate against a full-size long stock position.
• If K < S0 the structure is written in the money and the maximum profit can be negative, locking a loss on the stock if the call is assigned.
• The dividend early-exercise test governs the short leg. Assignment the day before ex-dividend removes both the stock and the dividend.

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Cash-secured put payoff algebra

One put written at strike K for premium P with K multiplied by the multiplier held in cash against assignment. The payoff is identical to a naked short put; only the capital treatment differs.

Field	Value
Formula	Payoff(S) = P - max(K - S, 0); capital committed = K * multiplier
Max profit	P
Max loss	K - P at S = 0
Breakeven	K - P
Effective purchase price if assigned	K - P
Worked	K = 95, P = 2.40. Capital 9,500.00. Max profit 240.00, which is 2.5263 percent of committed capital. Breakeven and effective purchase price both 92.60

• Securing the put with cash changes nothing about the payoff. It changes the leverage, and therefore the probability of a forced exit.
• Return on committed capital, not return on premium, is the comparable figure across strikes: 240.00 on 9,500.00 is not the same trade as 240.00 on 1,740.00 of naked-put requirement.
• If assigned, the resulting stock position is worth K - P per share against the original cash, so the assignment is economically neutral at the breakeven price.

Bull call spread payoff algebra

Long one call at K1 and short one call at K2 with K1 < K2, same expiration, for a net debit D. Maximum loss and maximum profit are both fixed at entry.

Field	Value
Formula	Payoff(S) = max(S - K1, 0) - max(S - K2, 0) - D
Max profit	K2 - K1 - D, realised for all S >= K2
Max loss	D, realised for all S <= K1
Breakeven	K1 + D
Risk/reward ratio	D : (K2 - K1 - D)
Worked	K1 = 100 at 3.20, K2 = 110 at 1.10, D = 2.10. Max profit 10 - 2.10 = 7.90 (790.00). Max loss 2.10 (210.00). Breakeven 102.10. At S = 105 payoff = 5.00 - 2.10 = 2.90

• The width K2 - K1 is the entire opportunity set. Paying more than the width for a vertical guarantees a loss at every S, and quoted mid prices occasionally imply exactly that in wide markets.
• Both legs are long calls, so neither carries early-assignment exposure. That is the structural difference from the equivalent bull put spread.
• Maximum profit requires S at or above K2 at expiration, not merely a move in the right direction.

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Bear call spread payoff algebra

Short one call at K1 and long one call at K2 with K1 < K2, same expiration, for a net credit C. The long call converts an unbounded short-call loss into the strike width less the credit.

Field	Value
Formula	Payoff(S) = C - [max(S - K1, 0) - max(S - K2, 0)]
Max profit	C, realised for all S <= K1
Max loss	K2 - K1 - C, realised for all S >= K2
Breakeven	K1 + C
Worked	K1 = 100 at 3.20, K2 = 110 at 1.10, C = 2.10. Max profit 2.10 (210.00). Max loss 10 - 2.10 = 7.90 (790.00). Breakeven 102.10. Identical to the 100/110 bull call spread reflected about zero

• A credit greater than the strike width is impossible without mispricing; if a quote shows one, the strikes or the multiplier have been misread.
• The short K1 call is in the money exactly when the position is losing, which is when early assignment is most likely and least convenient.
• Buying-power reduction equals the maximum loss: (K2 - K1 - C) times the multiplier.

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Bull put spread payoff algebra

Short one put at K2 and long one put at K1 with K1 < K2, same expiration, for a net credit C. Defined risk equal to the strike width less the credit.

Field	Value
Formula	Payoff(S) = C - [max(K2 - S, 0) - max(K1 - S, 0)]
Max profit	C, realised for all S >= K2
Max loss	K2 - K1 - C, realised for all S <= K1
Breakeven	K2 - C
Worked	K1 = 90 at 0.90, K2 = 95 at 2.05, C = 1.15. Max profit 1.15 (115.00). Max loss 5 - 1.15 = 3.85 (385.00). Breakeven 93.85

• Payoff is identical to the 90/95 bull call spread at every S. The difference is that the short put here is in the money when losing, so early assignment produces a long stock position.
• If the short leg is assigned and the long put is retained, the account holds stock plus a put, which is a protective put at a known worst case, not a new risk. The problem is the margin, not the payoff.
• The credit is collected at entry, so the maximum loss is the width less that credit, and buying power is reduced by exactly that amount.

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Bear put spread payoff algebra

Long one put at K2 and short one put at K1 with K1 < K2, same expiration, for a net debit D.

Field	Value
Formula	Payoff(S) = max(K2 - S, 0) - max(K1 - S, 0) - D
Max profit	K2 - K1 - D, realised for all S <= K1
Max loss	D, realised for all S >= K2
Breakeven	K2 - D
Worked	K1 = 90 at 0.90, K2 = 95 at 4.75, D = 3.85. Max profit 5 - 3.85 = 1.15 (115.00). Max loss 3.85 (385.00). Breakeven 91.15

• Both legs are puts held long and short at fixed strikes, so the debit version has no assignment exposure on the long leg and only a bounded one on the short.
• The paired bull put spread at the same strikes has breakeven 93.85 while this has 91.15. That is not an inconsistency: they are opposite positions, and each breakeven sits on the losing side of the other.

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Straddle payoff algebra

A call and a put at the same strike K and expiration. Long for a net debit D, short for a net credit C. Direction-neutral at entry, volatility-directional throughout.

Field	Value
Formula	Long payoff(S) = max(S - K, 0) + max(K - S, 0) - D = |S - K| - D
Long max profit	Unbounded above; K - D at S = 0
Long max loss	D, realised only at S = K exactly
Long breakevens	K + D and K - D
Short max profit	C at S = K
Short max loss	Unbounded above; K - C at S = 0
Required move to break even	D / K expressed as a fraction of the strike
Worked	K = 100, call 3.20 and put 2.80, D = 6.00. Breakevens 106.00 and 94.00, a required move of 6.00 percent of the strike in either direction. At S = 112 payoff = 12.00 - 6.00 = 6.00. Short straddle at C = 6.00 has the same two breakevens and max profit 600.00

• The two breakevens are symmetric about K only because the strike is shared. That symmetry disappears the moment strikes differ, which is why a strangle has an asymmetric profile.
• The straddle debit divided by the strike is a direct read of the move the market is charging for. Comparing it to the move the position needs is the whole trade.
• A long straddle carries roughly double the gamma and vega of either leg and near-zero delta at the money, because gamma and vega are identical for a call and a put at the same strike.

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Strangle payoff algebra

An out-of-the-money call at K2 and an out-of-the-money put at K1 with K1 < K2, same expiration. Long for a net debit D, short for a net credit C.

Field	Value
Formula	Long payoff(S) = max(S - K2, 0) + max(K1 - S, 0) - D
Long max profit	Unbounded above; K1 - D at S = 0
Long max loss	D, realised for all K1 <= S <= K2
Long breakevens	K2 + D and K1 - D
Short max profit	C, realised for all K1 <= S <= K2
Short max loss	Unbounded above; K1 - C at S = 0
Worked	K1 = 95 put at 1.40, K2 = 105 call at 1.60, D = 3.00. Breakevens 108.00 and 92.00. Max loss 3.00 (300.00) across the entire 95 to 105 band. Short version at C = 3.00 keeps the full 300.00 anywhere in that band

• The maximum loss on a long strangle is realised across a range, not at a point. That makes it more likely to be realised in full than a straddle maximum loss.
• A short strangle collects its maximum credit across a band, which is why its distribution of outcomes is heavily weighted to small wins and rare large losses.
• Widening the strikes lowers the debit and widens the breakevens simultaneously. The two effects work against each other and neither is free.

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Butterfly payoff algebra

Three strikes, equidistant, in a 1-2-1 ratio. Long call butterfly: buy K1, sell two K2, buy K3 with K3 - K2 = K2 - K1. Long put butterfly uses the mirrored puts and has the same expiration payoff.

Field	Value
Formula	Payoff(S) = max(S - K1, 0) - 2*max(S - K2, 0) + max(S - K3, 0) - D
Max profit	K2 - K1 - D, realised only at S = K2
Max loss	D, realised for S <= K1 and for S >= K3
Breakevens	K1 + D and K3 - D
Width invariant	K3 - K2 = K2 - K1 = W, so max profit = W - D
Worked	K1 = 95 at 6.40, K2 = 100 at 3.40 (two sold), K3 = 105 at 1.60. D = 6.40 - 6.80 + 1.60 = 1.20. Max profit 5 - 1.20 = 3.80 (380.00) at S = 100 exactly. Max loss 1.20 (120.00). Breakevens 96.20 and 103.80

• Maximum profit is a single point, not a range. The expected value of a butterfly is dominated by the probability of the band between the breakevens, not by the headline reward ratio.
• The long call butterfly and long put butterfly at identical strikes have identical expiration payoffs, so the choice between them is an assignment and liquidity decision.
• Broken-wing versions, where the wings are unequal, are no longer symmetric and no longer have max profit W - D. Recompute both breakevens from the payoff expression rather than reusing the symmetric formulas.

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Iron condor payoff algebra

Four strikes K1 < K2 < K3 < K4, same expiration: long put K1, short put K2, short call K3, long call K4, for a net credit C. A short put spread and a short call spread sharing one credit.

Field	Value
Formula	Payoff(S) = C - [max(K2 - S, 0) - max(K1 - S, 0)] - [max(S - K3, 0) - max(S - K4, 0)]
Max profit	C, realised for all K2 <= S <= K3
Max loss	max(K2 - K1, K4 - K3) - C
Breakevens	K2 - C and K3 + C
Profit band width	K3 - K2
Worked	90/95/105/110 with long put 0.55, short put 1.30, short call 1.45, long call 0.60. C = 1.30 + 1.45 - 0.55 - 0.60 = 1.60. Max profit 160.00 anywhere from 95 to 105. Max loss 5 - 1.60 = 3.40 (340.00). Breakevens 93.40 and 106.60

• Only one side can lose at expiration, so the maximum loss is the wider wing less the credit, not the sum of both wings.
• Unequal wings mean the two sides carry different maximum losses while sharing one credit. State which wing is wider before quoting a risk figure.
• Buying-power reduction equals the maximum loss, so a symmetric condor consumes the same capital as a single credit vertical of the same width while collecting two credits.

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Iron butterfly payoff algebra

Three strikes K1 < K2 < K3, equidistant, same expiration: long put K1, short put K2, short call K2, long call K3, for a net credit C. The zero-width-band limit of an iron condor.

Field	Value
Formula	Payoff(S) = C - |S - K2| bounded by the wings, i.e. C - min(|S - K2|, W) where W = K2 - K1 = K3 - K2
Max profit	C, realised only at S = K2
Max loss	W - C, where W = K2 - K1
Breakevens	K2 - C and K2 + C
Worked	95/100/105 with long put 1.30, short put 2.80, short call 3.20, long call 1.60. C = 2.80 + 3.20 - 1.30 - 1.60 = 3.10. Max profit 310.00 at S = 100. Max loss 5 - 3.10 = 1.90 (190.00). Breakevens 96.90 and 103.10

• The iron butterfly and the long call butterfly at the same three strikes have the same payoff shape. The iron version is entered for a credit and the call version for a debit, and C plus D equals the wing width when both are fairly priced.
• Both short legs sit at the same strike, which concentrates pin risk at a single price on expiration day.
• Breakevens are symmetric about K2 at plus and minus the credit, which makes the credit a direct read of the tolerated move.

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Condor payoff algebra

Four strikes K1 < K2 < K3 < K4 in a single option type, 1-1-1-1: buy K1, sell K2, sell K3, buy K4 for a net debit D. A butterfly with the peak stretched into a plateau.

Field	Value
Formula	Payoff(S) = max(S-K1,0) - max(S-K2,0) - max(S-K3,0) + max(S-K4,0) - D
Max profit	K2 - K1 - D, realised for all K2 <= S <= K3
Max loss	D, realised for S <= K1 and S >= K4
Breakevens	K1 + D and K4 - D
Worked	Calls 90/95/105/110 at 9.75, 6.40, 1.85, 0.40. D = 9.75 - 6.40 - 1.85 + 0.40 = 1.90. Max profit 5 - 1.90 = 3.10 (310.00) across 95 to 105. Max loss 1.90 (190.00). Breakevens 91.90 and 108.10

• Max profit assumes the inner strikes are the same distance apart as the outer pairs on each side. If K2 - K1 does not equal K4 - K3 the formula K2 - K1 - D is wrong on one side.
• The all-calls condor and the iron condor at the same four strikes carry the same expiration shape; one is a debit and one a credit.

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Calendar spread arithmetic

Short a near-dated option at K and long a longer-dated option at the same K, for a net debit D. There is no closed-form expiration payoff because the long leg is still alive when the short leg expires.

Field	Value
Formula	Value at near expiry = BS(S, K, T2 - T1, r, q, sigma2) - max(intrinsic of the near leg) - D
Max loss	D, realised when S moves far enough that both legs converge to the same intrinsic value
Max profit	Not closed-form; depends on sigma2 at near expiry
Position vega	Positive, since the longer leg has the larger vega
Position theta	Positive at the money, since the near leg decays faster
Worked	Sell the 30-day 100 call at 2.10, buy the 90-day 100 call at 4.00. D = 1.90 (190.00), which is the maximum loss. Profit at near expiry equals the residual value of the 60-day call less 1.90, so it cannot be stated without a volatility input

• A calendar is a position on the term structure, not on direction. The two legs can carry different implied volatilities and usually do.
• Maximum loss is bounded by the debit only if the two legs share a strike. Diagonals break that guarantee.
• Any figure quoted as a calendar max profit carries a hidden volatility assumption. Ask which sigma it used before comparing two quotes.

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Diagonal spread arithmetic

Short a near-dated option at K1 and long a longer-dated option at K2 with K1 not equal to K2. A calendar with a strike offset, so it carries directional exposure as well as term exposure.

Field	Value
Formula	Value at near expiry = BS(S, K2, T2 - T1, r, q, sigma2) - max(near-leg intrinsic at K1) - D
Max loss	Bounded by D only when the long leg strike is at least as favourable as the short leg strike for the option type held
Unbounded case	A short near-dated call at K1 against a long later call at K2 > K1 has unbounded loss above K1 until the long leg is in the money, and even then the loss is capped only at expiry of the long leg
Worked	Sell the 30-day 105 call at 1.10, buy the 90-day 100 call at 4.00. D = 2.90 (290.00). The long strike is lower than the short strike, so at near expiry the long leg intrinsic exceeds the short leg intrinsic by at least 5.00 for large S, and loss is bounded by the debit

• Whether a diagonal is defined-risk depends entirely on the strike order relative to the option type. Determine that before sizing.
• A diagonal where the short strike is more favourable than the long strike is not a defined-risk position, regardless of the debit paid.

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Ratio spread payoff algebra

Unequal quantities of long and short options in the same expiration. A 1x2 call ratio is long one call at K1 and short two at K2 with K1 < K2. The excess short contract is uncovered.

Field	Value
Formula	Call 1x2 payoff(S) = max(S-K1,0) - 2*max(S-K2,0) + N, where N is the net credit (negative for a debit)
Max profit	K2 - K1 + N at S = K2
Upper breakeven	2*K2 - K1 + N
Max loss (call ratio)	Unbounded above the upper breakeven
Put 1x2 payoff	max(K2-S,0) - 2*max(K1-S,0) + N
Put ratio max loss	2*K1 - K2 - N at S = 0
Worked	Call 1x2: buy 100 call at 3.20, sell two 105 calls at 1.90 each, N = +0.60 credit. Max profit 5 + 0.60 = 5.60 (560.00) at S = 105. Upper breakeven 210 - 100 + 0.60 = 110.60. Put 1x2: buy 100 put at 2.80, sell two 95 puts at 1.60 each, N = +0.40. Max profit 5.40 (540.00) at S = 95, max loss 190 - 100 - 0.40 = 89.60 (8,960.00) at S = 0, breakeven 89.60

• A net credit removes the loss on the far side but does nothing about the uncovered short. A credit ratio spread is still an undefined-risk position.
• The upper breakeven of a 1x2 call ratio is twice the short strike less the long strike, plus the credit. It is not the short strike plus the credit, and that error understates the risk band badly.
• Margin treats the excess short contract as naked, so buying-power reduction is far larger than the debit or credit suggests.

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Collar payoff algebra

Long the underlying at S0, long a put at K1 and short a call at K2 with K1 < K2, for a net credit or debit N (credit positive). Both tails are removed.

Field	Value
Formula	Payoff(S) = min(max(S, K1), K2) - S0 + N
Max profit	K2 - S0 + N, for all S >= K2
Max loss	S0 - K1 - N, for all S <= K1
Breakeven	S0 - N
Zero-cost condition	N = 0, i.e. the put and call premiums are equal
Worked	S0 = 98, long 95 put at 1.50, short 105 call at 2.10, N = +0.60 credit. Max profit 105 - 98 + 0.60 = 7.60 (760.00). Max loss 98 - 95 - 0.60 = 2.40 (240.00). Breakeven 97.40

• A collar with a credit has a breakeven below the entry price, which is the only sense in which the credit protects anything.
• The payoff between K1 and K2 is exactly the stock payoff shifted by N. The structure changes only the tails.
• Assignment on the short call before an ex-dividend date leaves the account holding a long put against no stock, which is a directionally opposite position to the one intended.

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Synthetic positions and their algebra

Combinations that replicate another instrument exactly at expiration. Each follows from put-call parity rather than from any separate rule.

Field	Value
Formula	Synthetic long stock = long call K + short put K; payoff(S) = S - K - D
Synthetic long stock	Long call K, short put K, net debit D. Payoff = S - K - D. Breakeven K + D
Synthetic short stock	Short call K, long put K, net credit C. Payoff = K + C - S. Breakeven K + C
Synthetic long call	Long stock at S0 + long put K
Synthetic long put	Short stock at S0 + long call K
Synthetic short call	Short stock at S0 + short put K
Synthetic short put	Long stock at S0 + short call K, i.e. the covered call
Worked	K = 100, call 3.20, put 2.80. Synthetic long: net debit 0.40, payoff S - 100.40, breakeven 100.40, so the effective purchase price is 100.40. Synthetic short at the same strikes: net credit 0.40, payoff 100.40 - S, breakeven 100.40

• The effective purchase price of a synthetic long is K plus the net debit, and it embeds the financing rate. Comparing it to spot is the cheapest available read on the implied borrow and carry.
• A synthetic carries no dividend entitlement. Any dividend the underlying pays over the life of the position is already in the option prices, not in the position.
• Synthetic stock positions are marked as options for margin, which can produce a very different requirement from the equivalent stock position.

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Box spread payoff algebra

A bull call spread and a bear put spread at the same two strikes and expiration: long call K1, short call K2, long put K2, short put K1. The expiration value is the strike width regardless of S.

Field	Value
Formula	Payoff at expiry = (K2 - K1) - D, for every S
Value at expiry	K2 - K1, deterministic
Profit	K2 - K1 - D
Implied financing rate	r_implied = ln((K2 - K1) / D) / T
Worked	K1 = 100, K2 = 110. Long 100 call 3.20, short 110 call 1.10, long 110 put 10.50, short 100 put 2.80. D = 9.80. Expiry value 10.00 for every S, so profit 0.20 (20.00) fixed. With T = 0.5, implied rate = ln(10 / 9.80) / 0.5 = 0.0404, i.e. 4.04 percent

• A box is a financing instrument, not a directional one. Its only variables are the rate implied by the price and the risk that the American short legs are exercised early.
• On American-style options a short box carries genuine early-assignment risk, and an assigned leg converts a deterministic payoff into an open stock position. Boxes on European-style index options do not have this exposure.
• A box quoted above the strike width implies a negative financing rate and is almost always a stale or crossed quote rather than an opportunity.

Also described at: Wikipedia · Wikidata

Jade lizard payoff algebra

A short put at K1 and a short call spread at K2 and K3 with K1 < K2 < K3, one expiration, for a net credit C. The upside risk is removed entirely when the credit exceeds the call spread width.

Field	Value
Formula	Payoff(S) = C - max(K1 - S, 0) - [max(S - K2, 0) - max(S - K3, 0)]
Max profit	C, realised for all K1 <= S <= K2
Upside outcome	C - (K3 - K2) for all S >= K3, which is a profit when C > K3 - K2
No-upside-risk condition	C > K3 - K2
Max loss	K1 - C at S = 0
Downside breakeven	K1 - C
Worked	Short 90 put at 2.20, short 105 call at 1.90, long 107.5 call at 1.10. C = 3.00, K3 - K2 = 2.50, so C exceeds the call spread width. Max profit 300.00 between 90 and 105. Above 107.5 the payoff is 3.00 - 2.50 = 0.50 (50.00), still a profit. Max loss 90 - 3.00 = 87.00 (8,700.00) at S = 0. Only breakeven 87.00

• The single-breakeven property is the defining feature. Verify C > K3 - K2 arithmetically at entry; the name does not guarantee it.
• All of the risk sits in the short put, so the position is a short put with a funded upside cap, and it should be sized as a short put.
• If the credit falls below the call spread width after a roll or an adjustment, upside risk reappears and the structure has a second breakeven at K2 + C.

Broken-wing butterfly payoff algebra

A butterfly with unequal wings. The symmetric formulas no longer apply: the maximum loss differs on the two sides, and on the wide side it is the wing imbalance less the net paid rather than the net paid alone. Every figure has to be recomputed from the payoff expression.

Field	Value
Formula	Payoff(S) = max(S - K1, 0) - 2*max(S - K2, 0) + max(S - K3, 0) - D, with K3 - K2 not equal to K2 - K1
Max profit	(K2 - K1) - D, at S = K2 exactly, unchanged from the symmetric case
Loss on the narrow side	D, for all S at or below K1
Loss on the wide side	(K3 - K2) - (K2 - K1) + D, for all S at or above K3
Breakevens	K1 + D and K2 + (K2 - K1) - D. The upper breakeven is no longer K3 - D
Worked	Buy the 95 call at 6.40, sell two 100 calls at 3.40, buy the 110 call at 1.10. Net debit D = 6.40 minus 6.80 plus 1.10 = 0.70. Brute-force scan: maximum profit 4.30 at S = 100.0000, which matches (100 minus 95) minus 0.70 = 4.30. Loss below 95 is 0.70 (70.00 per contract). Loss at and above 110 is 5.70, confirmed by evaluating the payoff at S = 120, 200 and 400, all of which return minus 5.70; the closed form gives (110 minus 100) minus (100 minus 95) plus 0.70 = 5.70. Breakevens 95.7000 and 104.3000. Note the upper breakeven is not 110 minus 0.70 = 109.30, which is what the symmetric formula would give - it is wrong by 5.00

• The maximum loss is eight times larger on the wide side than on the narrow side in the worked case, 5.70 against 0.70, on a structure whose name implies a defined and small risk. It is defined; it is not small.
• Applying the symmetric upper-breakeven formula K3 minus D gives 109.30 against the true 104.30, an error of 5.00 on a structure whose entire maximum profit is 4.30. This is the single most common error in the family.
• Widening one wing lowers the net cost and moves the loss, it does not remove it. The credit-versus-risk trade-off is visible only if both sides are computed separately.

Call backspread payoff algebra

One short call at a lower strike against two long calls at a higher strike, same expiration. The reverse of a call ratio spread: loss is defined and unbounded profit sits above the upper breakeven, with the worst point at the long strike.

Field	Value
Formula	Payoff(S) = 2*max(S - K2, 0) - max(S - K1, 0) + N, with K1 below K2 and N the net credit, negative for a debit
Max profit	Unbounded above, with slope plus 1 per share above K2 net of the short leg, i.e. plus 1 in total
Max loss	(K2 - K1) - N, realised at S = K2 exactly
Payoff below K1	N, the net credit, retained for all S at or below K1
Upper breakeven	K2 + (K2 - K1) - N
Worked	Sell one 100 call at 3.20, buy two 105 calls at 1.90. Net = minus 3.20 plus 3.80 = 0.60 debit, so N = minus 0.60. Brute-force scan at 0.0001 granularity from 0 to 400: payoff below 100 is minus 0.60 (a 60.00 loss per contract, the debit). Trough at S = 105.0000 of minus 5.60, matching (105 minus 100) plus 0.60 = 5.60. Payoff at 110 is minus 0.60, at 120 is plus 9.40. Single breakeven at 110.6000, matching 105 plus 5.00 minus (minus 0.60) = 110.60. This is the exact mirror of the 1x2 call ratio spread priced elsewhere on this site, whose maximum profit is 5.60 at S = 105 and whose breakeven is also 110.60

• The maximum loss sits at the long strike, not at either extreme. A position that finishes exactly where the long legs are struck is the worst outcome, which inverts the usual intuition that being near the long strike is good.
• Because the structure can be entered for a credit if the strikes are close enough, it is sometimes described as free. The 5.60 loss at S = 105 is unaffected by whether the entry was a credit or a debit; only its size changes.
• The backspread and the 1x2 ratio spread at the same strikes are exact mirrors, so their payoffs sum to zero at every price and they share one breakeven. Any table that shows them with different breakevens has an arithmetic error.