# options.wiki - full corpus

> A complete, machine-readable reference for listed equity and index options: explicit payoff, maximum profit, maximum loss and breakeven formulas for every standard structure, Greeks in closed form with verified worked values, volatility and probability arithmetic, exercise and assignment mechanics, contract conventions, and Regulation T margin treatment.

Reviewed: 2026-08-27
License: CC BY 4.0
Source: https://options.wiki
Change feed: https://options.wiki/changes.json

## Strategies
Reviewed: 2026-08-27
Canonical: https://options.wiki/strategies/ (JSON: https://options.wiki/strategies.json)

Formulas are stated per share. Multiply by the contract multiplier (100 for standard listed contracts) and the number of contracts for position-level figures. Breakevens are at expiration and ignore financing, dividends, and early assignment.

### 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.

Formula: BullCall(K1,K2) payoff == BullPut(K1,K2) payoff, for all S at expiry

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.

Formula: C - P = S*exp(-q*T) - K*exp(-r*T)

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.

### 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.

Formula: Payoff(S) = max(S - K, 0) - P

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.

### 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.

Formula: Payoff(S) = P - max(S - K, 0)

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.

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

Formula: Payoff(S) = max(K - S, 0) - P

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.

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

Formula: Payoff(S) = P - max(K - 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.

### 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.

Formula: Payoff(S) = min(S, K) - S0 + P

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.

### 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.

Formula: Payoff(S) = P - max(K - S, 0); capital committed = K * multiplier

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.

Formula: Payoff(S) = max(S - K1, 0) - max(S - K2, 0) - D

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.

### 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.

Formula: Payoff(S) = C - [max(S - K1, 0) - max(S - K2, 0)]

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.

### 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.

Formula: Payoff(S) = C - [max(K2 - S, 0) - max(K1 - S, 0)]

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.

### 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.

Formula: Payoff(S) = max(K2 - S, 0) - max(K1 - S, 0) - D

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.

### 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.

Formula: Long payoff(S) = max(S - K, 0) + max(K - S, 0) - D = |S - K| - D

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.

### 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.

Formula: Long payoff(S) = max(S - K2, 0) + max(K1 - S, 0) - D

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.

### 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.

Formula: Payoff(S) = max(S - K1, 0) - 2*max(S - K2, 0) + max(S - K3, 0) - D

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.

### 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.

Formula: Payoff(S) = C - [max(K2 - S, 0) - max(K1 - S, 0)] - [max(S - K3, 0) - max(S - K4, 0)]

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.

### 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.

Formula: Payoff(S) = C - |S - K2| bounded by the wings, i.e. C - min(|S - K2|, W) where W = K2 - K1 = K3 - K2

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.

### 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.

Formula: Payoff(S) = max(S-K1,0) - max(S-K2,0) - max(S-K3,0) + max(S-K4,0) - D

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.

### 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.

Formula: Value at near expiry = BS(S, K, T2 - T1, r, q, sigma2) - max(intrinsic of the near leg) - D

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.

### 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.

Formula: Value at near expiry = BS(S, K2, T2 - T1, r, q, sigma2) - max(near-leg intrinsic at K1) - D

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.

### 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.

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)

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.

### 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.

Formula: Payoff(S) = min(max(S, K1), K2) - S0 + N

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.

### 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.

Formula: Synthetic long stock = long call K + short put K; payoff(S) = S - K - D

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.

### 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.

Formula: Payoff at expiry = (K2 - K1) - D, for every S

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.

### 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.

Formula: Payoff(S) = C - max(K1 - S, 0) - [max(S - K2, 0) - max(S - K3, 0)]

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.

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

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.

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

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.

#### Single-leg positions
The four primitives. Every multi-leg structure below decomposes into these.

| Position | Construction | Max profit | Max loss | Breakeven at expiry |
|---|---|---|---|---|
| Long call | Buy 1 call at K for P | Unlimited | P | K + P |
| Long put | Buy 1 put at K for P | K - P | P | K - P |
| Short call (naked) | Sell 1 call at K for P | P | Unlimited | K + P |
| Short put (naked) | Sell 1 put at K for P | P | K - P | K - P |

#### Stock-plus-option overlays

| Position | Construction | Max profit | Max loss | Breakeven at expiry |
|---|---|---|---|---|
| Covered call | Long stock at S0, sell 1 call at K for P | K - S0 + P | S0 - P | S0 - P |
| Protective put | Long stock at S0, buy 1 put at K for P | Unlimited | S0 - K + P | S0 + P |
| Collar | Long stock at S0, buy put K1, sell call K2 for net N (credit positive) | K2 - S0 + N | S0 - K1 - N | S0 - N |
| Cash-secured put | Sell put at K for P, hold K in cash | P | K - P | K - P |

#### Vertical spreads
Two legs, same expiration, different strikes. K1 < K2 throughout. Every vertical has a maximum loss capped at the strike width net of the premium paid or received, which is why they are the standard defined-risk structure.

| Spread | Construction | Max profit | Max loss | Breakeven at expiry |
|---|---|---|---|---|
| Bull call (debit) | Buy call K1, sell call K2, net debit D | K2 - K1 - D | D | K1 + D |
| Bear call (credit) | Sell call K1, buy call K2, net credit C | C | K2 - K1 - C | K1 + C |
| Bull put (credit) | Sell put K2, buy put K1, net credit C | C | K2 - K1 - C | K2 - C |
| Bear put (debit) | Buy put K2, sell put K1, net debit D | K2 - K1 - D | D | K2 - D |

#### Volatility structures
Straddles and strangles are direction-neutral and volatility-directional. The long versions have defined risk and undefined reward; the short versions invert that.

| Structure | Construction | Max profit | Max loss | Breakevens at expiry |
|---|---|---|---|---|
| Long straddle | Buy call K and put K, net debit D | Unlimited above; K - D below | D | K + D and K - D |
| Short straddle | Sell call K and put K, net credit C | C | Unlimited above; K - C below | K + C and K - C |
| Long strangle | Buy call K2 and put K1, net debit D | Unlimited above; K1 - D below | D | K2 + D and K1 - D |
| Short strangle | Sell call K2 and put K1, net credit C | C | Unlimited above; K1 - C below | K2 + C and K1 - C |

#### Four-leg defined-risk structures
Wings converted into caps. All assume a single expiration.

| Structure | Construction | Max profit | Max loss | Breakevens at expiry |
|---|---|---|---|---|
| Iron condor | Sell put K2, buy put K1, sell call K3, buy call K4 (K1&lt;K2&lt;K3&lt;K4), net credit C | C | max(K2 - K1, K4 - K3) - C | K2 - C and K3 + C |
| Iron butterfly | Sell put K2 and call K2, buy put K1 and call K3, net credit C | C | (K2 - K1) - C | K2 - C and K2 + C |
| Long call butterfly | Buy call K1, sell 2 calls K2, buy call K3, equidistant, net debit D | K2 - K1 - D | D | K1 + D and K3 - D |
| Long put butterfly | Buy put K3, sell 2 puts K2, buy put K1, equidistant, net debit D | K2 - K1 - D | D | K1 + D and K3 - D |
| Long condor (calls) | Buy K1, sell K2, sell K3, buy K4, net debit D | K2 - K1 - D | D | K1 + D and K4 - D |

#### Time and ratio structures
These do not have closed-form expiration payoffs at the near leg because a longer-dated leg remains open. Maximum loss is stated where it is bounded.

| Structure | Construction | Risk profile | Note |
|---|---|---|---|
| Calendar spread | Sell near-dated option at K, buy longer-dated option at same K, net debit D | Max loss D; max profit not closed-form | Long vega, long theta on the spread. Value at near expiry depends on implied volatility of the remaining leg. |
| Diagonal spread | Sell near-dated K1, buy longer-dated K2 | Max loss bounded by net debit if long leg strike is favourable | A calendar with a directional tilt. |
| Call ratio spread | Buy 1 call K1, sell 2 calls K2 (K2 &gt; K1), net N | Unlimited loss above K2 | Undefined risk despite the long leg. One short call is uncovered. |
| Put ratio spread | Buy 1 put K2, sell 2 puts K1 (K1 &lt; K2), net N | Loss to zero below K1 | Maximum loss = 2K1 - K2 - N at S = 0. |
| Jade lizard | Sell put K1, sell call K2, buy call K3 (K1 &lt; K2 &lt; K3), net credit C | No upside risk if C &gt; K3 - K2 | Downside risk equals a short put: max loss K1 - C. |

#### Payoff expressions
The expiration payoff of every structure above as an explicit function of S, stated per share. D is a net debit, C a net credit, N a net credit that may be negative. Every maximum, minimum, and breakeven elsewhere on this page is derived from these expressions and nothing else.

| Structure | Payoff(S) |
|---|---|
| Long call | max(S - K, 0) - P |
| Short call | P - max(S - K, 0) |
| Long put | max(K - S, 0) - P |
| Short put | P - max(K - S, 0) |
| Covered call | min(S, K) - S0 + P |
| Protective put | max(S, K) - S0 - P |
| Cash-secured put | P - max(K - S, 0) |
| Collar | min(max(S, K1), K2) - S0 + N |
| Bull call spread | max(S-K1,0) - max(S-K2,0) - D |
| Bear call spread | C - max(S-K1,0) + max(S-K2,0) |
| Bull put spread | C - max(K2-S,0) + max(K1-S,0) |
| Bear put spread | max(K2-S,0) - max(K1-S,0) - D |
| Long straddle | |S - K| - D |
| Short straddle | C - |S - K| |
| Long strangle | max(S-K2,0) + max(K1-S,0) - D |
| Short strangle | C - max(S-K2,0) - max(K1-S,0) |
| Long call butterfly | max(S-K1,0) - 2*max(S-K2,0) + max(S-K3,0) - D |
| Long put butterfly | max(K3-S,0) - 2*max(K2-S,0) + max(K1-S,0) - D |
| Long condor (calls) | max(S-K1,0) - max(S-K2,0) - max(S-K3,0) + max(S-K4,0) - D |
| Iron condor | C - [max(K2-S,0) - max(K1-S,0)] - [max(S-K3,0) - max(S-K4,0)] |
| Iron butterfly | C - min(|S - K2|, K2 - K1) |
| Call ratio 1x2 | max(S-K1,0) - 2*max(S-K2,0) + N |
| Put ratio 1x2 | max(K2-S,0) - 2*max(K1-S,0) + N |
| Synthetic long stock | S - K - D |
| Synthetic short stock | K + C - S |
| Box spread | (K2 - K1) - D, constant in S |
| Jade lizard | C - max(K1-S,0) - [max(S-K2,0) - max(S-K3,0)] |

#### Worked examples, verified
Every figure below is computed from the payoff expression above at 0.0001 price granularity from S = 0 to S = 400. Premiums are stated inputs, not quotes. Dollar figures assume a 100 multiplier and one contract per leg unless the ratio says otherwise.

| Structure | Inputs | Net D/C | Max profit | Max loss | Breakeven(s) |
|---|---|---|---|---|---|
| Long call | K 100 at 3.20 | D 3.20 | Unbounded | 320.00 | 103.20 |
| Long put | K 100 at 2.80 | D 2.80 | 9,720.00 at S=0 | 280.00 | 97.20 |
| Short put | K 100 at 2.80 | C 2.80 | 280.00 | 9,720.00 at S=0 | 97.20 |
| Covered call | S0 98, K 105 at 2.10 | C 2.10 | 910.00 | 9,590.00 | 95.90 |
| Cash-secured put | K 95 at 2.40 | C 2.40 | 240.00 | 9,260.00 | 92.60 |
| Bull call spread | 100 at 3.20 / 110 at 1.10 | D 2.10 | 790.00 | 210.00 | 102.10 |
| Bear call spread | 100 at 3.20 / 110 at 1.10 | C 2.10 | 210.00 | 790.00 | 102.10 |
| Bull put spread | 95 at 2.05 / 90 at 0.90 | C 1.15 | 115.00 | 385.00 | 93.85 |
| Bear put spread | 95 at 4.75 / 90 at 0.90 | D 3.85 | 115.00 | 385.00 | 91.15 |
| Long straddle | K 100, 3.20 + 2.80 | D 6.00 | Unbounded | 600.00 | 94.00 and 106.00 |
| Long strangle | 95p 1.40 / 105c 1.60 | D 3.00 | Unbounded | 300.00 | 92.00 and 108.00 |
| Long call butterfly | 95 at 6.40 / 2x100 at 3.40 / 105 at 1.60 | D 1.20 | 380.00 | 120.00 | 96.20 and 103.80 |
| Long put butterfly | 105 at 6.40 / 2x100 at 3.40 / 95 at 1.60 | D 1.20 | 380.00 | 120.00 | 96.20 and 103.80 |
| Iron condor | 90/95/105/110 at 0.55/1.30/1.45/0.60 | C 1.60 | 160.00 | 340.00 | 93.40 and 106.60 |
| Iron butterfly | 95/100/100/105 at 1.30/2.80/3.20/1.60 | C 3.10 | 310.00 | 190.00 | 96.90 and 103.10 |
| Long call condor | 90/95/105/110 at 9.75/6.40/1.85/0.40 | D 1.90 | 310.00 | 190.00 | 91.90 and 108.10 |
| Call ratio 1x2 | 100 at 3.20 / 2x105 at 1.90 | C 0.60 | 560.00 | Unbounded | 110.60 |
| Put ratio 1x2 | 100 at 2.80 / 2x95 at 1.60 | C 0.40 | 540.00 | 8,960.00 at S=0 | 89.60 |
| Collar | S0 98, 95p 1.50, 105c 2.10 | C 0.60 | 760.00 | 240.00 | 97.40 |
| Synthetic long | 100c 3.20 / 100p 2.80 | D 0.40 | Unbounded | 10,040.00 at S=0 | 100.40 |
| Box spread 100/110 | 3.20 / 1.10 / 10.50 / 2.80 | D 9.80 | 20.00 fixed | None | No breakeven; payoff constant |
| Jade lizard | 90p 2.20, 105c 1.90, 107.5c 1.10 | C 3.00 | 300.00 | 8,700.00 at S=0 | 87.00 |

#### Synthetic equivalences
Each row is an identity at expiration, following from put-call parity. Strikes are shared within a row unless stated.

| Target exposure | Equivalent construction | Residual difference |
|---|---|---|
| Long stock | Long call K + short put K | Financing embedded in K + net debit; no dividend entitlement |
| Short stock | Short call K + long put K | No borrow required; no dividend obligation |
| Long call | Long stock + long put K | Requires full stock capital |
| Long put | Short stock + long call K | Requires a borrow |
| Short call | Short stock + short put K | Requires a borrow |
| Short put | Long stock + short call K (covered call) | Requires full stock capital; carries the dividend |
| Bull call spread K1/K2 | Bull put spread K1/K2 | Debit versus credit; short leg is ITM when losing in the credit version |
| Iron condor K1..K4 | Long condor K1..K4 in one option type | Credit versus debit; C + D equals the wing width when fairly priced |
| Iron butterfly K1/K2/K3 | Long butterfly K1/K2/K3 | Credit versus debit |
| Riskless bond maturing at K2 - K1 | Box spread K1/K2 | Early-assignment risk on American-style legs |

#### Broken-wing and backspread payoff arithmetic, verified
Both rows were evaluated by brute force from the payoff expression at 0.0001 price granularity from S = 0 to S = 400, the same engine used for the worked examples above. Premiums are stated inputs, not quotes. Figures are per share; multiply by 100 for one contract per leg.

| Structure | Construction | Net | Max profit | Max loss | Breakeven(s) |
|---|---|---|---|---|---|
| Broken-wing call butterfly | Buy call 95 at 6.40, sell 2 calls 100 at 3.40, buy call 110 at 1.10 | D 0.70 | 4.30 at S = 100 | 5.70 for all S at or above 110 | 95.70 and 104.30 |
| Call backspread 1x2 | Sell 1 call 100 at 3.20, buy 2 calls 105 at 1.90 | D 0.60 | Unbounded above | 5.60 at S = 105 | 110.60 |

## Pricing models
Reviewed: 2026-08-27
Canonical: https://options.wiki/models/ (JSON: https://options.wiki/models.json)

Every value in this section was recomputed from the stated formulas in double precision. The standard normal cumulative distribution is evaluated as N(x) = 0.5*(1 + erf(x/sqrt(2))), which is exact to machine precision; the Abramowitz and Stegun 26.2.17 rational approximation was run alongside it as a cross-check and agrees to within 7.5e-8 over the range -5 to +5, which is its published error bound. The reference scenario is S = 100, r = 0.04, q = 0, sigma = 0.20, T = 0.25 unless an entry states otherwise, and the Black-Scholes-Merton call at K = 100 on those inputs is 4.485236409 - every convergence figure below is an error against that number. Numeric inputs are chosen so the arithmetic is checkable; they are not observations of any market.

### Black-Scholes-Merton: the six inputs and where each one enters
The closed form takes six inputs and nothing else. Five are observable or contractual and one, sigma, is not. Every quoted implied volatility is the residual of that asymmetry: it is whatever number makes the other five reproduce the observed price.

Formula: d1 = [ln(S/K) + (r - q + sigma^2/2)*T] / (sigma*sqrt(T));  d2 = d1 - sigma*sqrt(T)

Homogeneity in S and K is why a per-unit price computed at S = 1, K = 1 can be multiplied by S to price any at-the-money option of the same tenor. Forward-starting structures are priced entirely on that property.,The drift term carries plus sigma-squared-over-two, which is why at-the-money d1 is positive and at-the-money call delta exceeds 0.50 even with zero rates.,There is no volatility of volatility, no jump term and no skew anywhere in these six inputs. Everything the market does that the model cannot represent gets pushed into sigma, one strike at a time.,The formula is the same at every strike. The reason a single sigma cannot reprice a whole expiration is not a defect in the arithmetic; it is that the process assumption is false and the arithmetic is faithfully reporting that.

### Black-Scholes-Merton closed forms with continuous dividend yield
The European call and put values under geometric Brownian motion with a continuous proportional dividend yield q. Both are combinations of two digital-style terms and reduce to the 1973 Black-Scholes form at q = 0.

Formula: C = S*exp(-q*T)*N(d1) - K*exp(-r*T)*N(d2);  P = K*exp(-r*T)*N(-d2) - S*exp(-q*T)*N(-d1)

The two terms are not the two halves of a probability. N(d2) is the risk-neutral probability of finishing in the money; N(d1) is a delta and a probability under a different measure. Reading either as a real-world chance is the single most common misuse of the formula.,Every arbitrage bound above is checkable from a quote screen in one subtraction. A price outside them is a stale or crossed quote, not an opportunity.,Put value is not monotone in r. The put term -K*T*exp(-r*T)*N(-d2) is negative, which is why a rate rise cheapens puts and is the mechanism behind interest-driven early exercise.

### The assumptions the closed form requires, and what each one being false costs
The derivation requires continuous frictionless hedging in a market with a constant rate, a constant volatility, a lognormal price with no jumps, and no transaction costs or position limits. Each assumption maps to a specific, identifiable pricing error rather than to a general caveat.

The model is used as a quoting convention, not as a belief. A trader who says a strike is at 28 volatility is naming a price in a unit that removes S, K, T and r from the comparison.,The assumption that is violated most and matters least is the constant rate. The assumption that is violated least and matters most is the absence of jumps.,None of these are reasons to reject the arithmetic. They are the reasons the same arithmetic returns a different sigma at every strike.

### Black-76 for options on futures and forwards
The forward-measure version of the same formula. It prices an option on a forward or futures price F directly, with no dividend yield and no spot, and discounts the whole payoff at the risk-free rate.

Formula: C = exp(-r*T)*[F*N(d1) - K*N(d2)];  P = exp(-r*T)*[K*N(-d2) - F*N(-d1)];  d1 = [ln(F/K) + sigma^2*T/2]/(sigma*sqrt(T))

A futures option quoted at a given volatility and an equity option quoted at the same volatility are not the same exposure, because one has forward-delta and the other spot-delta. The difference is the discount factor and it is not negligible on a long-dated contract.,Because there is no q, Black-76 is the natural form whenever the carry is already embedded in a traded forward. Trying to back a dividend yield out of a futures price and then use the spot form reintroduces an estimate that Black-76 never needed.,The forward-delta is bounded by exp(-r*T), not by 1. A deep in-the-money futures call cannot have a delta of one.

### Cox-Ross-Rubinstein binomial parameterisation
A lattice in which the underlying multiplies by u or by d each step, with the risk-neutral probability chosen so that the one-step expected growth equals the carry. The parameterisation is not unique; this is the one that matches the first two moments of the log return.

Formula: u = exp(sigma*sqrt(dt));  d = 1/u;  p = (exp((r - q)*dt) - d)/(u - d);  V = exp(-r*dt)*[p*V_up + (1 - p)*V_down]

Error halves when n doubles, so the scheme is first-order in the number of steps. Getting one more correct decimal costs ten times the work, which is why lattices are used for early exercise and almost never for European values that have a closed form.,The value oscillates as well as converging: at these inputs n = 1 overshoots by 0.98 and n = 2 undershoots by 0.45. Averaging n and n+1 is a cheap and standard way to kill most of the oscillation.,p is a risk-neutral probability, not a forecast. It contains the carry and nothing about the direction anyone expects.

### The trinomial lattice and its exact relation to the binomial
A lattice with three branches per step, up, middle and down, with the space step widened to sigma*sqrt(2*dt) so the probabilities remain admissible. Under the standard parameterisation it is not an independent method: it reproduces the binomial value at twice the number of steps, digit for digit.

Formula: dx = sigma*sqrt(2*dt);  pu = [(exp((r-q)*dt/2) - exp(-sigma*sqrt(dt/2))) / (exp(sigma*sqrt(dt/2)) - exp(-sigma*sqrt(dt/2)))]^2;  pd = [(exp(sigma*sqrt(dt/2)) - exp((r-q)*dt/2)) / (exp(sigma*sqrt(dt/2)) - exp(-sigma*sqrt(dt/2)))]^2;  pm = 1 - pu - pd

The middle branch is what makes a trinomial useful in practice: a barrier or a strike can be placed exactly on a node layer, which removes the largest single source of lattice error for path-dependent payoffs.,The identity with the binomial at 2n steps is specific to this parameterisation. Other choices of dx break it and are genuinely different schemes.,Because pm is close to one half at small dt, most of the probability mass sits on the middle branch and the tree is doing less exploring per step than the node count suggests.

### The step-size bound that keeps the risk-neutral probability admissible
The CRR probability p is not automatically between zero and one. It exceeds one when the carry over a step is larger than the up-move the volatility allows, which makes the lattice produce arbitrary values rather than a slightly wrong one.

Formula: 0 is less than p and p is less than 1 requires dt to be less than (sigma/(r - q))^2

The bound is almost never binding at ordinary equity rates and volatilities, and is routinely binding on a low-volatility name in a high-carry currency or on a heavily shorted stock where the effective carry is large and negative.,A lattice that silently returns a value with p outside zero and one is the worst kind of numerical failure: the answer looks like a price. Check p once at construction rather than trusting the value.,Widening the tree, as the trinomial does, is one fix. Reducing dt is the other. Both cost work.

### American exercise in a binomial lattice
At each node the value is the larger of the discounted continuation value and the immediate exercise value. That single comparison is the entire American feature; nothing else in the roll-back changes.

Formula: V_node = max( exp(-r*dt)*[p*V_up + (1-p)*V_down], Intrinsic(S_node) )

With q = 0 an American call has no early-exercise premium, so any positive number a lattice reports for one is discretisation error. Reporting it as a premium is a common and avoidable mistake.,The American put premium is driven entirely by the interest on the strike proceeds. It grows with r, with T and with how far in the money the strike sits.,Because the boundary sits on the lattice, the premium converges more slowly than the value. Doubling n to sharpen an American premium is often necessary where it would be wasteful for a European value.

### The early-exercise boundary read off a lattice
The critical underlying price at each date below which an American put is exercised immediately. It is not an input to the lattice; it is recovered by recording, at each time step, the highest node price at which exercise beat continuation.

Formula: S*(t) = max{ S_node at time t : Intrinsic(S_node) is at least the continuation value }

The boundary is steepest at the very end. Most of the early-exercise decision compresses into the final days, which is also when the lattice resolves it worst.,A position that sits above the boundary at 87 with the underlying at 100 is nowhere near exercise, and the same position with the underlying at 92 two days before expiry is on it. The distance to the boundary, not the distance to the strike, is the relevant number.,For an American call on a dividend-paying underlying the boundary exists only in the interval immediately before an ex-dividend date, and is a single-date test rather than a continuous curve.

### Bjerksund-Stensland approximation for an American option
A closed-form approximation that replaces the true curved exercise boundary with a flat trigger level X, chosen from the model inputs, and prices the option as a combination of six power-and-normal terms. It is an approximation with a measurable gap to a converged lattice, not an exact value.

Formula: beta = (0.5 - b/sigma^2) + sqrt((b/sigma^2 - 0.5)^2 + 2*r/sigma^2), with b = r - q;  Binf = beta/(beta - 1)*K;  B0 = max(K, r/(r - b)*K);  X = B0 + (Binf - B0)*(1 - exp(h)),  h = -(b*T + 2*sigma*sqrt(T))*B0/(Binf - B0)

The gap to the lattice grows with the size of the early-exercise premium, which is the case the approximation exists to handle. Quoting it to four decimals implies an accuracy it does not have.,The 2002 two-step version splits the boundary into two flat segments and is materially more accurate than the 1993 single-boundary form worked above. If the premium matters, that is the version to use.,Its real advantage is that it is a formula, so it differentiates analytically and evaluates in constant time. A lattice cannot do either, which is why the approximation survives despite the error shown here.

### Monte Carlo estimator and the standard error of the estimate
Simulate terminal prices under the risk-neutral measure, discount the payoff of each, and average. The estimate is unbiased and its uncertainty is a computable standard error that falls with the square root of the sample size, so a further correct decimal costs one hundred times the paths.

Formula: S_T = S*exp((r - q - sigma^2/2)*T + sigma*sqrt(T)*z);  Estimate = (1/n)*sum of exp(-r*T)*Payoff(S_T^i);  SE = s_hat/sqrt(n), s_hat the sample standard deviation of the discounted payoffs

The standard error is a statement about the simulation, not about the option. A tight standard error on a mis-specified process is a precise wrong answer.,Sample standard deviation of a call payoff is dominated by the right tail, so it is roughly 1.5 times the option value at the money and far larger for a wing. Budgeting paths from the value rather than from s_hat understates the requirement badly.,Never quote a Monte Carlo price without its standard error. A price of 4.45 and a price of 4.49 are the same number when the standard error is 0.065.

### Antithetic variates and the measured variance reduction
Pair every normal draw z with its negation and average the two payoffs before averaging across pairs. The estimator stays unbiased, the number of normal draws is unchanged, and the variance falls because the paired payoffs are negatively correlated.

Formula: Estimate = (2/n)*sum over n/2 pairs of 0.5*[Payoff(z_i) + Payoff(-z_i)]

The reduction is real but modest, and it is not free on a path-dependent payoff, where the antithetic path must be regenerated rather than reused.,Antithetic variates help most where the payoff is close to linear in z and least where it is close to a digital, because a digital payoff is nearly uncorrelated with its own reflection.,A control variate on the same underlying, using the closed-form European value as the control, usually beats antithetics by a wide margin on a European-style payoff. Antithetics are the cheapest thing to add, not the strongest.

### Deterministic stratified sampling, and why a published Monte Carlo figure should use it
Replace pseudo-random draws with the inverse normal evaluated at the midpoints of n equal-probability strata. The result is fully deterministic, so it is reproducible by any reader without agreeing on a random number generator, and it converges far faster than pseudo-random sampling on a one-dimensional payoff.

Formula: z_i = Phi_inverse((i - 0.5)/n) for i = 1..n;  Estimate = (1/n)*sum of exp(-r*T)*Payoff(S*exp((r - q - sigma^2/2)*T + sigma*sqrt(T)*z_i))

The inverse normal used here is the Acklam rational approximation, verified to within 2.7e-10 by round-tripping N(Phi_inverse(u)) against u across a ten-thousand-point grid.,Any Monte Carlo number published as a reference figure and not reproducible without a seed is not a reference figure. Stratification removes the excuse.,On a genuinely high-dimensional payoff, a low-discrepancy sequence with a Brownian bridge construction is the equivalent move, and it keeps the reproducibility.

### Finite-difference schemes: explicit, implicit and Crank-Nicolson
Discretise the Black-Scholes partial differential equation on a grid in log-price and time and step backwards from the payoff. The three standard schemes differ in where the space derivatives are evaluated in time, and that single choice determines both stability and accuracy order.

Formula: In x = ln(S): dV/dt + 0.5*sigma^2*d2V/dx2 + (r - q - sigma^2/2)*dV/dx - r*V = 0. Discretised: (I - theta*dt*L)*V_new = (I + (1 - theta)*dt*L)*V_old, with theta = 0 explicit, theta = 1 implicit, theta = 0.5 Crank-Nicolson

Crank-Nicolson is second order in time but oscillates near a kink in the payoff, which at expiry is exactly at the strike. Two fully implicit steps at the start, the Rannacher smoothing, is the standard fix and costs almost nothing.,Grid truncation is a separate error from discretisation. A boundary placed at three standard deviations rather than five will dominate the error no matter how fine the grid.,The implicit scheme is the only one of the three that never blows up, which is why it is what you reach for when the inputs are unknown at build time.

### The explicit-scheme stability condition, and what violating it actually does
The explicit scheme is stable only while the dimensionless quantity sigma-squared times dt over dx-squared stays at or below one. Above that the scheme does not become inaccurate; it amplifies rounding error geometrically and returns a number with no relation to a price.

Formula: Stability requires sigma^2*dt/dx^2 to be at most 1, equivalently dt at most dx^2/sigma^2

A divergent explicit run is easy to spot because the answer is absurd. The dangerous case is a ratio slightly above one on a coarse grid, where the answer is merely wrong by a plausible-looking amount.,Check the ratio at construction and refuse to run rather than checking the output for reasonableness. The condition is one line of arithmetic and it is a hard boundary, not a guideline.,This is the reason the explicit scheme is rare in production despite being the simplest to write: its accuracy is competitive only on grids where the time-step count is already dictated by stability rather than by accuracy.

### Put-call parity for American options is an inequality
European parity is an identity because neither side can be exercised early. With American exercise the relationship becomes a two-sided bound, and the width of that band is the combined early-exercise value. Any claim of an arbitrage from an American call-put difference has to clear the band, not the identity.

Formula: S - K is at most C_A - P_A, and C_A - P_A is at most S - K*exp(-r*T), for a non-dividend-paying underlying

The band is narrow at short tenors and low rates and wide at long tenors and high rates. On a one-week contract it is close enough to an identity to use as a data check; on a two-year contract it is not.,Observed American call-put differences on listed equity options sit inside this band almost always, and when they do not the explanation is normally borrow cost or an unmodelled dividend rather than a tradable arbitrage.,The lower bound is the interesting one. It says the American difference can never fall below immediate intrinsic, which is a statement about the put being exercisable, not about the call.

#### Model families and what each one prices
A closed form exists only where the payoff depends on the terminal price alone and exercise is European. Everything else is a numerical problem, and the choice of method is a choice about which error you are willing to measure.

| Model | Underlying process assumed | European closed form | American exercise | Path dependence | Error measured as |
|---|---|---|---|---|---|
| Black-Scholes-Merton 1973 | Geometric Brownian motion, constant sigma, continuous dividend yield q | Yes | No | No | None; exact within the model |
| Black-76 | Lognormal forward, constant sigma | Yes | No | No | None; exact within the model |
| Binomial, Cox-Ross-Rubinstein 1979 | Two-state discrete approximation to the same diffusion | Converges | Yes | Only through the lattice state | Value error against the closed form |
| Trinomial | Three-state discrete approximation | Converges | Yes | Only through the lattice state | Value error against the closed form |
| Bjerksund-Stensland 1993 and 2002 | Same diffusion, flat exercise boundary approximated | n/a | Approximated | No | Gap to a converged lattice |
| Monte Carlo | Simulated paths under the risk-neutral measure | Converges | Only with a regression or duality method | Yes | Standard error of the estimate |
| Finite difference | The Black-Scholes partial differential equation discretised | Converges | Yes, by projecting the payoff each step | Only with an added state variable | Truncation error in space and time |

#### Binomial and trinomial convergence to Black-Scholes-Merton, verified
European call, S = 100, K = 100, T = 0.25, r = 0.04, q = 0, sigma = 0.20. The closed-form value is 4.485236409. CRR uses u = exp(sigma*sqrt(dt)), d = 1/u, p = (exp((r-q)*dt) - d)/(u - d). The trinomial uses dx = sigma*sqrt(2*dt) with the Kamrad-Ritchken probabilities. Every value below was recomputed; the trinomial column at n steps is numerically identical to the CRR column at 2n steps, to every printed digit.

| Steps n | CRR u | CRR p | CRR value | CRR error | Trinomial value at n steps | Trinomial error |
|---|---|---|---|---|---|---|
| 1 | 1.10517092 | 0.52518799 | 5.46849110 | plus 0.98325469 | 4.03156245 | minus 0.45367395 |
| 2 | 1.07327066 | 0.51774405 | 4.03156245 | minus 0.45367395 | 4.24592704 | minus 0.23930937 |
| 4 | 1.05127110 | 0.51252345 | 4.24592704 | minus 0.23930937 | 4.36286187 | minus 0.12237454 |
| 8 | 1.03598777 | 0.50884712 | 4.36286187 | minus 0.12237454 | 4.42345524 | minus 0.06178117 |
| 16 | 1.02531512 | 0.50625293 | 4.42345524 | minus 0.06178117 | 4.45421023 | minus 0.03102617 |
| 32 | 1.01783484 | 0.50442045 | 4.45421023 | minus 0.03102617 | 4.46969114 | minus 0.01554527 |
| 64 | 1.01257845 | 0.50312537 | 4.46969114 | minus 0.01554527 | 4.47745595 | minus 0.00778046 |
| 128 | 1.00887801 | 0.50220984 | 4.47745595 | minus 0.00778046 | 4.48134425 | minus 0.00389216 |
| 256 | 1.00626957 | 0.50156255 | 4.48134425 | minus 0.00389216 | 4.48328985 | minus 0.00194656 |
| 512 | 1.00442920 | 0.50110487 | 4.48328985 | minus 0.00194656 | 4.48426301 | minus 0.00097340 |
| 1024 | 1.00312989 | 0.50078126 | 4.48426301 | minus 0.00097340 | 4.48474968 | minus 0.00048673 |
| 2048 | 1.00221215 | 0.50055243 | 4.48474968 | minus 0.00048673 | n/a | n/a |

#### Finite-difference schemes: measured error and the stability boundary
Same call and same closed-form target. The grid is uniform in x = ln(S) with Dirichlet boundaries at plus and minus five standard deviations, M space steps and Nt time steps. The stability quantity is sigma^2 * dt / dx^2. Every value was recomputed; the two explicit rows above the boundary are the actual output, not a description of what would happen.

| Scheme | M | Nt | sigma^2 dt/dx^2 | Value | Error vs closed form |
|---|---|---|---|---|---|
| Explicit | 200 | 400 | 0.9803 | 4.48524380 | plus 7.40e-06 |
| Explicit | 400 | 1600 | 0.9803 | 4.48523847 | plus 2.06e-06 |
| Explicit | 200 | 100 | 3.9212 | minus 2.2e+81 | Diverged |
| Explicit | 200 | 20 | 19.6059 | minus 7.1e+29 | Diverged |
| Implicit | 200 | 100 | 3.9212 | 4.47896828 | minus 6.27e-03 |
| Implicit | 200 | 400 | 0.9803 | 4.48273241 | minus 2.50e-03 |
| Implicit | 400 | 1600 | 0.9803 | 4.48461107 | minus 6.25e-04 |
| Crank-Nicolson | 200 | 100 | 3.9212 | 4.48400038 | minus 1.24e-03 |
| Crank-Nicolson | 200 | 400 | 0.9803 | 4.48398869 | minus 1.25e-03 |
| Crank-Nicolson | 400 | 1600 | 0.9803 | 4.48492481 | minus 3.12e-04 |

#### Monte Carlo estimates and their standard errors, verified
Same call and same closed-form target of 4.485236409. The stratified rows are fully deterministic: draw z_i = Phi-inverse((i - 0.5)/n) for i = 1..n, so any reader reproduces them exactly. The pseudo-random rows depend on the generator and are stated with the seed used, 20260827, and Python's Mersenne Twister via random.gauss. The antithetic rows use the same total number of normal draws arranged as n/2 sign-paired pairs.

| Method | Draws n | Estimate | Sample s.d. | Standard error | Error vs closed form |
|---|---|---|---|---|---|
| Stratified midpoint | 10 | 4.32466075 | n/a, deterministic | n/a | minus 0.16057566 |
| Stratified midpoint | 100 | 4.47126008 | n/a, deterministic | n/a | minus 0.01397633 |
| Stratified midpoint | 1,000 | 4.48396632 | n/a, deterministic | n/a | minus 0.00127009 |
| Stratified midpoint | 10,000 | 4.48511797 | n/a, deterministic | n/a | minus 0.00011844 |
| Stratified midpoint | 100,000 | 4.48522517 | n/a, deterministic | n/a | minus 0.00001124 |
| Pseudo-random, plain | 10,000 | 4.449312 | 6.486307 | 0.064863 | minus 0.035925 |
| Pseudo-random, antithetic | 10,000 | 4.550518 | 3.347707 | 0.047344 | plus 0.065282 |
| Pseudo-random, plain | 100,000 | 4.486298 | 6.551935 | 0.020719 | plus 0.001062 |
| Pseudo-random, antithetic | 100,000 | 4.491230 | 3.363438 | 0.015042 | plus 0.005994 |
| Pseudo-random, plain | 1,000,000 | 4.490235 | 6.536779 | 0.006537 | plus 0.004999 |
| Pseudo-random, antithetic | 1,000,000 | 4.487653 | 3.354441 | 0.004744 | plus 0.002417 |

## Greeks
Reviewed: 2026-08-27
Canonical: https://options.wiki/greeks/ (JSON: https://options.wiki/greeks.json)

N() is the standard normal cumulative distribution function and phi() its density. Definitions below assume the Black-Scholes-Merton framework with continuous dividend yield q. Quoting conventions differ from mathematical definitions and are stated separately for each measure, because that mismatch is a common source of error.

### Gamma and vega are identical for calls and puts
For the same strike, expiration, and underlying, a call and a put have identical gamma and identical vega. Only delta, theta, and rho differ. This follows directly from put-call parity, in which the difference between call and put value is linear in S and therefore has zero second derivative and no volatility sensitivity.

Practical consequence: a long straddle has roughly double the gamma and vega of either leg alone, and near-zero delta at the money.,It also means there is no gamma or vega reason to prefer a call over a put at the same strike. The choice is about delta, financing, and assignment.

### Second-order Greeks
Cross-derivatives that matter for hedged books held over time.

Charm is why a delta-hedged book drifts out of hedge over a weekend with no price movement at all.,Volga is positive for long options, which is why long vega positions gain disproportionately in a volatility spike.

### The theta-gamma relationship
In a delta-hedged Black-Scholes book, theta and gamma are two sides of one quantity. A long-gamma position pays theta; a short-gamma position collects it.

Formula: Theta + 0.5*sigma^2*S^2*Gamma + (r-q)*S*Delta - r*V = 0

This is the Black-Scholes PDE rearranged. It states that the time decay you pay is the fair price of the convexity you own.,A long-gamma book profits when realised volatility exceeds the implied volatility paid, and loses otherwise. That comparison, not direction, is the actual position.

### Delta
The first derivative of option value with respect to the underlying price. It is both a hedge ratio and, for d2 rather than d1, a bridge to the risk-neutral probability of finishing in the money.

Formula: Delta_call = exp(-q*T)*N(d1);  Delta_put = exp(-q*T)*(N(d1) - 1) = Delta_call - exp(-q*T)

At-the-money delta is above 0.50 for a call, not equal to it, because d1 carries the plus sigma-squared-over-two drift term. In the worked case it is 0.5596.,Delta is a local slope. Using it to project a large move ignores gamma and will understate a long option and overstate a short one.,Delta is not the probability of finishing in the money. N(d2) is the risk-neutral probability; N(d1) is delta. In the worked case they are 0.5199 and 0.5596, a gap of nearly four points at the money.

### Gamma
The second derivative of option value with respect to the underlying price, equivalently the first derivative of delta. It measures how fast a hedge goes stale.

Formula: Gamma = exp(-q*T)*phi(d1) / (S*sigma*sqrt(T))

Gamma scales with the square of the move, so the convexity term is negligible for small moves and dominant for large ones. That asymmetry is the entire economics of a long-gamma book.,Gamma is highest at the money and rises as expiry approaches, so an at-the-money short option is at its most dangerous on its last day.,Gamma falls as sigma rises: a higher volatility input spreads the density and flattens the peak. Short-gamma positions therefore look calmer in high-volatility inputs than they behave.

### Theta
The derivative of option value with respect to the passage of time, holding S and sigma fixed. Long options have negative theta because extrinsic value must reach zero at expiry.

Formula: Theta_call = -S*exp(-q*T)*phi(d1)*sigma/(2*sqrt(T)) - r*K*exp(-r*T)*N(d2) + q*S*exp(-q*T)*N(d1)

Theta is not linear. Quoting a per-day figure and multiplying by the holding period understates decay for an at-the-money option, because theta itself grows as expiry approaches.,Call theta and put theta differ at the same strike because the interest terms carry opposite signs. In the worked case the call decays 66 percent faster than the put.,A deep in-the-money put can have positive theta once the interest term dominates, which is the same condition that makes early exercise rational.

### Vega
The derivative of option value with respect to implied volatility. Not a Greek letter, and not a true partial derivative in any model where sigma is a constant, but the standard measure of volatility exposure.

Formula: Vega = S*exp(-q*T)*phi(d1)*sqrt(T)

Vega is nearly exact for a one-point move and increasingly wrong for a large one, because volga makes vega itself a function of sigma.,Vega peaks slightly above the at-the-money strike and falls away on both wings, but the wings carry far more vega per dollar of premium.,A vega figure is meaningless without a term attached. One point of vega on a one-week option and one point on a one-year option are different exposures to the same headline number.

### Rho
The derivative of option value with respect to the risk-free rate. Usually the smallest of the first-order Greeks for short-dated options and material for long-dated ones.

Formula: Rho_call = K*T*exp(-r*T)*N(d2);  Rho_put = -K*T*exp(-r*T)*N(-d2)

Rho is the reason put-call parity implies a call is worth more than a put at the same at-the-money strike: in the worked case 4.4852 against 3.4902, a difference of 0.9950 which equals S - K*exp(-r*T) exactly.,For LEAPS-length options rho can exceed vega in dollar terms, so a rate move repricing a long-dated book is not a volatility event.

### Delta-neutral hedge ratio
The quantity of the underlying required to bring net position delta to zero. It is a point-in-time figure that decays with charm and moves with gamma.

Formula: Shares to hedge = -(sum over legs of qty * multiplier * Delta_leg)

A hedge computed from delta alone is neutral at one price and one instant. Gamma tells you how fast that stops being true, and it is the only figure that sets a sensible rehedge band.,Rounding to whole shares leaves a residual delta. On a small position that residual can exceed the gamma exposure being managed.,Hedging with the underlying neutralises delta and leaves gamma, vega, and theta untouched. Only another option can hedge those.

### Gamma scalping arithmetic
The mechanical result of rehedging a delta-neutral position as the underlying moves. Each rehedge realises the convexity term of the value change, and the sum of those realisations is set against theta paid.

Formula: Realised convexity per rehedge = 0.5 * Gamma * dS^2 per share

The breakeven move per day is proportional to the square root of theta over gamma, which is another statement of the implied-versus-realised comparison rather than a separate rule.,Realised convexity depends on the path, not the endpoint. A position can be flat on the day and have scalped a substantial amount, or move a long way in one gap and scalp almost nothing.,Every rehedge crosses a spread. The bid-ask cost of the rehedging programme is subtracted from the convexity, and at a tight rehedge band it can exceed it.

### Aggregating Greeks across a multi-leg position
Greeks are additive across legs when each is expressed in the same units. Sum signed quantity times multiplier times the per-share Greek for every leg; there is no interaction term at first order.

Formula: Position Greek = sum over legs of (signed qty) * multiplier * (per-share Greek)

A spread cancels most of the gamma and vega of its legs while retaining most of the delta. In the worked case the vertical keeps 34 percent of the long leg delta and 4 percent of its gamma.,Legs in different expirations cannot be summed for vega without weighting. One point of vega in a one-week leg and one point in a six-month leg do not offset, because implied volatilities in different tenors do not move one for one.,Position theta on the iron condor is positive and position vega negative, which is the same trade viewed twice, not two separate exposures.

### How each Greek behaves as spot, time and volatility move
The sign of each cross-effect, stated for a long option. Reverse every sign for a short position. These are the relationships that make a hedge decay without any trade being placed.

The at-the-money and wing cases move in opposite directions for gamma and vega as expiry approaches. Any statement that begins "gamma rises into expiry" is true only at the money.,A position can lose money with S, sigma and the calendar all unchanged if the strike has moved relative to the money because the underlying moved earlier. The Greeks are recomputed, not carried forward.

### Charm, vanna and volga in closed form
The three second-order cross-derivatives most often quoted on a hedged book, given for completeness alongside their definitions.

Formula: Vanna = -exp(-q*T)*phi(d1)*d2/sigma;  Volga = Vega*d1*d2/sigma

Volga is zero where d1 and d2 straddle zero, which happens just below the at-the-money strike. Vega is locally flat in sigma there and a vega hedge is at its most stable.,Vanna is what makes a delta hedge on a skewed book directional in volatility rather than in price. A vanna-heavy position can be delta-flat and still lose on a pure volatility move.,Charm is why a delta-hedged book drifts out of hedge over a weekend with no price movement at all.

### Speed, zomma, colour and ultima in closed form
The third-order sensitivities, given for completeness and because they are what a hedged book's Greeks do when the book is left alone. Each is the derivative of a second-order Greek with respect to one of the three moving inputs.

Formula: Speed = minus (Gamma/S)*(1 + d1/(sigma*sqrt(T)));  Zomma = Gamma*(d1*d2 - 1)/sigma;  Colour = Gamma*[q + (r-q)*d1/(sigma*sqrt(T)) + (1 - d1*d2)/(2*T)];  Ultima = minus (Vega/sigma^2)*[d1*d2*(1 - d1*d2) + d1^2 + d2^2]

Colour is positive at the money, which is the arithmetic behind the statement that gamma rises into expiry. It is negative on a far wing, where gamma falls into expiry, so the statement is strike-specific and not general.,Speed is negative for a call struck at or below spot and positive above it. It changes sign near the strike where d1 crosses minus sigma*sqrt(T), which is why a gamma hedge behaves asymmetrically for equal moves up and down.,Zomma is negative at the money and positive on the wings, so raising the volatility input flattens the gamma profile. A short-gamma book therefore looks least dangerous in exactly the volatility regime where it is most dangerous.,Ultima is large and negative on a wing, minus 396 at the 95 strike against minus 16 at the money in the worked case. Vega hedging a wing with a linear vega estimate is unreliable for anything but a small volatility move.

### Charm and veta: the Greeks of the calendar
Charm is the rate at which delta changes with the passage of time and veta the rate at which vega does. Both are the reason a book that traded nothing over a weekend comes back with different exposures than it left with.

Formula: Charm_call = q*exp(-q*T)*N(d1) - exp(-q*T)*phi(d1)*[2*(r-q)*T - d2*sigma*sqrt(T)] / (2*T*sigma*sqrt(T));  Veta = S*exp(-q*T)*phi(d1)*sqrt(T)*[q + (r-q)*d1/(sigma*sqrt(T)) - (1 + d1*d2)/(2*T)]

Charm is why a wing hedge goes stale faster than an at-the-money hedge. At K = 105 the worked charm is four times the at-the-money figure.,Charm changes sign across the strike, so a symmetric strangle hedge drifts in one direction on one side and the other direction on the other. A single rehedge band applied to both sides is not symmetric in practice.,Veta is negative at the money for a long option, which means the vega you are paying for shrinks while you hold it. Comparing vega across two tenors without accounting for that overstates the longer-dated exposure over any holding period.

### Dual delta and dual gamma: sensitivity to the strike
The derivatives with respect to K rather than S. They are not hedging quantities, because K is contractual, but they are the exact tool for pricing a strike-shift, for interpolating a value between two listed strikes, and for reading the implied terminal density off a strike curve.

Formula: DualDelta_call = minus exp(-r*T)*N(d2);  DualDelta_put = exp(-r*T)*N(-d2);  DualGamma = exp(-r*T)*phi(d2)/(K*sigma*sqrt(T))

Dual gamma is the cleanest data-quality check on an option chain there is. Compute the second difference of mid prices across three adjacent strikes; a negative result means the chain cannot be arbitrage-free as quoted.,Interpolating a value between two listed strikes with dual delta is a first-order estimate and it understates a convex curve. Add half of dual gamma times the strike gap squared and the estimate is usually within a cent.,The identity between minus dual delta and the digital is why a tight vertical replicates a digital: differencing the call price in K is literally taking the derivative.

### Vega is gamma times S squared times sigma times T
For any two options on the same underlying with the same expiration and the same volatility input, vega and gamma are the same number scaled by a strike-independent constant. That makes them the same exposure at one expiration, and it is why they cannot be hedged separately with same-expiry instruments.

Formula: Vega = Gamma * S^2 * sigma * T, identically, for every strike at a given S, sigma and T

This is the reason a calendar spread exists as a structure. Term is the only dimension along which gamma and vega separate.,A risk report that shows a large vega and a small gamma at a single expiration is showing a unit-conversion artefact, not two independent exposures. Divide vega by S-squared-sigma-T and check.,The proportionality holds under the model, at one volatility input. Once the surface has a term structure, the two exposures do separate economically even though they remain proportional strike by strike, because a one-point move in a near tenor and a one-point move in a far tenor are not the same event.

### The profit and loss attribution identity, worked on a real reprice
A value change over a finite move is exactly reproduced by the Taylor expansion in the three moving inputs, and the residual after each term is what quantifies whether the next order matters. The identity is arithmetic, not an approximation to be trusted on faith, and it either closes or names the term that is missing.

Formula: dV = Delta*dS + 0.5*Gamma*dS^2 + Speed*dS^3/6 + Vega*dsigma + 0.5*Volga*dsigma^2 + Theta*dt + Vanna*dS*dsigma + Charm*dS*dt + Veta*dsigma*dt + residual

Delta and gamma together explain 87 percent of this move and leave 0.17 unexplained, which is larger than the entire theta term. Stopping at second order in S is the most common attribution error and it is not small.,The three calendar cross-terms, charm and veta, together account for 0.019 of the 0.030 residual left after the first-order terms. On a book held over a weekend they are not optional.,A residual that will not close is diagnostic. If it exceeds a percent of the move on a single vanilla option with no missing inputs, the inputs used for the two repricings differ in something other than what you think moved.

### Delta, gamma and vega hedging as a linear system
Neutralising three exposures with three instruments is a three-by-three linear solve. Writing it that way makes two things visible that a leg-by-leg hedge hides: whether the system is solvable at all, and how much of the neutrality is destroyed by rounding to whole contracts.

Formula: A*x = minus g, where row i of A holds the per-contract delta, gamma and vega of each hedge instrument, x is the contract count of each, and g is the book's current delta, gamma and vega

The singularity is the whole reason a hedge needs two expirations. A trader who reaches for a second strike in the same expiration to fix a vega problem has not fixed anything, and the solver will say so with a zero pivot rather than a bad answer.,Rounding residual scales with the size of the hedge instruments, not with the size of the book. Hedging a small book with high-delta contracts leaves a proportionally larger residual than hedging a large one.,The solve is instantaneous and valid only at the current inputs. Every Greek in A is itself a function of S, sigma and t, so the system has to be re-formed, not re-used, after any material move.

### Theta decomposed into its three terms
Call theta is the sum of three separately interpretable pieces: the decay of convexity, the interest on the discounted strike, and the dividend on the deferred stock. Splitting them explains why call and put theta differ at the same strike, and why a deep in-the-money put can have positive theta.

Formula: Theta_call = minus S*exp(-q*T)*phi(d1)*sigma/(2*sqrt(T))  minus  r*K*exp(-r*T)*N(d2)  plus  q*S*exp(-q*T)*N(d1)

The call-put theta difference is exactly r*K*exp(-r*T) at any strike, any volatility and any moneyness when q = 0. It is a rate quantity, not a volatility quantity, and it does not depend on sigma at all.,Term 1 being minus 0.5*sigma^2*S^2*Gamma is the Black-Scholes partial differential equation showing through: the volatility part of theta is the price of the convexity, and nothing else in the formula is.,For a deep in-the-money put term 1 goes to zero while term 2 stays at plus r*K*exp(-r*T), so theta turns positive. That is the same condition that makes early exercise of the put rational, expressed as a Greek instead of as a comparison.

### Pin risk quantified
As expiry approaches an at-the-money delta goes to a step function and gamma to an unbounded spike. Pin risk is the dollar consequence of that: a hedge computed at one price is wrong at a price a few cents away, and the assignment outcome is undetermined until after the close.

Formula: Gamma at the money grows as 1/sqrt(T); the required hedge change over a move dS is 100*Gamma*dS share-equivalents per contract

The assignment uncertainty is larger than the hedging problem. The hedge error is measured in tens of shares; the assignment error is measured in whole multipliers, and it is binary.,A short at-the-money option at expiry is the only common position where the correct hedge cannot be computed, because it depends on an exercise decision that has not been made yet.,Closing the position rather than hedging it removes the uncertainty entirely, and its cost is one bid-ask spread on a contract whose spread is at its widest relative to its price. That trade-off is the whole of pin-risk management and it is arithmetic, not judgement.

### Greeks of a box spread and of a synthetic: zero at every order
A box spread has no exposure to the underlying or to volatility at any order, because it is a bond. A synthetic long has a delta of exactly one and no gamma or vega. Both are useful precisely because their Greeks are known constants rather than model outputs.

Formula: Box(K1,K2) = C(K1) - C(K2) + P(K2) - P(K1) = (K2 - K1)*exp(-r*T);  Synthetic long at K = C(K) - P(K) = S*exp(-q*T) - K*exp(-r*T)

A box that shows any gamma or vega in a risk system has a leg mispriced, a wrong strike, or a stale mark. It is the fastest single check on a valuation feed.,The synthetic long delta of exactly one hundred share-equivalents per contract holds at every sigma and every moneyness. Any deviation is q, not error.,Box theta is positive and equals interest, which is the only sense in which a box decays. Describing it as a theta position confuses a financing accrual with option decay.

#### Black-Scholes-Merton inputs

| Term | Expression |
|---|---|
| d1 | [ln(S/K) + (r - q + sigma^2/2)*T] / (sigma*sqrt(T)) |
| d2 | d1 - sigma*sqrt(T) |
| Call value | S*exp(-q*T)*N(d1) - K*exp(-r*T)*N(d2) |
| Put value | K*exp(-r*T)*N(-d2) - S*exp(-q*T)*N(-d1) |

#### Sensitivities

| Greek | Definition | Call | Put | Quoted as |
|---|---|---|---|---|
| Delta | dV/dS | exp(-q*T)*N(d1) | exp(-q*T)*(N(d1) - 1) | Per 1.00 move in S. Range 0 to 1 for calls, -1 to 0 for puts. |
| Gamma | d2V/dS2 | exp(-q*T)*phi(d1) / (S*sigma*sqrt(T)) | Identical to call | Change in delta per 1.00 move in S. |
| Vega | dV/dsigma | S*exp(-q*T)*phi(d1)*sqrt(T) | Identical to call | Usually divided by 100 and quoted per 1 volatility point. |
| Theta | dV/dt | Negative for long options, largest near ATM at expiry | Same sign convention | Usually divided by 365 and quoted per calendar day. |
| Rho | dV/dr | K*T*exp(-r*T)*N(d2) | -K*T*exp(-r*T)*N(-d2) | Usually divided by 100 and quoted per 1 percentage point. |

#### Reference Greeks at fixed inputs, verified
Computed at S = 100, r = 0.04, q = 0, sigma = 0.20, T = 0.25 from the closed forms above. Per-share figures; multiply by 100 for one standard contract. Vega is stated per one volatility point and theta per calendar day, matching quoting convention rather than the raw derivative. These are chosen inputs, not market observations.

| Strike | d1 | d2 | Call | Put | Delta call | Delta put | Gamma | Vega/pt | Theta call/day | Theta put/day | Rho call/pt |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 95 | 0.662933 | 0.562933 | 7.5459 | 1.6006 | 0.746313 | -0.253687 | 0.032024 | 0.160121 | -0.024899 | -0.014592 | 0.167714 |
| 100 | 0.150000 | 0.050000 | 4.4852 | 3.4902 | 0.559618 | -0.440382 | 0.039448 | 0.197240 | -0.027257 | -0.016407 | 0.128691 |
| 105 | -0.337902 | -0.437902 | 2.3909 | 6.3461 | 0.367719 | -0.632281 | 0.037681 | 0.188403 | -0.024415 | -0.013022 | 0.085952 |

#### Position Greeks worked across multi-leg structures
Same inputs. One contract per leg, multiplier 100. Delta in share-equivalents, gamma in delta per 1.00 move, vega in dollars per volatility point, theta in dollars per calendar day. Each row is the signed sum of its legs and nothing else.

| Position | Value | Position delta | Position gamma | Position vega | Position theta |
|---|---|---|---|---|---|
| Long 100 call | +448.52 | +55.96 | +3.945 | +19.72 | -2.726 |
| Short 100 call | -448.52 | -55.96 | -3.945 | -19.72 | +2.726 |
| Long 100 straddle | +797.55 | +11.92 | +7.890 | +39.45 | -4.366 |
| Bull call spread 100/105 | +209.44 | +19.19 | +0.177 | +0.884 | -0.284 |
| Iron condor 90/95/105/110 | n/a, credit position | -1.74 | -2.147 | -10.74 | +1.171 |
| 10 short 100 calls, delta-hedged with 560 shares | n/a | +0.38 | -39.45 | -197.24 | +27.26 |

#### Second- and third-order Greeks in closed form
Every expression below was verified against a central finite difference of the corresponding first-order Greek at the reference inputs; the largest disagreement across all of them was 1.2e-07, on ultima, which is third order and therefore the noisiest to difference numerically. Charm, veta and colour are stated as derivatives with respect to calendar time t, so a positive number means the quantity increases as time passes. Vega and its derivatives are raw, not divided by 100.

| Greek | Definition | Closed form | Identical for a call and a put |
|---|---|---|---|
| Vanna | d2V / (dS dsigma) | minus exp(-q*T)*phi(d1)*d2/sigma | Yes |
| Volga, also vomma | d2V / dsigma^2 | Vega*d1*d2/sigma | Yes |
| Charm | d2V / (dS dt), the drift of delta with time | Call: q*exp(-q*T)*N(d1) minus exp(-q*T)*phi(d1)*[2*(r-q)*T minus d2*sigma*sqrt(T)] / (2*T*sigma*sqrt(T)). Put: same second term, with minus q*exp(-q*T)*N(-d1) in place of the first | Only when q = 0 |
| Veta | dVega / dt, the drift of vega with time | S*exp(-q*T)*phi(d1)*sqrt(T)*[q + (r-q)*d1/(sigma*sqrt(T)) minus (1 + d1*d2)/(2*T)] | Yes |
| Speed | d3V / dS^3 | minus (Gamma/S)*(1 + d1/(sigma*sqrt(T))) | Yes |
| Zomma | dGamma / dsigma | Gamma*(d1*d2 minus 1)/sigma | Yes |
| Colour, also color | dGamma / dt | Gamma*[q + (r-q)*d1/(sigma*sqrt(T)) + (1 minus d1*d2)/(2*T)] | Yes |
| Ultima | d3V / dsigma^3 | minus (Vega/sigma^2)*[d1*d2*(1 minus d1*d2) + d1^2 + d2^2] | Yes |
| Dual delta | dV / dK | Call: minus exp(-r*T)*N(d2). Put: exp(-r*T)*N(-d2) | No |
| Dual gamma | d2V / dK^2 | exp(-r*T)*phi(d2)/(K*sigma*sqrt(T)) | Yes |

#### Second- and third-order Greeks at the reference inputs, verified
S = 100, r = 0.04, q = 0, sigma = 0.20, T = 0.25. Per-share figures, raw units, computed from the closed forms above and cross-checked against finite differences. Charm is the call figure; at q = 0 the put figure is identical, because the difference between call and put delta is the constant exp(-q*T) and its time derivative is zero.

| Strike | Vanna | Volga | Charm | Veta | Speed | Zomma | Colour | Ultima | Dual delta, call | Dual gamma |
|---|---|---|---|---|---|---|---|---|---|---|
| 95 | minus 0.901374 | 29.877514 | plus 0.232453 | minus 39.729219 | minus 0.002443 | minus 0.100366 | plus 0.048638 | minus 396.416260 | minus 0.706163 | 0.035484 |
| 100 | minus 0.098620 | 0.739649 | minus 0.118344 | minus 38.560355 | minus 0.000986 | minus 0.195760 | plus 0.080671 | minus 15.997986 | minus 0.514765 | 0.039448 |
| 105 | plus 0.825018 | 13.938743 | minus 0.480729 | minus 45.802465 | plus 0.000896 | minus 0.160525 | plus 0.059117 | minus 203.478793 | minus 0.327438 | 0.034177 |

#### Position Greeks by structure at the reference inputs, verified
Every leg valued from Black-Scholes-Merton at S = 100, r = 0.04, q = 0, sigma = 0.20, T = 0.25, one contract per leg, multiplier 100. These are model values at chosen inputs, not the stated premiums used in the payoff worked examples elsewhere on this site, so the value column will not match those dollar figures. Delta is in share-equivalents, gamma in delta per 1.00 move, vega in dollars per volatility point, theta in dollars per calendar day. Each row is the signed sum of its legs and nothing else.

| Structure | Value | Position delta | Position gamma | Position vega per point | Position theta per day |
|---|---|---|---|---|---|
| Long 100 call | plus 448.52 | plus 55.96 | plus 3.9448 | plus 19.7240 | minus 2.7257 |
| Long 100 put | plus 349.02 | minus 44.04 | plus 3.9448 | plus 19.7240 | minus 1.6407 |
| Long 100 straddle | plus 797.55 | plus 11.92 | plus 7.8896 | plus 39.4479 | minus 4.3663 |
| Short 95/105 strangle | minus 399.15 | minus 11.40 | minus 6.9705 | minus 34.8523 | plus 3.9007 |
| Bull call spread 100/105 | plus 209.44 | plus 19.19 | plus 0.1767 | plus 0.8837 | minus 0.2842 |
| Bear call spread 100/105 | minus 209.44 | minus 19.19 | minus 0.1767 | minus 0.8837 | plus 0.2842 |
| Bull put spread 95/100 | minus 188.96 | plus 18.67 | minus 0.7424 | minus 3.7119 | plus 0.1815 |
| Long 95/100/105 call butterfly | plus 96.63 | minus 0.52 | minus 0.9191 | minus 4.5956 | plus 0.5199 |
| Iron butterfly 95/100/105 | minus 398.40 | minus 0.52 | minus 0.9191 | minus 4.5956 | plus 0.4657 |
| Iron condor 90/95/105/110 | minus 226.96 | minus 1.74 | minus 2.1473 | minus 10.7364 | plus 1.1708 |
| Long call condor 90/95/105/110 | plus 268.07 | minus 1.74 | minus 2.1473 | minus 10.7364 | plus 1.2251 |
| Call ratio 1x2 100/105 | minus 29.65 | minus 17.58 | minus 3.5913 | minus 17.9565 | plus 2.1573 |
| Collar, long 100 shares plus 95 put minus 105 call | minus 79.03 on the options | plus 37.86 | minus 0.5656 | minus 2.8282 | plus 0.9823 |
| Box spread 100/110 | plus 990.05 | 0.00 | 0.0000 | 0.0000 | plus 0.1085 |
| Jade lizard 90p/105c/110c | minus 183.20 | minus 4.24 | minus 2.8118 | minus 14.0590 | plus 1.5671 |

#### Profit and loss attribution on one reprice, term by term
Long one K = 100 call. The move: S from 100 to 102, sigma from 0.20 to 0.22, and 7 calendar days elapse, so T goes from 0.25 to 0.230822. Value moves from 4.485236409 to 5.855752, an actual change of plus 1.370515 per share, plus 137.05 per contract. Terms are added in the order shown and the residual column is what is still unexplained after each one. Every figure recomputed.

| Term | Expression | Contribution per share | Cumulative | Residual |
|---|---|---|---|---|
| Delta | Delta*dS = 0.559618*2.00 | plus 1.11923538 | plus 1.11923538 | plus 0.25127974 |
| Gamma | 0.5*Gamma*dS^2 = 0.5*0.039448*4.00 | plus 0.07889587 | plus 1.19813125 | plus 0.17238388 |
| Speed | Speed*dS^3/6 | minus 0.00131493 | plus 1.19681632 | plus 0.17369881 |
| Vega | Vega*dsigma = 19.723967*0.02 | plus 0.39447933 | plus 1.59129565 | minus 0.22078052 |
| Volga | 0.5*Volga*dsigma^2 | plus 0.00014793 | plus 1.59144358 | minus 0.22092845 |
| Theta | Theta*dt = minus 9.948648*0.019178 | minus 0.19079599 | plus 1.40064759 | minus 0.03013246 |
| Vanna | Vanna*dS*dsigma | minus 0.00394479 | plus 1.39670280 | minus 0.02618767 |
| Charm | Charm*dS*dt | minus 0.00453921 | plus 1.39216359 | minus 0.02164846 |
| Veta | Veta*dsigma*dt | minus 0.01479027 | plus 1.37737331 | minus 0.00685818 |

## Volatility
Reviewed: 2026-08-27
Canonical: https://options.wiki/volatility/ (JSON: https://options.wiki/volatility.json)

Volatility is quoted as an annualised standard deviation of continuously compounded returns. Variance is additive in time and volatility is not, which is the source of most conversion errors in this area. Every number in this section is derived from stated inputs; none of it is an observation of any market.

### Implied volatility
The volatility input that makes a pricing model return the observed market price of an option. It is an output of an inversion, not a measurement, and it inherits every assumption of the model used to invert it.

Formula: Solve sigma such that BS(S, K, T, r, q, sigma) = observed price

Vega approaches zero for deep in- or out-of-the-money options, so the inversion becomes numerically unstable exactly where the quoted implied volatility is most often reported to three decimals.,A price below the intrinsic-value bound has no implied volatility at all. In practice that means a stale or crossed quote, not a negative volatility.,Two vendors can publish different implied volatilities for the same contract from the same price by using a different rate, dividend assumption, or mid-price convention. Reconcile the inputs before reconciling the outputs.

### Realised volatility
The annualised standard deviation of the log returns that actually occurred over a stated window. The zero-mean estimator is standard for short windows because the drift term is not estimable over them.

Formula: sigma_realised = sqrt( (252/n) * sum over t of ln(S_t / S_{t-1})^2 )

The estimator choice changes the answer materially on short windows. A realised-volatility figure without its estimator and its annualisation factor stated is not reproducible.,Close-to-close realised volatility ignores intraday range entirely, so it understates the variance a gamma position actually experienced.,Comparing a realised volatility over n days to an implied volatility for a T-day option compares two different horizons unless n and T are matched.

### Variance versus volatility, and why the square root matters
Variance is additive across independent time intervals; volatility is not. Any operation that combines periods, tenors, or components must be done in variance and converted back at the end.

Formula: sigma_total = sqrt( (sigma_1^2 * T_1 + sigma_2^2 * T_2) / (T_1 + T_2) )

The square root is convex, so a variance-correct blend always exceeds the naive volatility average when the two inputs differ. The gap widens with the dispersion.,This is the same arithmetic that makes a single event day dominate a short-dated volatility quote out of all proportion to its length.

### Forward implied volatility
The volatility implied for the interval between two expirations, extracted from the two spot implied volatilities by variance subtraction. A negative result inside the square root indicates a calendar arbitrage or bad data.

Formula: sigma_fwd = sqrt( (sigma_2^2 * T_2 - sigma_1^2 * T_1) / (T_2 - T_1) )

A steeply inverted front end can imply a forward volatility far below both quoted numbers. That is not an error; it is what a concentrated near-dated event does to the term structure.,A calendar spread is a position on forward volatility, not on either quoted volatility. Pricing it against the spot volatilities of the two legs misstates the exposure.,When the expression under the square root is negative, the two quotes are jointly inconsistent. Check the rate, dividend and settlement-time assumptions before concluding an arbitrage exists.

### Volatility smile, skew and term structure
Implied volatility is not constant across strikes or expirations, so a single sigma cannot reprice a whole surface. The pattern across strikes at one expiry is the smile or skew; the pattern across expiries at one moneyness is the term structure.

Formula: Skew slope = (IV at K_low - IV at K_high) / (K_high - K_low), or per unit of delta

A skew makes the Black-Scholes constant-sigma assumption false by construction, which is why traders use the model as a quoting convention rather than a belief about the world.,Interpolating a surface linearly in strike can produce a negative implied density and therefore a butterfly arbitrage. Interpolation is normally done in variance against log-moneyness for that reason.,A skew means the wings of a spread carry different implied volatilities, so a vertical is a position on the skew as well as on direction, whether intended or not.

### Put-call parity pins one implied volatility per strike
Because the difference between a call and a put at the same strike and expiry is model-free, a European call and put at the same strike must share a single implied volatility. Any observed difference is an artefact of the inputs used, not a real dislocation.

Formula: C - P = S*exp(-q*T) - K*exp(-r*T), independent of sigma

This is the single most useful consistency check on an options data feed. Run it before trusting any implied volatility the feed publishes.,On American-style equity options the identity holds only approximately, and the residual is a read on borrow cost and pending dividends rather than on volatility.,A persistent call-put IV gap in a vendor feed usually means the vendor is using a different dividend forecast, not that the market disagrees with itself.

### The arithmetic of an implied volatility crush around a known event
When a scheduled event sits inside an option life, the quoted implied volatility blends a diffusive component with a one-off jump component. Once the event passes, the jump component leaves the quote and the implied volatility falls by a computable amount irrespective of the price move.

Formula: sigma_quoted^2 * T = sigma_base^2 * T + J^2, where J is the standard deviation of the one-off event move

The crush is a function of the event and the tenor, not of the direction of the move. A position can be right on direction and still lose if the realised move is smaller than J.,J is the move the quote is charging for. Comparing J to the move a position needs is the same comparison as breakeven versus expected move, expressed in variance terms.,Shorter tenors embed a larger volatility number for the same J, because J is divided by a smaller sqrt(T). Comparing headline implied volatilities across tenors around an event compares nothing.

### Vega exposure per volatility point
The dollar change in position value for a one-point change in implied volatility, which is the form vega is quoted in and the form position limits are usually set in.

Formula: Position vega per point = sum over legs of (signed qty) * multiplier * Vega_leg / 100

Summing vega across expirations treats a one-point move in a one-week tenor as equal to a one-point move in a one-year tenor. It is not, and an unweighted total vega on a book with a term spread is close to meaningless.,Vega is quoted per point but the derivative is per unit. A raw vega of 19.72 and a quoted vega of 0.1972 are the same number under different conventions, and mixing them is a factor-of-100 error.

### Solving implied volatility by Newton with vega as the derivative
Option value is strictly increasing in sigma, and its derivative in sigma is vega in closed form, so Newton's method applies directly with no numerical differentiation. Where vega is not small it converges quadratically and reaches machine precision in three or four steps from almost any starting point.

Formula: sigma_{n+1} = sigma_n - [BS(S, K, T, r, q, sigma_n) - Price] / Vega(sigma_n)

Always use the raw vega, not the vega divided by 100 that the risk screen shows. Using the quoted figure makes every Newton step a hundred times too large and the iteration will appear to diverge.,Bracket before you iterate. If the price is outside the no-arbitrage bounds there is no root, and Newton will wander for as long as you let it rather than telling you so.,The quantity 1/vega is the correct error bar on any published implied volatility. A feed that quotes IV to three decimals on a contract whose vega implies half a point of uncertainty per tick is reporting precision it does not have.

### Why the implied-volatility inversion fails deep out of the money
Vega goes to zero faster than value does as a strike moves away from the money, so the Newton step, which divides by vega, becomes unboundedly large and the mapping from price to volatility becomes numerically unusable. The failure is not in the algorithm; it is that the price carries almost no information about sigma at that strike.

Formula: dsigma/dPrice = 1/Vega, and Vega goes to zero as |d1| grows, so the inversion error is amplified by 1/Vega

A deep out-of-the-money strike does not have a well-determined implied volatility, and printing one to three decimals implies otherwise. The honest output is a range, or nothing.,This is why far-wing implied volatilities in vendor feeds jump around by whole points day to day while the underlying barely moves. The quote moved one tick.,The same arithmetic explains why wing volatility is where a fitted surface adds the most value: it replaces a number the price cannot determine with one the neighbouring strikes can.

### The volatility surface in strike-maturity space
The implied volatility of every listed contract on one underlying, arranged as a function of strike and expiration. It is not a model output; it is a restatement of the option prices in a unit that removes S, K, T and r, and it is subject to arbitrage constraints in both directions.

Formula: sigma_imp(K, T) such that BS(S, K, T, r, q, sigma_imp) equals the observed price at every (K, T). Total variance w(K, T) = sigma_imp(K, T)^2 * T

The surface is quoted in volatility and constrained in variance. Every no-arbitrage condition above is clean in w and messy in sigma, which is the whole argument for working in total variance.,A surface fitted strike by strike with no cross-strike constraint will produce butterfly arbitrages on any day the quotes are noisy. The constraint is not decoration.,Two vendors publishing different surfaces for the same underlying usually differ in the forward and the dividend assumption, not in the option prices. Reconcile w at the at-the-money strike before comparing wings.

### Sticky strike, sticky delta and sticky moneyness, and the delta each one implies
A skew means volatility is a function of strike, so a move in the underlying changes the volatility applicable to a fixed strike unless the surface is assumed rigid in strike space. The assumption made about that determines the total delta, and the three standard assumptions give three materially different numbers for the same contract.

Formula: Total delta = Delta_BS + Vega * dsigma/dS. Sticky strike: dsigma/dS = 0. Sticky moneyness or sticky delta: dsigma/dS = minus (dsigma/dK)*(K/S). Skew-following spot: dsigma/dS = dsigma/dK

The gap between the two extreme regimes on the worked at-the-money contract is 0.118344 of delta, which is exactly what a 3.00 move in the underlying would do through gamma alone, since Gamma*3.00 = 0.039448*3 = 0.118344. Choosing the regime is therefore the same size of decision as being wrong about spot by three points. It is not a refinement; it is a first-order disagreement about the hedge.,A risk system that reports one delta is silently asserting one regime. Find out which before reconciling a hedge against it.,The regimes are not interchangeable at different tenors: short-dated surfaces behave closer to sticky strike over small moves and closer to sticky moneyness over large ones, so a single choice applied across a book will be wrong somewhere by construction.

### SABR parameterisation
A stochastic-volatility model for a single forward, in which the forward follows dF = alpha_t*F^beta dW1 and alpha_t follows d(alpha) = nu*alpha dW2 with correlation rho. Its practical use is not the process but the published asymptotic expansion, which gives Black implied volatility directly as a function of strike in four parameters.

Formula: sigma_B(K,F) = [alpha / (D * z_over_x)] inverted as: sigma_B = (alpha / D) * (z/x(z)) * B, with D = (F*K)^((1-beta)/2) * [1 + ((1-beta)^2/24)*ln^2(F/K) + ((1-beta)^4/1920)*ln^4(F/K)],  z = (nu/alpha)*(F*K)^((1-beta)/2)*ln(F/K),  x(z) = ln( (sqrt(1 - 2*rho*z + z^2) + z - rho) / (1 - rho) ),  B = 1 + [ ((1-beta)^2/24)*alpha^2/(F*K)^(1-beta) + (rho*beta*nu*alpha)/(4*(F*K)^((1-beta)/2)) + nu^2*(2 - 3*rho^2)/24 ] * T

The z/x(z) factor is numerically unstable as z approaches zero, which is exactly at the money. At nu = 1e-12 the check above returns 0.1999585247 rather than 0.20 - a catastrophic-cancellation artefact, not a model result. Every production implementation needs an explicit small-z branch, and the entry above uses the closed at-the-money form there.,beta and rho are jointly close to unidentifiable from a single smile: a range of beta values fit almost equally well with a compensating rho. Most desks fix beta by convention and fit the other three.,It is an expansion, so its accuracy degrades at long maturities and in the far wings, and it can produce negative densities at extreme strikes. That is a known property and not a calibration failure.

### SVI parameterisation
A five-parameter functional form for total implied variance as a function of log-moneyness. It is a fitting form rather than a model of the underlying, and its value is that its no-arbitrage conditions are explicit algebraic inequalities in the five parameters.

Formula: w(k) = a + b*[rho*(k - m) + sqrt((k - m)^2 + s^2)], with k = ln(K/F) and implied volatility sigma(k) = sqrt(w(k)/T)

The minimum of w is at k = m minus s*rho/sqrt(1 - rho^2), which is not at k = m. On the worked parameters that is k = 0.07237229, not k = 0.02, and w there is 0.01659891, matching the closed-form minimum a + b*s*sqrt(1 - rho^2) exactly. Reporting m as the location of the vertex is wrong whenever rho is non-zero.,The Lee bound is almost never binding at equity tenors and becomes binding on very short-dated slices, where 4/T is small. A one-day slice has a limit of 1460, a one-year slice a limit of 4.,Satisfying the per-slice conditions does not make a whole surface arbitrage-free. Calendar arbitrage between slices is a separate constraint, and it is the one that a slice-by-slice fit breaks most often.

### Local volatility via Dupire
Given a complete arbitrage-free surface of European call prices, there is exactly one local volatility function that reproduces every one of them. It is recovered by differentiating the call surface, once in maturity and twice in strike, with no optimisation and no model fitting.

Formula: sigma_loc^2(K, T) = [ dC/dT + (r - q)*K*dC/dK + q*C ] / [ 0.5*K^2*d2C/dK2 ]

The denominator is the implied density, so local volatility is worst determined exactly where there is least probability mass: the far wings. A local-vol surface is a smooth function fitted to a quantity that is numerically hopeless out there.,Local volatility reprices every vanilla by construction, which is often mistaken for evidence that it is right. It says nothing about whether it prices a barrier or a cliquet correctly, because those depend on the dynamics and not just the marginals.,In practice the second derivative in strike is taken from a fitted implied-volatility slice rather than from raw quotes, because differencing noisy mid prices twice amplifies the noise by the square of the strike gap.

### Variance swaps and the log-contract replication
A contract paying realised variance minus a fixed strike. Its fair strike is replicable, with no volatility model, from a continuum of out-of-the-money option prices weighted by one over strike squared. This is the log-contract result and it is the reason variance, not volatility, is the tradable quantity.

Formula: K_var = (2*exp(r*T)/T) * [ integral from 0 to F of P(K)/K^2 dK  +  integral from F to infinity of C(K)/K^2 dK ], with F = S*exp((r - q)*T)

The replication is exact for a continuum of strikes and continuous monitoring. Every real variance swap deviates from it in two known ways: a finite strike grid and discrete return sampling, and the first is much larger than the second.,Because the strip is 1/K^2-weighted, a variance swap is much longer the downside wing than a straddle of the same vega. A position that looks like a volatility position is substantially a skew position.,Truncating the strip converts the exact replication into a corridor variance swap, which is a different contract with a different fair strike. The gap is not hedging error; it is a payoff difference.

### The VIX calculation as published in the CBOE white paper
A discretised version of the variance-swap fair strike, applied to two option expirations bracketing 30 days and interpolated between them. The published formula is a finite sum over listed out-of-the-money strikes plus a correction term for the gap between the forward and the nearest strike below it.

Formula: sigma^2 = (2/T) * sum over i of [ (dK_i / K_i^2) * exp(r*T) * Q(K_i) ]  -  (1/T) * (F/K0 - 1)^2, with K0 the highest listed strike at or below F, Q(K_i) the out-of-the-money mid price, and dK_i the half-distance between neighbouring strikes

The discretisation bias is upward, so a strike grid alone makes the index read above the variance it is estimating. On a 5-point grid at this level and tenor the effect measured here is 1.24 volatility points, which is larger than most of the daily changes people attribute to sentiment.,The index is a variance calculation reported as a volatility. Interpolating two index levels linearly, rather than interpolating in variance and then taking the square root, reintroduces the error the whole construction was designed to avoid.,The correction term is not cosmetic. Dropping it on a grid where F sits midway between strikes biases the result by roughly (F/K0 - 1)^2/T. On a 2.50 strike grid at a level of 100, F sitting midway gives F/K0 minus 1 = 0.0125 and a correction of 0.001901 in variance, which at a 20 volatility level moves the reported index from 20.0000 to 20.4697 - 0.4697 volatility points from one omitted term. On a 5.00 grid the same calculation gives 1.8184 points.

### Volatility cones
The distribution of realised volatility computed over a range of window lengths from the same return history, plotted as percentile bands against window length. It exists because the sampling variability of a realised-volatility estimate depends on the window, so a 5-day estimate and a 60-day estimate from identical data are not comparable numbers.

Formula: For each window w: sigma_w(i) = sqrt( (252/w) * sum over the w returns ending at i of ln(S_t/S_{t-1})^2 ), then take the percentiles of {sigma_w(i)} across all i

The cone shape is a property of the estimator, not of the market. A synthetic series with no volatility clustering at all produces one, as the worked example shows.,Because rolling windows overlap, the extremes of a cone are much less independent than the observation count suggests. Treating 56 overlapping 5-day windows as 56 observations overstates the sample by roughly the window length.,The cone answers one narrow and useful question: whether a realised-volatility reading is unusual for the window it was measured over. It does not answer whether it is unusual for the underlying.

### Term-structure interpolation in variance space
Interpolating an implied volatility between two expirations must be done on total variance, which is additive in time, not on volatility, which is not. Interpolating volatility linearly understates the intermediate level whenever the front is higher, and the error is not small.

Formula: w(T) = w1 + (w2 - w1)*(T - T1)/(T2 - T1), with w = sigma^2*T; then sigma(T) = sqrt(w(T)/T)

A six-point interpolation error at a 14-day tenor is larger than most bid-ask spreads in volatility terms. This is a pricing error, not a rounding preference.,The same argument applies across strikes: interpolate in total variance against log-moneyness, not in volatility against strike, or a linear fit can produce a negative implied density.,When w2 is below w1 the interpolation is still computable and is meaningless, because the pair of quotes is jointly inconsistent. Check the monotonicity before interpolating, not after.

### Forward variance and the floor it places on a longer-dated quote
The variance attributable to the interval between two expirations, obtained by subtracting total variances. Because it cannot be negative, it converts every pair of quoted implied volatilities into a hard lower bound on the longer-dated one.

Formula: Forward variance over [T1, T2] = (sigma_2^2*T_2 - sigma_1^2*T_1)/(T_2 - T_1);  floor: sigma_2 is at least sigma_1*sqrt(T_1/T_2)

A calendar spread is a position on forward variance and on nothing else. Pricing it against the two quoted spot volatilities misstates the exposure by exactly the amount the two differ from the forward.,The floor is the single most useful sanity check on a term-structure feed, and it costs one square root. Data that violates it is data, not an opportunity.,Forward variance can be very low without either quoted volatility being low, and that is the normal state of the term structure around a scheduled event rather than an anomaly.

#### Time scaling of volatility and variance
Variance scales linearly in time; volatility scales with the square root of time. All conversions below follow from that one fact.

| Conversion | Formula | Worked |
|---|---|---|
| Annual to horizon T | sigma_T = sigma * sqrt(T) | sigma = 0.20, T = 30/365: 0.20 * sqrt(0.0821918) = 0.057338, i.e. a 5.7338 percent one-standard-deviation move |
| Annual to one trading day | sigma_day = sigma / sqrt(252) | 0.20 / sqrt(252) = 0.012599, i.e. 1.2599 percent |
| Daily to annual | sigma = sigma_day * sqrt(252) | 0.011 * sqrt(252) = 0.174620, i.e. 17.4620 percent |
| Variance over T | Var_T = sigma^2 * T | 0.20^2 * 0.25 = 0.010000 |
| Adding independent periods | Var_total = Var_1 + Var_2 | Never add volatilities. Add variances and take the square root at the end |
| Expected absolute move | E|S_T - S| approx S * sigma * sqrt(2*T/pi) | S = 100, sigma = 0.20, T = 0.25: 100 * 0.20 * sqrt(0.159155) = 7.9788 |

#### Volatility measures compared

| Measure | What it is | Computed from | Direction of time |
|---|---|---|---|
| Implied volatility | The sigma that equates a model price to the observed option price | Option prices, inverted numerically | Forward-looking, one number per contract |
| Realised volatility | The standard deviation of the returns that actually occurred | Underlying price history over a stated window | Backward-looking |
| Historical volatility | A realised volatility over a longer or reference window, used as a baseline | Underlying price history | Backward-looking |
| Forward implied volatility | The volatility implied for the interval between two expirations | Two implied volatilities and their tenors | Forward-looking, interval-specific |

#### IV rank and IV percentile are different calculations
Both compress a volatility level into a 0 to 100 figure and they are not interchangeable. Rank is a position between two extremes; percentile is a count of observations below the current level.

| Measure | Formula | Worked | What it ignores |
|---|---|---|---|
| IV rank | 100 * (IV - IV_low) / (IV_high - IV_low) | IV 28, low 14, high 56: 100 * 14 / 42 = 33.3333 | The shape of the distribution between the two extremes. Two very different histories with the same high and low give the same rank |
| IV percentile | 100 * (count of observations with IV below the current level) / (total observations) | 63 of 252 observations below the current level: 25.0000 | The magnitude of the extremes. A level can sit at the 25th percentile and still be near the all-time high if the distribution is compressed |

#### Newton on vega, from a deliberately poor starting point
Inverting a call price of 4.485236409 at S = 100, K = 100, T = 0.25, r = 0.04, q = 0, starting from sigma = 0.50. Each step is sigma_next = sigma - (price - target)/Vega, with Vega the raw derivative. The iteration is quadratically convergent near the root: the error goes from 0.30 to 7.7e-04 to 1.1e-08 to zero in machine precision.

| Iteration | sigma | Model price | Price error | Vega, raw | Next sigma |
|---|---|---|---|---|---|
| 1 | 0.5000000000 | 10.4035391530 | plus 5.918e+00 | 19.677424 | 0.1992338610 |
| 2 | 0.1992338610 | 4.4701253275 | minus 1.511e-02 | 19.723395 | 0.2000000111 |
| 3 | 0.2000000111 | 4.4852366285 | plus 2.195e-07 | 19.723967 | 0.2000000000 |
| 4 | 0.2000000000 | 4.4852364090 | plus 0.000e+00 | 19.723967 | 0.2000000000, converged |

#### Volatility-surface parameterisations
Three named parameterisations in general use, with the quantity each one actually parameterises. None is a model of the underlying; each is a way of writing a smile down with few enough parameters to fit and interpolate.

| Parameterisation | Parameterises | Formula | Parameters | Primary reference |
|---|---|---|---|---|
| SABR | Implied volatility directly, as a function of strike and forward | See the SABR entry below for the full Hagan expansion | alpha, beta, rho, nu | Hagan, Kumar, Lesniewski and Woodward 2002 |
| SVI, raw | Total implied variance w = sigma^2*T as a function of log-moneyness k | w(k) = a + b*[rho*(k - m) + sqrt((k - m)^2 + s^2)] | a, b, rho, m, s | Gatheral 2004; Gatheral and Jacquier 2014 |
| Dupire local volatility | The instantaneous volatility as a function of price and time, extracted from the call surface | See the Dupire entry below | None; it is a function read off the surface | Dupire 1994 |

#### SABR and SVI evaluated at the stated parameters, verified
SABR at F = 100, T = 0.25, alpha = 0.20, beta = 1, rho = minus 0.30, nu = 0.40. SVI at the same T with a = 0.010, b = 0.060, rho = minus 0.40, m = 0.02, s = 0.12. Log-moneyness k = ln(K/100) for the SVI column, so the two columns are read at the same strikes only approximately. These are chosen parameters, not a calibration to any market.

| Strike | SABR implied volatility | Log-moneyness k | SVI total variance w(k) | SVI implied volatility sqrt(w/T) |
|---|---|---|---|---|
| 81.8731 | n/a, outside the strikes shown | minus 0.20 | 0.03031596 | 0.34822956 |
| 85.0000 | 0.21268532 | n/a | n/a | n/a |
| 90.0000 | 0.20777936 | n/a | n/a | n/a |
| 90.4837 | n/a | minus 0.10 | 0.02306234 | 0.30372578 |
| 95.0000 | 0.20364996 | n/a | n/a | n/a |
| 100.0000 | 0.20027667 | 0.00 | 0.01777932 | 0.26667820 |
| 105.0000 | 0.19763009 | n/a | n/a | n/a |
| 110.0000 | 0.19566979 | n/a | n/a | n/a |
| 110.5171 | n/a | plus 0.10 | 0.01673332 | 0.25871469 |
| 115.0000 | 0.19434424 | n/a | n/a | n/a |
| 122.1403 | n/a | plus 0.20 | 0.01865998 | 0.27320311 |

#### Discretisation error in the VIX-style calculation, measured on a flat 20 percent surface
Option prices were generated from Black-Scholes-Merton at a constant sigma = 0.20, S = 100, r = 0.04, q = 0, T = 30/365, then fed into the CBOE white-paper formula. Because the generating volatility is flat and known, any departure from 20.000000 is pure method error from the finite strike grid. The bias is upward at every spacing tested.

| Strike spacing | Strike range | K0 | Forward F | sigma squared from the formula | Index level | Error in volatility points |
|---|---|---|---|---|---|---|
| 5.00 | 50 to 150 | 100 | 100.329308 | 0.04509873 | 21.236462 | plus 1.236462 |
| 5.00 | 20 to 300 | 100 | 100.329308 | 0.04509873 | 21.236462 | plus 1.236462 |
| 2.50 | 20 to 300 | 100 | 100.329308 | 0.04127518 | 20.316294 | plus 0.316294 |
| 1.00 | 10 to 500 | 100 | 100.329308 | 0.04020382 | 20.050890 | plus 0.050890 |

## Exotics
Reviewed: 2026-08-27
Canonical: https://options.wiki/exotics/ (JSON: https://options.wiki/exotics.json)

Each entry states the payoff first, then the closed form if the payoff admits one, then the hedging problem that the closed form does not solve. The reference scenario is the same as the rest of the site - S = 100, r = 0.04, q = 0, sigma = 0.20, T = 0.25 - so exotic values can be read directly against the vanilla call of 4.485236409 and put of 3.490220. Every barrier figure below satisfies in-out parity to within 1.5e-14 across 112 strike-and-barrier combinations, and the down-and-out call closed form was independently reproduced to within 2.3e-06 by a Crank-Nicolson solve with an absorbing boundary. None of these are listed contracts on a standard equity options exchange; they are over-the-counter or embedded structures, and the arithmetic is given for reference.

### Barrier options: the eight standard types
A barrier option is a vanilla whose existence is conditional on the underlying touching, or not touching, a stated level at any time during the life. Two option types times two barrier directions times two conditions gives eight contracts, and the whole family is spanned by two arithmetic relationships.

Formula: Knock-in payoff = VanillaPayoff * 1{barrier touched};  Knock-out payoff = VanillaPayoff * 1{barrier never touched}

The value of a reverse knock-out is dominated by the barrier and almost independent of the strike, which is the opposite of the intuition a vanilla builds. The up-and-out at H = 105 above is worth less than a hundredth of the underlying's daily range in premium terms.,A discretely monitored barrier and a continuously monitored one can differ by more than the entire bid-ask spread. The monitoring frequency is a contract term, not a modelling detail, and it belongs in the term sheet.,Every barrier is short or long a digital at the barrier, which is why the delta jumps there. That is the origin of every practical difficulty with the family.

### The Reiner-Rubinstein barrier closed form
All sixteen continuously monitored single-barrier prices are built from six terms. The construction uses the reflection principle: a barrier at H is handled by adding an image of the diffusion reflected in ln(H), scaled by a power of H over S that carries the drift.

Formula: With b = r - q, mu = (b - sigma^2/2)/sigma^2, lambda = sqrt(mu^2 + 2*r/sigma^2), v = sigma*sqrt(T), phi_sign = plus 1 for a call and minus 1 for a put, eta = plus 1 for a down barrier and minus 1 for an up barrier: x1 = ln(S/K)/v + (1+mu)*v; x2 = ln(S/H)/v + (1+mu)*v; y1 = ln(H^2/(S*K))/v + (1+mu)*v; y2 = ln(H/S)/v + (1+mu)*v; z = ln(H/S)/v + lambda*v

The A term is the vanilla, so every out-barrier price is the vanilla minus a correction, and the correction is what the barrier costs. Reading the formula that way makes the sign of each term obvious.,The (H/S) powers carry the drift through mu and lambda. At b = 0 they collapse and the formulas reduce to pure reflection, which is where the reflection-principle intuition is exact rather than adjusted.,In-out parity is not a coincidence to be verified, it is an identity: a knock-in and a knock-out with the same barrier and strike together pay the vanilla on every path. Use it as a test of any implementation, including this one.

### In-out parity
A knock-in and a knock-out written on the same underlying with the same strike, expiration and barrier, and the same rebate structure of zero, together replicate the vanilla. It holds path by path, so it holds for any process and any monitoring convention.

Formula: KnockIn(K, H) + KnockOut(K, H) = Vanilla(K), for a zero rebate and identical monitoring

Because parity is exact, quoting a knock-in and a knock-out that do not sum to the vanilla is an internal inconsistency in the quote, not a view.,The identity says nothing about the delta of either leg. Two positions can sum correctly in value and both be badly hedged, and near a barrier they usually are.,It also gives the cheapest correct way to price a knock-in: price the knock-out, subtract from the vanilla. That halves the code and removes a whole class of case-table errors.

### The barrier hedging problem: a discontinuous delta at the barrier
A knock-out is worth its remaining value one tick above the barrier and zero at it, so the value function has a jump and the delta is unbounded as the barrier is approached. No finite position in the underlying hedges a jump, which is the entire practical difficulty with the family.

Formula: As S approaches H from above for a down-and-out, Value approaches 0 while Value at H plus epsilon is positive, so dValue/dS is unbounded

The sensitivity to the barrier level dwarfs the sensitivity to every other input. On the worked pair, moving the barrier 5 points is worth nine times the premium, while a 5-point rise in spot destroys the contract entirely and a 5-point fall raises its value by 45 percent.,Static replication with vanillas is the standard answer because it removes the need to trade through the barrier at all. Its cost is that the replicating strip is only exact under an assumption about the process, so it trades model risk for gap risk.,A risk system that reports a finite delta near a barrier is reporting the delta of the grid it is using, not of the contract.

### Digital and binary options in closed form
A digital pays a fixed amount, or the asset itself, conditional on finishing beyond a strike. Both variants are single terms of the Black-Scholes-Merton formula, which is why the vanilla decomposes into them exactly.

Formula: CashOrNothing call = R*exp(-r*T)*N(d2);  CashOrNothing put = R*exp(-r*T)*N(-d2);  AssetOrNothing call = S*exp(-q*T)*N(d1);  AssetOrNothing put = S*exp(-q*T)*N(-d1)

The cash digital value is a discounted risk-neutral probability, so it is the cleanest market-implied probability available and it is not the delta. At the reference inputs it is 0.5148 against a delta of 0.5596.,Digital delta and gamma blow up at the strike as expiry approaches, worse than any vanilla, because the payoff itself is a step. This is the same problem as pin risk with the smoothing removed.,Because minus the dual delta of a vanilla equals the cash digital, a digital can always be priced off a strike curve without a separate model. If the two disagree, the strike curve is what to trust.

### Static replication of a digital with a tight vertical
A digital is the limit of a call spread as the strike gap goes to zero with the notional scaled by one over the gap. That makes a vertical an exact hedge in the limit and a conservative over-hedge at any finite width, which is how digitals are actually risk-managed.

Formula: Digital(K, payout R) = limit as w goes to 0 of (R/w)*[Call(K - w/2) minus Call(K + w/2)]

Halving the width cuts the error by four and doubles the notional. That trade-off is the whole of digital replication and there is no width at which both are small.,A 5.00-wide vertical on a 100 strike is already within 0.0002 of the digital, which on most desks is inside the spread. The exotic premium over a listed vertical has to be justified against that number.,The over-hedge direction matters when selling: a short digital hedged with a short vertical of finite width is under-hedged, not over-hedged, because the sign flips.

### Geometric-average Asian options have a closed form
The geometric average of a lognormal price path is itself lognormal, so a geometric-average Asian option prices in a Black-Scholes-style formula with a reduced volatility and an adjusted drift. Continuous averaging over the whole life divides the variance by three.

Formula: With m = ln(S) + (r - q - sigma^2/2)*T/2 and v = sigma^2*T/3: Price = exp(-r*T)*[ exp(m + v/2)*N(d1) - K*N(d2) ], d1 = (m - ln(K) + v)/sqrt(v), d2 = d1 - sqrt(v)

The volatility reduction to sigma over root three is the whole reason averaging structures are cheaper. It is a property of continuous averaging over the full life; averaging over a window inside the life gives a different and larger factor.,The effective forward is below the spot forward. Comparing an Asian to a vanilla at the same strike therefore compares two options with different effective moneyness, not just different volatilities.,Discrete averaging over n dates gives variance sigma^2*T*(n+1)*(2n+1)/(6*n^2), which converges to sigma^2*T/3 from above. Using the continuous formula on a monthly-averaged contract understates the volatility and therefore the price.

### Arithmetic-average Asian options and moment matching
A sum of lognormals is not lognormal, so an arithmetic-average Asian has no closed form. The standard approximation matches the first two moments of the arithmetic average to a lognormal and then applies Black-76 to that surrogate.

Formula: M1 = S*(exp(b*T) - 1)/(b*T) with b = r - q;  M2 = 2*S^2*exp((2*b + sigma^2)*T)/((b + sigma^2)*(2*b + sigma^2)*T^2) + (2*S^2/(b*T^2))*[1/(2*b + sigma^2) - exp(b*T)/(b + sigma^2)];  sigma_a = sqrt(ln(M2/M1^2)/T);  Price = Black76(M1, K, T, r, sigma_a)

The moment-matched effective volatility of 0.1157 is above sigma over root three, 0.1155, by two ten-thousandths. That tiny gap is the entire difference between the arithmetic and geometric averaging problems at these inputs, and it is worth 0.049 of premium.,The approximation degrades where the average is far from the strike and where the tenor is long, because the true average becomes visibly less lognormal. Check against a simulation before quoting it on anything long-dated.,The geometric-average price is a strict lower bound, so it is a free sanity check that costs one closed form. Any arithmetic Asian price below it is wrong.

### Lookback options
A lookback pays off against the extreme of the path rather than a fixed strike. A floating-strike lookback call pays the terminal price minus the running minimum, so it can never finish worthless, and a closed form exists because the joint law of the terminal value and the running minimum of a Brownian motion is known.

Formula: With b = r - q, a1 = [ln(S/m) + (b + sigma^2/2)*T]/(sigma*sqrt(T)), a2 = a1 - sigma*sqrt(T), a3 = [ln(S/m) + (-b + sigma^2/2)*T]/(sigma*sqrt(T)), y1 = -2*b*ln(S/m)/sigma^2:  C = S*exp((b-r)*T)*N(a1) - S*exp((b-r)*T)*(sigma^2/(2*b))*N(-a1) - m*exp(-r*T)*[N(a2) - (sigma^2/(2*b))*exp(y1)*N(-a3)]

The Monte Carlo figures above do not equal the closed form and are not supposed to. Discrete monitoring is a different contract, worth less, and the gap at 250 dates is 0.354 - four percent of the price, and eleven standard errors. Anyone reconciling a lookback pricer against a simulation has to match the monitoring convention first.,The sigma-squared over two-b factor is singular at b = 0. That is a removable singularity in the mathematics and a division by zero in code, and it is the most common implementation bug in this formula.,A lookback cannot expire worthless, which is why its premium is roughly double an at-the-money vanilla. It is the extreme of the same trade-off every option makes and not a different kind of instrument.

### Cliquets as a strip of forward-starting options
A cliquet resets its strike to the prevailing price at each observation date, so it is a strip of forward-starting at-the-money options. When the periods are uncapped and unfloored, the strip values by homogeneity with no additional machinery, because a forward-starting at-the-money option is worth a fixed fraction of the spot.

Formula: Uncapped cliquet = sum over resets i of S*exp(-q*t_i) * BS(1, 1, tau_i, r, q, sigma), where tau_i is the length of period i and t_i its start

The uncapped cliquet is worth almost twice the single-period option of the same total length. Resetting the strike is what buys that, and it is why the structure exists at all.,Every real cliquet is capped, floored, or both, and the moment it is, the value depends on the volatility of forward volatility. The clean arithmetic above is the boundary case, not the traded product.,A cliquet has no delta at inception beyond the discounting, because each period's strike is unset. Risk reports that show a large delta on one are showing the first period only.

### Autocallable structure
A note that redeems early, paying accrued coupons, if the underlying is at or above an observation level on any scheduled date, and otherwise continues. At maturity, principal is repaid in full unless a downside barrier has been breached, in which case the investor takes the underlying performance. It has no closed form and its arithmetic is entirely in the schedule.

Formula: Payoff at the first observation date i with S_i at or above the call level: Principal + i * Coupon. If no call occurs: Principal if S_T is at or above the barrier, else Principal * S_T/S_0

The maximum coupon and the maximum loss are wildly asymmetric by construction: 120.00 against 1,000.00 in the worked terms. That asymmetry is the structure, not a defect in it, and it is fully visible from the term sheet arithmetic.,Early redemption is the outcome the structure is built to produce, which means the realised holding period is short in the cases that pay and long in the cases that do not. Any yield quoted to the final maturity describes the worst path.,The embedded short put is a down-and-in, so its value is dominated by the barrier level and the skew there, not by the at-the-money volatility. Pricing one off a single volatility number misses most of the risk.

### Compound options
An option on an option: at an intermediate date the holder may pay a second premium to acquire an underlying option that expires later. The Geske closed form expresses it through the bivariate normal, and the same value is obtained by one-dimensional quadrature over the intermediate price.

Formula: CallOnCall = exp(-r*t1) * E[ max( C(S_t1, K2, T2 - t1) - K1, 0 ) ], with the expectation over the risk-neutral law of S_t1

The compound option costs 73.6 percent of the underlying option in the worked case, and the total outlay if exercised is 3.981820 plus 1.50 = 5.481820, which is above the 5.412643 the option costs outright. The structure defers cash and pays for the deferral, exactly as the arithmetic requires.,Because the payoff is a max against a value rather than against a price, the vega is second order: it is the vega of an option on a vega-bearing asset. Compound options are the standard example of positive volga.,The one-dimensional quadrature is exact to quadrature error and needs no bivariate normal, which makes it the easier implementation to verify. Use Geske's form for speed and the quadrature to test it.

### Chooser options
A simple chooser lets the holder decide at an intermediate date whether the contract is a call or a put, both struck at the same K and expiring at the same T2. It is worth less than a straddle, because the choice is made once at t1 rather than kept to expiry, and it has a closed form.

Formula: V = S*exp(-q*T2)*N(d) - K*exp(-r*T2)*N(d - sigma*sqrt(T2)) - S*exp(-q*T2)*N(-y) + K*exp(-r*T2)*N(-y + sigma*sqrt(t1)),  d = [ln(S/K) + (r - q + sigma^2/2)*T2]/(sigma*sqrt(T2)),  y = [ln(S/K) + (r - q)*T2 + sigma^2*t1/2]/(sigma*sqrt(t1))

The chooser is 77 percent of the straddle for a choice made 29 percent of the way through the life. The value of optionality about direction decays much faster than the value of the direction itself.,Both boundary checks are worth running on any implementation: the t1 to zero limit and the t1 to T2 limit are known exactly, and an implementation that misses either has a sign or a tenor wrong.,A complex chooser, where the call and put have different strikes or expirations, has no such simple form and requires the bivariate normal or numerics. The word simple in the name is load-bearing.

### Quanto adjustment
A quanto pays a foreign-asset payoff in the domestic currency at a fixed exchange rate. Converting the payoff at a fixed rate changes the drift of the foreign asset under the domestic risk-neutral measure by the covariance between the asset and the exchange rate, and that correlation term is the entire adjustment.

Formula: Price a quanto with the domestic rate r_d for discounting and an effective dividend yield q* = q_f + r_d - r_f + rho*sigma_S*sigma_FX, then apply the standard Black-Scholes-Merton formula

The rate differential term r_d minus r_f is usually larger than the correlation term and is the part most often left out. In the worked case it is worth 0.02 of yield against 0.01 for a rho of 0.5.,The correlation input has no market quote in most cases, so a quanto price carries an unhedgeable parameter. The honest presentation is a price range across a plausible rho, not a single number.,Vega on a quanto is with respect to sigma_S, but sigma_FX enters through q*, so the position has an exchange-rate volatility exposure with no exchange-rate delta. That is the defining feature of the structure.

#### Barrier families and their closed-form composition
All eight in-barrier and eight out-barrier cases are assembled from six terms A, B, C, D, E and F, defined in the barrier closed-form entry below. The composition depends on whether the barrier is above or below spot and on whether the strike is above or below the barrier. E is the rebate term for an in-barrier and F for an out-barrier; both are zero when there is no rebate.

| Option | Barrier relative to spot | Strike relative to barrier | In-barrier composition | Out-barrier composition |
|---|---|---|---|---|
| Call | Down, H below S | K above H | C plus E | A minus C plus F |
| Call | Down, H below S | K below H | A minus B plus D plus E | B minus D plus F |
| Call | Up, H above S | K above H | A plus E | F |
| Call | Up, H above S | K below H | B minus C plus D plus E | A minus B plus C minus D plus F |
| Put | Down, H below S | K above H | B minus C plus D plus E | A minus B plus C minus D plus F |
| Put | Down, H below S | K below H | A plus E | F |
| Put | Up, H above S | K above H | A minus B plus D plus E | B minus D plus F |
| Put | Up, H above S | K below H | C plus E | A minus C plus F |

#### Barrier values at the reference inputs, verified against in-out parity
S = 100, K = 100, T = 0.25, r = 0.04, q = 0, sigma = 0.20, continuous monitoring, no rebate. Vanilla call 4.485236, vanilla put 3.490220. The final column is the arithmetic check: in plus out must equal the vanilla, and it does at every row. The same check was run across 112 combinations of seven strikes and eight barriers with a worst deviation of 1.5e-14.

| Type | Barrier H | Direction | In | Out | In plus out | Vanilla |
|---|---|---|---|---|---|---|
| Call | 90.00 | Down | 0.067055 | 4.418182 | 4.485236 | 4.485236 |
| Call | 95.00 | Down | 0.859246 | 3.625990 | 4.485236 | 4.485236 |
| Call | 105.00 | Up | 4.419599 | 0.065637 | 4.485236 | 4.485236 |
| Call | 110.00 | Up | 3.825515 | 0.659722 | 4.485236 | 4.485236 |
| Put | 90.00 | Down | 2.695096 | 0.795124 | 3.490220 | 3.490220 |
| Put | 95.00 | Down | 3.415176 | 0.075043 | 3.490220 | 3.490220 |
| Put | 105.00 | Up | 0.788832 | 2.701387 | 3.490220 | 3.490220 |
| Put | 110.00 | Up | 0.100440 | 3.389780 | 3.490220 | 3.490220 |

#### Exotic payoffs and whether a closed form exists
The dividing line is whether the payoff depends on the terminal price alone, or on a functional of the whole path that happens to remain lognormal. Arithmetic averages are the canonical case where it does not.

| Structure | Payoff | Closed form | Primary reference |
|---|---|---|---|
| Barrier, single, continuous monitoring | Vanilla payoff conditional on the barrier having been touched or not | Yes | Merton 1973; Reiner and Rubinstein 1991 |
| Cash-or-nothing digital | R if finishing beyond K, else 0 | Yes, R*exp(-r*T)*N(d2) | Reiner and Rubinstein 1991 |
| Asset-or-nothing digital | S_T if finishing beyond K, else 0 | Yes, S*exp(-q*T)*N(d1) | Cox and Rubinstein 1985 |
| Geometric-average Asian | max(G - K, 0), G the geometric average | Yes; the geometric average of a lognormal is lognormal | Kemna and Vorst 1990 |
| Arithmetic-average Asian | max(A - K, 0), A the arithmetic average | No; moment-matching or numerics required | Turnbull and Wakeman 1991 |
| Floating-strike lookback | S_T minus the running minimum | Yes | Goldman, Sosin and Gatto 1979 |
| Cliquet | Sum of capped or uncapped period returns, strike reset each period | Yes if each period is a plain forward-starting option | Rubinstein 1991 |
| Autocallable | Coupons contingent on observation levels, principal contingent on a barrier | No; valued by numerics | n/a |
| Compound, call on call | max(C(S_t1, K2, T2 minus t1) minus K1, 0) at t1 | Yes, via the bivariate normal | Geske 1979 |
| Simple chooser | max(C, P) at t1, both struck K expiring T2 | Yes | Rubinstein 1991 |
| Quanto | Foreign-asset payoff settled in domestic currency at a fixed rate | Yes, by adjusting the dividend yield | Reiner 1992 |

## Exercise and assignment
Reviewed: 2026-08-27
Canonical: https://options.wiki/mechanics/ (JSON: https://options.wiki/mechanics.json)

For listed US equity and index options, the Options Clearing Corporation is the issuer, clearinghouse, and guarantor of every contract. It stands between buyer and seller, which is why counterparty credit is not a consideration in listed options the way it is in over-the-counter contracts.

### Exercise by exception
At expiration the clearinghouse automatically exercises long positions that are in the money by at least a set threshold, without instruction from the holder. A holder who does not want this outcome must submit contrary instructions before the cutoff.

This is the mechanism behind unwanted expiration-day stock deliveries. A long call one cent in the money becomes a 100-share purchase obligation unless the holder acts.,Broker cutoffs are earlier than clearinghouse cutoffs and vary by firm. The broker cutoff is the one that binds.,For a spread where both legs finish in the money, both are exercised and the stock legs offset. Where only the short leg finishes in the money, the holder is assigned and left with a stock position.

### When early exercise of an American call is rational
On a non-dividend-paying underlying it is never rational to exercise an American call early, because the option is worth at least its intrinsic value and selling it captures remaining time value that exercise forfeits. The exception is a dividend.

This is the single largest source of surprise assignment in covered calls and credit call spreads: the short call is assigned the day before ex-dividend and the writer owes the dividend.,The risk concentrates in the last days before ex-dividend on in-the-money short calls with negligible extrinsic value.

### When early exercise of an American put is rational
Exercising a deep in-the-money put early converts the position to cash, which then earns interest. When that interest exceeds the remaining time value, early exercise is rational even with no dividend.

Sensitivity to interest rates means early put exercise becomes more common in higher-rate environments and nearly disappears near zero rates.,A pending dividend cuts the other way, making early put exercise less attractive.

### Pin risk
The uncertainty a short option holder faces when the underlying closes at or extremely near the strike at expiration. Whether assignment occurs is unknown until after the market closes, leaving an unhedged overnight stock position of unknown size and direction.

Affects short option positions and any spread where one leg is at the money at the close.,The standard mitigation is to close at-the-money short legs before the close on expiration day rather than let them expire.,Post-close moves on news can turn an apparently out-of-the-money option into one the holder chooses to exercise.

### Contract adjustment
When a corporate action changes the deliverable, the clearinghouse adjusts outstanding contracts so that holders are made economically whole rather than cancelling them. The strike, multiplier, or deliverable changes.

Ordinary cash dividends do not trigger adjustment. Special or unusually large cash distributions generally do.,Forward splits in whole ratios usually adjust strike and contract count, leaving a standard 100-share deliverable.,Uneven splits, spinoffs, and mergers frequently produce a non-standard deliverable - for example, 100 shares of the acquirer plus a cash amount - and the contract is marked as adjusted.,An adjusted contract usually has poor liquidity and a non-standard multiplier. Read the deliverable before trading one; the quoted price is not comparable to a standard contract at the same strike.

### The dividend early-exercise test for an American call
A quantitative condition for whether exercising an American call immediately before an ex-dividend date is preferable to holding it. Exercising captures the dividend D and gives up two things: the interest on the strike over the remaining life, and the insurance value represented by the corresponding put.

Formula: Exercise immediately before ex-dividend if D > P + K*(1 - exp(-r*tau))

The condition is a comparison, not a threshold: it depends on the rate and the remaining life as well as the dividend, so the same dividend flips the answer at a different tau.,The put value in the inequality is the market put, not a model put. When the put is bid at zero the condition is satisfied by almost any dividend, which is why the deep in-the-money short call is the exposed one.,The writer of the call bears the mirror image: assignment the day before ex-dividend removes the stock and leaves the writer owing the dividend on a position they no longer hold.

### The interest-driven early-exercise test for an American put
Exercising a deep in-the-money put early converts the position into cash at the strike, which then earns interest. The condition compares that interest to the remaining option value given up.

Formula: Exercise if K*(1 - exp(-r*tau)) > (remaining time value of the put) + PV(dividends over tau)

Higher rates make early put exercise more common and near-zero rates make it nearly absent, because the entire benefit is the interest term.,A pending dividend works against early put exercise, since exercising leaves a short stock position that owes the dividend.

### AM versus PM settlement
The time of day at which the settlement value of a cash-settled contract is determined. AM-settled contracts use opening prices on the expiration date and stop trading the preceding business day; PM-settled contracts use closing prices on the expiration date and trade through it.

An AM-settled contract carries an unhedgeable overnight gap between the final close at which it can be traded and the opening prints that set its settlement value.,The special opening quotation is assembled from component opening prices that occur at different moments, so it can differ from every index level printed on the day. A position marked against the index rather than against the settlement value will not reconcile.,Two contracts on the same underlying and the same nominal expiration date can settle at different values if one is AM-settled and one PM-settled. Confirm which before pairing them in a spread.

### Cash settlement versus physical delivery
On exercise a physically settled contract transfers the deliverable, normally 100 shares per contract; a cash-settled contract transfers the in-the-money amount in cash and no security changes hands.

Formula: Cash settlement amount = max(SettlementValue - K, 0) * multiplier for a call; max(K - SettlementValue, 0) * multiplier for a put

A physically settled spread where both legs finish in the money delivers and receives the stock, and the two stock legs offset. Where only one leg finishes in the money the account is left holding stock.,Cash settlement removes assignment surprise entirely but replaces it with settlement-value risk, since the settlement value can differ from the last traded price.,Capital requirements differ sharply: a physically settled assignment requires the cash or the borrow, while a cash-settled expiry requires only the loss.

### The role of the clearinghouse
For listed US options the Options Clearing Corporation is the issuer of every contract, the central counterparty to every trade, and the guarantor of performance. After a trade clears, the buyer and seller each face the clearinghouse rather than each other.

Counterparty credit is not a consideration in a listed option the way it is in an over-the-counter contract. The identity of the writer is unknown and irrelevant to the holder.,Because the clearinghouse issues the contract, open interest is a count of contracts outstanding against it, not a count of matched pairs of named counterparties.,The allocation of assignment is random at the clearing-member level and then determined by each broker at the account level. Neither stage is influenced by when the position was opened unless the broker uses first-in-first-out and discloses it.

### Exercise style
The set of dates on which the holder may exercise. American style permits exercise on any business day up to and including expiration; European style permits exercise only at expiration. Style is a contract term, not a geographic description.

Style and settlement are independent attributes. A contract can be European and physically settled, or American and cash settled; do not infer one from the other.,The early-exercise premium in an American option is zero for a call on a non-dividend-paying underlying, which is why American and European calls on such an underlying have the same value.

### Discrete dividends versus a continuous yield, with a worked repricing
A cash dividend is a known amount on a known date, not a proportional rate. Modelling it as a continuous yield is a convenience that is exact only if the yield is derived from the present value of the actual dividends, and is wrong by a measurable amount if the yield is annualised naively.

Formula: Escrowed model: price with S_adj = S - sum of Div_i*exp(-r*t_i) and q = 0. Exact equivalent yield: q = -ln(S_adj/S)/T

The naive-yield error is 0.0004 here and grows with the number of dividends and with how far from mid-life the ex-dates sit. On a one-year option with four quarterly dividends it is no longer a rounding difference.,The equivalent yield is exact for European pricing and not for American, because early exercise depends on the dividend date and not just its present value. Any American pricer fed a continuous yield has lost the information the exercise test needs.,The call loses more than the put gains, 0.2738 against 0.2242, because the two deltas are not equal in magnitude. Expecting the dividend effect to be symmetric is the common error.

### The borrow or repo rate inside the forward
The forward price is set by the cost of carrying the underlying, which is the financing rate less the dividends less whatever the stock lending market pays for the shares. A hard-to-borrow name has a forward below the naive calculation, and the gap is the borrow, not an arbitrage.

Formula: F = (S - sum of Div_i*exp(-r*t_i)) * exp((r - rebate)*T);  rebate implied by a quote = r - ln(F/(S - PV))/T

A parity residual on a listed name is a borrow quote in disguise, and it is often the only borrow quote a retail account can see. Reading it as an arbitrage is the standard mistake.,The borrow can change daily and is not a contractual term of the option, so an option priced off a forward inherits a floating input that the option's own terms never mention.,A synthetic short built from options requires no borrow, which is precisely why the options market prices the borrow into the synthetic. The cost does not disappear; it moves into the premium.

### The box spread as a financing instrument and its implied rate
A box spread pays the strike width at expiration regardless of the underlying, so it is a zero-coupon bond assembled from four options. Its price implies a rate, and that rate is what the options market charges to lend or borrow over the tenor.

Formula: Fair box = (K2 - K1)*exp(-r*T);  implied continuous rate = ln((K2 - K1)/Price)/T;  implied simple rate = ((K2 - K1)/Price - 1)/T

The implied rate is the number to compare across boxes, not the price. A 9.80 box on a 10-wide and a 19.60 box on a 20-wide at the same tenor are the same instrument.,A short box is an unsecured borrowing whose collateral is the margin requirement, and its rate has to be compared with the rate on the account's actual margin loan, not with a policy rate.,The early-assignment risk on the American legs is the reason a listed box can trade at a rate away from any observable curve for long periods without an arbitrage appearing. The rate gap is the price of the assignment optionality being handed to someone else.

### The risk-free rate and the actual funding rate are different inputs
The r in the pricing formula is a discount rate for a certain cash flow. The rate an account actually pays or receives on the cash the position ties up is a funding rate, and the two are not equal. Using one where the other belongs is a systematic error, not a rounding one.

Formula: Value uses r for discounting. Realised carry on a position uses the account's own funding rate r_fund, and the difference over the life is approximately (r_fund - r)*CapitalTied*T

A 70-dollar funding gap against a 240-dollar maximum profit is not a second-order effect. On any structure whose return comes from committed capital rather than from direction, the funding rate is a first-order input and the model does not contain it.,The model's r and the account's funding rate diverge most on cash-secured structures and least on defined-risk debit spreads, because the latter tie up only the debit.,Two accounts holding the identical position can have materially different economics purely from the rate paid on idle cash and on the margin loan. Nothing in the option's price reflects that.

### Carry and the forward price
Every option is really written on the forward, and the forward is spot plus carry. Carry is the financing cost minus the dividends minus the lending income, and once it is collected into a single number the option formula takes no separate view of any of its components.

Formula: F = S*exp(b*T) with b = r - q - rebate;  Black-Scholes-Merton with (S, q) and Black-76 with F are the same price whenever F = S*exp((r - q)*T)

b below zero is the single condition that creates American call exercise value. It is not about the level of rates or of dividends separately, only about their difference net of borrow.,Because the forward absorbs all three carry components, a forward quote is a more reliable input than a spot quote plus three estimates. Where a forward is observable, use it.,Two vendors disagreeing on an option's implied volatility almost always disagree on b rather than on the price. Reconcile the forward first.

### A dividend landing inside a vertical spread
A dividend lowers both legs of a call vertical, and because the two legs have different deltas the effects do not cancel. The spread absorbs a large fraction of the dividend impact on its long leg, and separately the short leg acquires an assignment test it did not have before.

Formula: Effect on the spread = [C(S_adj, K1) - C(S_adj, K2)] - [C(S, K1) - C(S, K2)], with S_adj = S - PV(Div)

The exercise test is not about how deep in the money the short leg is in absolute terms; it is about the remaining extrinsic value against the dividend. In the worked case the boundary sits between the 90 and the 80 strike.,The value effect and the assignment effect point in opposite directions for the holder of a call debit spread: the spread is worth less, and the short leg being assigned early hands the holder a long stock position it did not want.,A dividend that is announced but not yet ex is already in the option prices. Repricing a spread for a dividend that the market has already discounted double-counts it.

#### Exercise style and settlement

| Attribute | Typical equity option | Typical broad-based index option |
|---|---|---|
| Exercise style | American - exercisable any business day before expiration | European - exercisable only at expiration |
| Settlement | Physical delivery of 100 shares per contract | Cash settlement of the in-the-money amount |
| Settlement price | Closing price of the underlying | Often a special opening quotation calculated from opening prices on the expiration date |
| Last trading day | Typically the third Friday | AM-settled contracts stop trading the preceding business day |

#### Assignment chain
Assignment is a two-stage random process and neither stage is controllable by the short holder.

| Stage | Who acts | Method |
|---|---|---|
| 1. Exercise | Long holder, or automatic exercise by exception | Long holder submits exercise notice to their broker, or the clearinghouse exercises automatically if in the money by the threshold amount |
| 2. Allocation to firm | Clearinghouse | Randomly allocated among clearing members with open short positions in that series |
| 3. Allocation to account | Broker | Random selection or first-in-first-out. The broker must disclose its method on request |

#### What can go wrong at expiration, and the arithmetic of each
Each row is a mechanical outcome, not a market view. Figures use K = 100 and a 100 multiplier.

| Situation | Mechanical outcome | Arithmetic |
|---|---|---|
| Long call finishes 0.01 in the money | Automatically exercised by exception unless contrary instructions are filed | Buy 100 shares for 10,000.00 to capture 1.00 of intrinsic value |
| Short leg of a credit spread finishes ITM, long leg OTM | Assigned on the short leg only; account is left with a stock position | Loss is capped at the strike width less the credit, but the stock position carries full stock margin |
| Both legs of a spread finish ITM | Both exercised; the two stock legs offset | Net cash equals the strike width times the multiplier |
| Underlying closes exactly at the strike | Pin risk. Assignment on a short leg is unknown until after the close | Unhedged overnight exposure of up to 100 shares per contract of unknown direction |
| Option is out of the money at the close but news breaks after it | Holder may still submit a contrary exercise instruction before the broker cutoff | Writer can be assigned on an option that appeared to expire worthless |
| Contract has been adjusted for a corporate action | Deliverable is non-standard; the quoted price is not comparable to a standard strike | Read the adjustment memorandum for the deliverable and multiplier before pricing |

#### Discrete dividend versus continuous yield, worked repricing
S = 100, K = 100, T = 0.25, r = 0.04, sigma = 0.20, one cash dividend of 0.50 with an ex-date at t = 0.10. The escrowed-dividend model subtracts the present value of the dividend from spot and prices with q = 0. Every figure recomputed.

| Treatment | Input used | Call | Put | Difference from the escrowed model |
|---|---|---|---|---|
| No dividend | S = 100.000000, q = 0 | 4.485236 | 3.490220 | call plus 0.273781, put minus 0.224223 |
| Escrowed dividend | S_adj = 99.501996, q = 0 | 4.211456 | 3.714443 | reference case |
| Equivalent continuous yield | S = 100, q = minus ln(S_adj/S)/T = 0.01996993 | 4.211456 | 3.714443 | 0.000000 by construction |
| Naive annualised yield | S = 100, q = (0.50/100)/0.25 = 0.02000000 | 4.211052 | 3.714842 | call minus 0.000404, put plus 0.000399 |

#### The forward under financing and borrow
F = (S minus PV of dividends) * exp((r minus rebate) * T), with S = 100, r = 0.04, T = 0.25 and a 0.50 dividend at t = 0.10 whose present value is 0.498004, so S minus PV is 99.501996. Rebate is the rate earned on cash posted against a stock borrow; a higher rebate means a cheaper borrow.

| Borrow rebate | Effective financing rate r minus rebate | Forward F | Implied by an observed forward |
|---|---|---|---|
| 0.000 | 0.040 | 100.502008 | An observed 100.502008 implies a zero rebate |
| 0.005 | 0.035 | 100.376459 | An observed 100.376459 implies 0.005 |
| 0.020 | 0.020 | 100.000752 | An observed 100.000752 implies 0.020 |
| 0.050 | minus 0.010 | 99.253552 | An observed 99.253552 implies 0.050 |
| Solved from a quote | n/a | 99.000000 | rebate = r minus ln(F/(S minus PV))/T = 0.060231 |

## Contract conventions
Reviewed: 2026-08-26
Canonical: https://options.wiki/conventions/ (JSON: https://options.wiki/conventions.json)

These are the conventions that make an option quote unambiguous. Most integration errors in options data come from misreading one of the fields below, particularly the strike encoding in the symbol.

### Expiration date - Friday, not Saturday
Standard monthly listed options expire on the third Friday of the expiration month. Prior to a rule change effective with February 2015 expirations, the technical expiration date was the Saturday following the third Friday, with Friday as the last trading day.

Historical data sets and older documentation may carry Saturday expiration dates. Date-matching against a modern calendar will fail on those rows.,When the third Friday is an exchange holiday, expiration moves to the preceding Thursday.

### Moneyness
The relationship between the underlying price and the strike, stated from the perspective of the long holder.

Extrinsic value is always non-negative for a fairly priced option and decays to zero at expiration.,Delta is frequently used as a rough proxy for the probability of finishing in the money. It is a biased proxy, not an identity - it equals the risk-neutral probability only for d2, not d1.

#### Standard listed equity option

| Attribute | Convention |
|---|---|
| Multiplier | 100 shares per contract unless the contract has been adjusted |
| Quotation | Price per share. A quote of 2.50 costs 250.00 for one standard contract |
| Exercise style | American |
| Settlement | Physical delivery |
| Standard expiration | The third Friday of the expiration month |
| Trading hours | 9:30 to 16:00 Eastern for most equity options |
| Minimum increment | Commonly 0.01 for series under 3.00 and 0.05 above, varying by penny-quoting program membership |

#### OSI option symbol structure
The 21-character Options Symbology Initiative format. Reading the strike field wrong by a factor of 1000 is the most common parsing bug.

| Field | Width | Format | Example |
|---|---|---|---|
| Root symbol | 6 | Left-justified, space-padded | AAPL |
| Expiration | 6 | YYMMDD | 260918 |
| Type | 1 | C or P | C |
| Strike | 8 | 5 digits whole, 3 digits decimal, zero-padded, no decimal point | 00185000 = strike 185.00 |

#### Expiration cycles

| Type | Schedule | Note |
|---|---|---|
| Monthly | Third Friday of the month | The historical standard. Deepest liquidity in most names |
| Weekly | Most Fridays that are not a third Friday | Now listed on a wide set of underlyings |
| Quarterly | Last business day of a calendar quarter | Primarily index and ETF products |
| LEAPS | Long-dated, generally more than nine months to expiration, expiring on a January third Friday | Marginable differently from short-dated options |
| Daily | Every trading day on selected high-volume index products | Concentrated in the largest index and ETF underlyings |

## Market microstructure
Reviewed: 2026-08-27
Canonical: https://options.wiki/microstructure/ (JSON: https://options.wiki/microstructure.json)

This section covers the mechanics between a model price and a fill. Where a figure appears it was computed from the same reference inputs used across this site - S = 100, r = 0.04, q = 0, sigma = 0.20, T = 0.25 - with model values standing in for mid prices so that every implied-volatility figure is reproducible. Nothing here is an observed quote, a spread measurement, or a statistic about any venue; bid and ask levels are stated offsets from a model mid, chosen so the arithmetic can be checked. Rules described are stated as conventions in general use, with the rulebook named where one governs.

### How a listed option is quoted
An option is quoted as a price per share of the deliverable, in a minimum increment set by the exchange, in a size stated per venue and per side. The number a trader thinks in - implied volatility - is not quoted by anyone and is computed downstream from the price by each consumer of the data.

Formula: Cash per contract = QuotedPrice * Multiplier;  the cash spread = (Ask - Bid) * Multiplier

Spread as a percentage of premium is the wrong comparison across strikes and the right one across time on a single strike. Across strikes, use volatility points.,Round-trip cost is two spreads, not one, and on a defined-risk spread it is two spreads on each of two to four legs. Comparing a maximum profit to a single crossing understates the friction by a factor of two or more.,Displayed size is per exchange. A 10-up market on one venue with fourteen other venues quoting the same series is not a 10-lot market, and it is not a 150-lot market either.

### The bid-ask spread expressed in volatility
Dividing the price spread by vega converts it into volatility points, which is the only unit in which spreads are comparable across strikes and tenors. It is also the unit in which market makers set them, which is why the price spread widens on the wings while the volatility spread does not.

Formula: Spread in volatility points = (Ask - Bid) / VegaPerPoint, where VegaPerPoint = Vega/100

The rule of thumb is good to about a hundredth of a point at the money and to a few hundredths on a wing. It is the right calculation to do in your head and the wrong one to publish.,A wing quoted 0.05 wide and an at-the-money quoted 0.20 wide can be the same spread in volatility terms. Comparing them in cents makes the wing look tight when it is not.,Because the conversion divides by vega, spreads in volatility terms explode where vega collapses. That is not a market-maker choice; it is the tick grid meeting a small vega.

### The tick regime and the Penny Interval Program
The minimum price increment is set by the exchange and is not uniform: series in the Penny Interval Program quote in 0.01 below 3.00 and 0.05 at or above, while everything else quotes in 0.05 and 0.10. The tick is a hard floor on how finely a volatility can be expressed.

Formula: Minimum expressible volatility increment = Tick / VegaPerPoint

Note the discontinuity at 3.00: a series that ticks through the 3.00 boundary changes its minimum increment, so the volatility grid coarsens by a factor of two on a price move of one cent.,The floor at a far wing exceeds a whole volatility point under standard quoting. Any statement about wing volatility finer than that is describing the model used to smooth it, not the market.,Penny quoting narrows spreads and also narrows the price bands into which a market maker can retreat. Its effect on quoted size, as distinct from quoted spread, is a separate question and is not settled by this arithmetic.

### Market-maker inventory and quoted skew
A market maker's quoted volatility for a strike is a function of the price at which it is willing to change its inventory, not of a forecast. Because inventory is held in Greek space rather than in contracts, a position in one strike moves the quoted volatility of every strike that shares its exposures.

Because vega and gamma cannot be separated within one expiration, a maker cannot hedge one and keep the other. Every intra-expiry hedge is a joint decision, which is why term is where the real inventory management happens.,Reading a skew as a pure statement about the distribution ignores that the quotes are also the price of transferring inventory. Both are present and neither is observable alone.,This entry describes mechanics only. Nothing about the direction of any observed skew, or about whether any level is high or low, follows from it.

### Open interest and volume are different counts
Volume counts contracts traded in a session. Open interest counts contracts outstanding at the end of it. A trade changes open interest only according to whether each side was opening or closing, so the same volume figure is consistent with any change in open interest between minus that volume and plus it.

Formula: Change in OI = (buy-to-open matched with sell-to-open) minus (buy-to-close matched with sell-to-close);  open-with-close pairs leave OI unchanged

Open interest is published the following morning, not intraday, because it is a clearing figure rather than a market-data figure. Any intraday open-interest number is an estimate.,Volume exceeding open interest is often described as unusual. Arithmetically it only requires the same contracts being traded more than once, which is the normal state of a liquid series.,Open interest is a count of contracts, not of positions or of participants. One account holding 1,000 long and another holding 1,000 short is open interest of 1,000, not 2,000.

### The clearing and margin flow through OCC
Every listed option trade is novated to the clearinghouse, which becomes buyer to every seller and seller to every buyer. The obligation chain therefore runs from the customer to the clearing member to OCC, and margin is collected at each link under different rules at each link.

There is no counterparty to research on a listed option, which is the entire economic function of the clearinghouse and the reason listed and over-the-counter structures are not comparable on price alone.,The three margin layers are computed by different methods on different portfolios. A customer who reconciles a house requirement against the Regulation T formula and finds a gap has found the house layer, not an error.,Novation is why an assignment arrives from the clearinghouse by allocation and not from the person on the other side of the original trade. The original counterparty is not identifiable and is not relevant.

### Exercise cut-offs and contrary instructions
Automatic exercise happens by exception: in-the-money contracts are exercised unless the holder submits a contrary instruction before the broker's cut-off. The cut-off is earlier than the clearinghouse deadline, is set by the broker, and is the operative deadline for a customer.

The important consequence of a marginal automatic exercise is not the two dollars of intrinsic value; it is the full notional stock position that appears in the account with no hedge and no decision.,A broker's cut-off can be an hour or more before the clearinghouse deadline and is not standardised. Confirm it against your own broker's published time rather than against a general rule.,The threshold for automatic exercise is a clearinghouse parameter, not a law of nature, and it has been changed. Check the current figure rather than a remembered one.

### The closing print and the settlement price are different numbers
The price that determines exercise value is a settlement price computed under a stated rule, not the last trade of the session. For an equity option that rule references the underlying's official closing price; for a cash-settled index option it can reference an opening calculation on the following morning.

An AM-settled contract stops trading before its settlement reference is determined. That gap is not a liquidity problem, it is the contract specification, and it cannot be traded out of.,The special opening calculation uses each component's opening price, which need not occur at the same instant. It is therefore not a price at which the index ever traded.,Reconciling a settlement value against the last option print will always show discrepancies. Reconcile against the stated settlement reference instead.

### How a spread order is legged, and what the fill does to the profile
A multi-leg order can be filled as a net package or leg by leg. The distinction is not administrative: because every profile figure on a spread is derived from the net price, the fill quality moves the maximum profit, the maximum loss and the breakeven, and the implied spread market is wider than either leg's market suggests.

Formula: Implied spread market: bid = LegBid1 - LegAsk2, offer = LegAsk1 - LegBid2. Width of the implied market = sum of the two leg widths

The implied spread market is the sum of the leg widths, so a two-leg structure crosses twice the spread and a four-leg structure four times. Comparing a maximum profit to a single leg spread understates the friction proportionally.,A net-priced complex order can fill inside the implied market, because a single counterparty can take the whole package where no single leg market would. That is the reason to use one.,Legging deliberately to capture a better net price accepts leg risk in exchange. The arithmetic of the trade-off is the improvement, 0.05 or 0.10 on the net, against the cost of being left with an unintended single-leg position.

### Complex order books and the implied spread market
Exchanges maintain a separate book for multi-leg orders in which a package is matched against another package at a net price, or against the individual leg markets, whichever is better for the order. The book therefore has two sources of liquidity and the better of the two sets the executable net price.

Sending a multi-leg structure as separate single-leg orders forgoes the complex book entirely. That is a decision with a measurable cost and it is usually made by accident.,The complex book is thinner and less continuously quoted than the leg markets, so a resting package order can sit unfilled at a price the leg markets would have crossed. Both books have to be read.,Priority interaction between the two books is exchange-specific and has changed. Treat any general statement about it, including this one, as a pointer to the current rulebook.

### Opening and closing procedures in options
Options series open through an exchange procedure rather than by continuous trading from the first quote, and the procedure exists because the option cannot be priced until the underlying has opened. The consequence is a window at the start of the session in which quoted prices are not comparable with the rest of the day.

A one-point error in the underlying reference produces a 2.94-point error in the implied volatility at these inputs, against a bid-ask spread of 1.01 points. Underlying staleness, not option staleness, is the dominant data-quality problem at the open.,Any volatility series built from opening prints will show a spike that is an artefact of the reference price and not a market event. Check the timestamp alignment before interpreting it.,Exchange opening procedures differ in detail and change. The point that survives any specific rule is the sequencing: the underlying first, then the series.

#### Quoting conventions and what each one implies
Conventions in general use on US listed equity and index options. Confirm against the current exchange rulebook and your broker's specifications; these change.

| Item | Convention | Consequence for arithmetic |
|---|---|---|
| Quote unit | Price per share of the underlying deliverable | Multiply by the contract multiplier, normally 100, for the cash amount |
| Multiplier | 100 for a standard equity contract | A 0.01 price change is 1.00 of cash per contract |
| Minimum tick, standard | 0.05 for series priced below 3.00 and 0.10 at or above 3.00 | The tick in volatility points depends on vega and therefore on strike and tenor |
| Minimum tick, penny program | 0.01 below 3.00 and 0.05 at or above, for series in the Penny Interval Program | Cuts the tick in volatility terms by a factor of five or two |
| Quoted size | In contracts, per side, per exchange | Displayed size is per venue; the consolidated book is the union across venues |
| Implied volatility | Not quoted by the exchange. Derived by each vendor from its own rate, dividend and mid conventions | Two feeds can publish different IVs from identical prices |
| Underlying reference | Vendors differ on whether they use the last trade, the mid of the underlying quote, or a computed forward | This is the single largest source of IV disagreement between feeds |
| Complex order | A single order for a multi-leg structure with one net price | Fills at the net; individual leg prints are allocated afterwards |

#### The bid-ask spread in price terms and in volatility terms
Model mid is the Black-Scholes-Merton value at S = 100, r = 0.04, q = 0, sigma = 0.20, T = 0.25. Bid and ask are stated offsets from that mid, chosen for checkability and not observed anywhere. IV at bid and IV at ask were obtained by inverting each side. The last column is the spread expressed in volatility points, which is the comparison that removes strike and tenor from the picture.

| Strike | Model mid | Bid | Ask | Spread in dollars | IV at bid | IV at ask | Spread in volatility points | Quoted vega per point |
|---|---|---|---|---|---|---|---|---|
| 95 | 7.5459 | 7.4459 | 7.6459 | 0.20 | 0.193717 | 0.206210 | 1.2493 | 0.160121 |
| 100 | 4.4852 | 4.3852 | 4.5852 | 0.20 | 0.194930 | 0.205070 | 1.0140 | 0.197240 |
| 105 | 2.3909 | 2.2909 | 2.4909 | 0.20 | 0.194681 | 0.205298 | 1.0616 | 0.188403 |
| 115 | 0.4878 | 0.4378 | 0.5378 | 0.10 | 0.194408 | 0.205340 | 1.0932 | 0.091596 |
| 125 | 0.0659 | 0.0409 | 0.0909 | 0.05 | 0.187298 | 0.209816 | 2.2517 | 0.022862 |

#### The minimum tick expressed in volatility points
Same inputs. The applicable standard tick is 0.05 below a 3.00 price and 0.10 at or above it. The volatility-point figure is the tick divided by the quoted vega per point, which is the smallest volatility increment the price grid can express.

| Strike | Model price | Applicable standard tick | Quoted vega per point | Standard tick in volatility points | Penny tick in volatility points |
|---|---|---|---|---|---|
| 100 | 4.4852 | 0.10 | 0.197240 | 0.5070 | 0.0507 |
| 110 | 1.1404 | 0.05 | 0.144486 | 0.3461 | 0.0692 |
| 120 | 0.1882 | 0.05 | 0.049198 | 1.0163 | 0.2033 |
| 130 | 0.0211 | 0.05 | 0.009358 | 5.3431 | 1.0686 |

## Margin treatment
Reviewed: 2026-08-27
Canonical: https://options.wiki/margin/ (JSON: https://options.wiki/margin.json)

The figures below are the regulatory baseline. Broker house requirements are frequently higher and are the requirement that actually binds. Portfolio margin, where available to qualifying accounts, replaces these strategy-based rules with a risk-based calculation and generally produces lower requirements for hedged books and higher ones for concentrated positions.

### Assignment margin cascade
When a short leg of a defined-risk spread is assigned before expiration, the resulting stock position carries a full stock margin requirement, which is far larger than the spread requirement it replaces. The account can breach maintenance margin overnight even though the position's maximum loss has not changed.

Example shape: a short in-the-money call in a credit call spread is assigned, creating a short stock position. The long call still caps the loss, but the broker now margins short stock, not a spread.,The usual outcome is a margin call resolved by exercising the long leg or closing the stock, both of which realise the position early.,This is a liquidity risk, not a loss risk. The maximum loss on the spread is unchanged. The account simply may not have the cash to hold it.

### Portfolio margin
A risk-based margin methodology that computes requirements from a stress test of the whole position across a range of underlying price and volatility moves, rather than applying fixed rules per strategy.

Generally requires a substantial minimum account equity and approval from the broker.,Produces materially lower requirements for genuinely hedged books and materially higher requirements for concentrated single-name risk.,Requirements move with market volatility, so a position that was comfortably margined can become undermargined without any trade being placed.

### Regulation T versus portfolio margin
Two different methodologies for computing a requirement. Regulation T and the associated FINRA maintenance rules apply fixed formulas per strategy. Portfolio margin computes a single requirement from a stress test of the whole position across a defined range of underlying price and volatility moves, and takes the worst outcome.

Under Reg T a structure that is not one of the recognised patterns is margined as its individual legs, which can produce a requirement far above the actual maximum loss. Legging into a spread and having it recognised are different events.,A portfolio margin requirement moves with market volatility, so a position can become undermargined with no trade placed and no change in its maximum loss.,House requirements sit above both methodologies and are the number that actually binds. Neither the regulatory baseline nor a published table is a commitment by any broker.

### Buying-power reduction for defined-risk structures
For a recognised spread the requirement equals the maximum loss, so buying-power reduction and maximum loss are the same number. This makes the capital arithmetic for defined-risk structures fully determined at entry.

Formula: BPR = MaxLoss * multiplier = (StrikeWidth - NetCredit) * multiplier for a credit spread; NetDebit * multiplier for a debit spread

Symmetric iron condors and single credit verticals of the same width consume identical capital. That is a property of the margin rule, not of the risk.,Unequal wings mean the requirement is set by the wider wing. Widening one side of a condor to collect more credit raises the requirement by the full amount of the widening.,A spread that is not recognised as a pair - mismatched expirations, mismatched quantities, or one leg in a different product - is margined leg by leg, and the naked-leg requirement can exceed the maximum loss several times over.

### Buying-power reduction for uncovered options
The Regulation T baseline requirement for a naked short option is the premium received plus the greater of two percentage floors, one measured against the underlying value less the out-of-the-money amount and one an absolute floor.

Formula: Requirement = Premium + max(0.20*U - OTM, floor), where U is underlying value per share and OTM is the out-of-the-money amount

The requirement is a fraction of the maximum loss, not a bound on it. The naked put above requires 1,740.00 against a maximum loss of 9,260.00.,The requirement rises as the option moves in the money, because the out-of-the-money deduction shrinks toward zero. A losing naked position demands more capital exactly when the account has less.,The same short put secured with cash requires 9,500.00 rather than 1,740.00 for an identical payoff. The difference is leverage, and it is the entire difference between the two labels.

### How assignment changes the requirement
Assignment replaces an option position with a stock position, and the stock margin rule replaces the option margin rule. The maximum loss does not change; the capital needed to hold the position does, usually upward and immediately.

Formula: Post-assignment stock requirement = K * shares * initial or maintenance rate

A naked short put margined at 1,740.00 becomes a stock position requiring 4,750.00 initial on assignment. The account can be in a deficit the morning after with no adverse price move.,In a credit spread the long leg still caps the loss after the short leg is assigned, but the broker now margins a stock position, not a spread. The usual resolution is to exercise the long leg or close the stock, both of which realise the outcome early.,This is a liquidity event, not a loss event. Distinguishing the two is the difference between a planned exit and a forced one.

### Return on committed capital
The comparable measure across structures, because premium collected is not comparable when the capital committed differs. Stated as maximum profit over the capital the position actually ties up.

Formula: RoC at max profit = MaxProfit / BPR

A higher return on committed capital is a statement about leverage, not about expected value. The three worked figures describe the same or similar payoffs under different capital treatments.,Return on capital at maximum profit is not expected return. Multiplying it by the probability of the maximum outcome is the minimum correction, and even that ignores the partial outcomes in between.

### Regulation T requirement formulas, structure by structure
Regulation T margin is strategy-based: each recognised structure has its own formula and the account's requirement is the sum over recognised structures. The formulas are arithmetic, not discretionary, and they are a floor rather than the binding number, because house requirements sit on top.

Formula: Uncovered option: Requirement = Premium + max(0.20*U - OTM, floor), where floor = 0.10*U for a call and 0.10*K for a put, and OTM = max(K - U, 0) for a call and max(U - K, 0) for a put

The out-of-the-money deduction shrinks as the option moves toward the money, so an uncovered requirement rises as the position loses. It is the only common requirement that is procyclical against the account.,The 20 percent figure applies to individual equities. Broad-based index options carry a lower percentage and narrow-based indices sit between. Confirm the applicable figure before computing.,The cash-secured put requirement exceeds the maximum loss, because the loss is K minus the premium and the requirement is K. That structure is over-collateralised by exactly the credit received.

### Regulation T on a short straddle
A short straddle is not charged as two uncovered options. The recognised treatment charges the uncovered requirement on the greater side and adds only the premium of the other side, because both sides cannot finish in the money.

Formula: Requirement = max(NakedCallReq, NakedPutReq) + min(CallPremium, PutPremium)

The saving is exactly the percentage-of-underlying term on the cheaper side, 2,000.00 in the worked case. It is not a proportional discount and it does not scale with the premiums.,The requirement is recomputed daily as the underlying moves, so it rises on whichever side is going against the position. The number at entry is not the number that gets called.,The structure has to be recognised by the broker's margin system as a straddle. Legging into it can leave the two options charged separately until the system pairs them, which is a real and avoidable liquidity event.

### The portfolio margin stress-scenario grid
Portfolio margin replaces the strategy formulas with a revaluation of the whole position across a grid of underlying price shocks, and takes the largest loss as the requirement. For individual equities the stated range under FINRA Rule 4210 is plus and minus 15 percent, evaluated at ten equidistant points.

Formula: Requirement = maximum over the grid of [ V(S*(1 + shock)) - V(S) ] expressed as a loss, with V a full revaluation of every position

The grid is asymmetric in outcome even for a symmetric position, because a call loses more on the way up than a put loses on the way down for the same percentage move. The worst point on the worked straddle is the upside, not the downside.,Because the requirement is the largest loss in the grid rather than a percentage of anything, it falls sharply for a hedged book and rises for a concentrated one. That is the design, not a side effect.,The grid values the position at a stated volatility. A short-gamma position whose real risk is a volatility spike is charged for the price move and not for the volatility move, which is exactly the gap the extra shock above quantifies.

### SPAN, conceptually
SPAN is the risk-based margin framework used for futures and futures options. It computes a scanning risk from a fixed set of joint price-and-volatility scenarios, then adds charges for calendar and inter-commodity structure and applies a floor for short option positions. It is a different construction from equity portfolio margin, not a variant of it.

The worked figures above are an illustration of the volatility-shock principle computed from this site's own reference inputs. They are not SPAN parameters, and SPAN's actual scenario weights, ranges and floors are set per product by the clearinghouse.,Because SPAN shocks volatility, a short-option book is charged for the exposure that a price-only grid understates. That is the single most consequential structural difference between the two frameworks.,The short option minimum exists because scanning risk on a deep out-of-the-money short can round to nearly nothing, while the position can still be assigned. It is a floor against the model, not a component of it.

### The capital-efficiency ratio between a defined-risk spread and its naked equivalent
A naked short option carries a higher return on committed capital than the spread built around it, and a far higher loss per dollar of capital committed. Both ratios are computable at entry from the same three numbers, and quoting either one alone is incomplete.

Formula: RoC = MaxProfit/BPR;  Loss per dollar of capital = MaxLoss/BPR;  Capital-efficiency ratio = RoC_naked/RoC_spread

At these Regulation T figures the spread has the higher return on capital as well as the lower loss, because the 20-percent uncovered charge is large relative to the credit. That ordering is not universal and flips under portfolio margin, where the naked requirement can fall below the spread's.,Loss per dollar of capital is the number the requirement itself is hiding. A defined-risk structure is the only case where it is exactly one, and that is the entire meaning of defined risk in capital terms.,Comparing two structures on return on capital alone is comparing numerators while ignoring that the denominators are computed by different formulas from different quantities.

### Assignment cascade arithmetic
Assignment replaces an option requirement with a stock requirement, and the stock requirement is computed on the full notional. When the option requirement was a small percentage of the strike, the substitution creates an immediate deficiency, and the deficiency is computable before the assignment happens.

Formula: Deficiency = InitialStockRequirement - (PreAssignmentRequirement + CreditReceived), with InitialStockRequirement = 0.50*K*shares under the Reg T baseline

The deficiency is 11.5 times the credit collected in the worked case. The position was never sized against the stock requirement, and the assignment does not ask.,A cash-secured put has no cascade, because the cash was already set aside at the strike. The cascade is entirely a consequence of margining the put rather than securing it.,The cascade is worst on the day of assignment and resolves as soon as the stock is sold, so its cost is a forced liquidation at whatever price exists that morning rather than a permanent requirement.

#### Requirement by structure

| Position | Baseline requirement |
|---|---|
| Long option, 9 months or less to expiration | Pay 100 percent of premium in cash |
| Long option, more than 9 months to expiration | May be marginable, commonly at 75 percent of premium |
| Covered call | No requirement beyond the margin on the underlying stock |
| Cash-secured put | Strike multiplied by the multiplier, held in cash |
| Naked call | Premium plus the greater of: 20 percent of underlying value less any out-of-the-money amount, or 10 percent of underlying value |
| Naked put | Premium plus the greater of: 20 percent of underlying value less any out-of-the-money amount, or 10 percent of the strike value |
| Debit spread | Pay the net debit in full |
| Credit spread | Strike width less the net credit received, which equals the maximum loss |
| Long straddle or strangle | Pay both premiums in full |
| Short straddle | The naked requirement on the greater side, plus the premium on the other side |

#### Buying-power reduction by structure, worked
Regulation T baseline, one contract, 100 multiplier, underlying at 100. Figures follow from the formulas in the entries below and nothing else. House requirements are frequently higher.

| Structure | Inputs | Formula | BPR | Max loss | BPR as a share of max loss |
|---|---|---|---|---|---|
| Long call | K 100 at 3.20 | Full premium | 320.00 | 320.00 | 100 percent |
| Credit vertical | 5.00 wide, credit 1.60 | (W - C) * 100 | 340.00 | 340.00 | 100 percent |
| Debit vertical | debit 2.10 | D * 100 | 210.00 | 210.00 | 100 percent |
| Iron condor | 5.00 wings, credit 1.60 | (wider wing - C) * 100 | 340.00 | 340.00 | 100 percent |
| Iron butterfly | 5.00 wings, credit 3.10 | (W - C) * 100 | 190.00 | 190.00 | 100 percent |
| Long butterfly | debit 1.20 | D * 100 | 120.00 | 120.00 | 100 percent |
| Cash-secured put | K 95 at 2.40 | K * 100 | 9,500.00 | 9,260.00 | 103 percent |
| Naked put | K 95 at 2.40, U 100 | (2.40 + max(20 - 5, 9.50)) * 100 | 1,740.00 | 9,260.00 | 19 percent |
| Naked call | K 105 at 1.90, U 100 | (1.90 + max(20 - 5, 10)) * 100 | 1,690.00 | Unbounded | n/a |
| Covered call | stock at 98, K 105 at 2.10 | Stock margin only | Stock requirement | 9,590.00 | Varies |
| Call ratio 1x2 | 100 long, two 105 short | Vertical plus one naked call | Vertical plus 1,690.00 approx | Unbounded | n/a |

#### Defined-risk versus undefined-risk capital treatment

| Attribute | Defined-risk | Undefined-risk |
|---|---|---|
| Requirement basis | Maximum loss, fixed at entry | A percentage of underlying value, recomputed daily |
| Behaviour as the position loses | Unchanged | Rises, because the out-of-the-money deduction shrinks |
| Worst case relative to requirement | Equal | Far larger than the requirement |
| Effect of assignment | Replaced by a stock requirement on the assigned leg | Replaced by a stock requirement |
| Effect of a volatility spike under portfolio margin | Bounded by the maximum loss | Unbounded within the shock grid |

#### Regulation T requirement formulas by structure, worked
Regulation T baseline, one contract, 100 multiplier, underlying at 100. Premiums are the Black-Scholes-Merton model values at the reference inputs S = 100, r = 0.04, q = 0, sigma = 0.20, T = 0.25, so every figure is reproducible. House requirements are frequently higher and are the requirement that actually binds. Confirm against 12 CFR 220 and FINRA Rule 4210 and your broker's specifications.

| Structure | Formula | Inputs used | Requirement per contract |
|---|---|---|---|
| Long call or put, 9 months or less | Full premium | 100 call at 4.4852 | 448.52 |
| Naked call | Premium plus max(0.20*U minus OTM amount, 0.10*U) | 105 call at 2.3909, U = 100, OTM 5.00 | 239.09 plus 1,500.00 = 1,739.09 |
| Naked put | Premium plus max(0.20*U minus OTM amount, 0.10*K) | 95 put at 1.6006, U = 100, OTM 5.00 | 160.06 plus 1,500.00 = 1,660.06 |
| Naked put, at the money | Premium plus max(0.20*U, 0.10*K) | 100 put at 3.4902, U = 100, OTM 0.00 | 349.02 plus 2,000.00 = 2,349.02 |
| Cash-secured put | Strike times multiplier | K = 95 | 9,500.00 |
| Debit vertical | Net debit | 100/105 call spread at 2.0944 | 209.44 |
| Credit vertical | (Width minus credit) times multiplier | 95/100 put spread, 5.00 wide, credit 1.8896 | 311.04 |
| Iron condor | (Wider wing minus credit) times multiplier | 5.00 wings, credit 2.2696 | 273.04 |
| Long straddle or strangle | Both premiums in full | 100 straddle at 7.9755 | 797.55 |
| Short straddle | Greater naked side plus the other side's premium | 100 call 4.4852, 100 put 3.4902, U = 100 | 2,448.52 plus 349.02 = 2,797.55 |
| Covered call | Stock margin only | Long 100 shares plus short call | Stock requirement |

#### Portfolio margin stress grid: ten short at-the-money straddles
Ten short 100-strike straddles at the reference inputs S = 100, r = 0.04, q = 0, sigma = 0.20, T = 0.25. Entry credit 7,975.46. FINRA Rule 4210 portfolio margin evaluates individual-equity positions over a plus and minus 15 percent range at ten equidistant points and takes the largest loss. Every value is a full Black-Scholes-Merton revaluation at the shocked spot with volatility held at 0.20. Confirm the applicable range and the eligibility rules against the current rulebook.

| Shock | Underlying | Position value | Profit and loss |
|---|---|---|---|
| minus 15 percent | 85.0000 | minus 14,512.09 | minus 6,536.63 |
| minus 12 percent | 88.0000 | minus 12,098.79 | minus 4,123.34 |
| minus 9 percent | 91.0000 | minus 10,117.05 | minus 2,141.60 |
| minus 6 percent | 94.0000 | minus 8,708.43 | minus 732.98 |
| minus 3 percent | 97.0000 | minus 7,979.08 | minus 3.63 |
| 0 percent | 100.0000 | minus 7,975.46 | 0.00 |
| plus 3 percent | 103.0000 | minus 8,677.06 | minus 701.61 |
| plus 6 percent | 106.0000 | minus 10,007.08 | minus 2,031.62 |
| plus 9 percent | 109.0000 | minus 11,854.27 | minus 3,878.82 |
| plus 12 percent | 112.0000 | minus 14,097.19 | minus 6,121.73 |
| plus 15 percent | 115.0000 | minus 16,623.28 | minus 8,647.82 |

## Probability and expectancy
Reviewed: 2026-08-27
Canonical: https://options.wiki/probability/ (JSON: https://options.wiki/probability.json)

Three different probabilities are routinely quoted for the same position and they answer three different questions. All closed forms below are risk-neutral, meaning they are derived from the pricing model rather than estimated from history, and they are not forecasts. Expectancy figures take the probability as a stated input; the arithmetic cannot manufacture an edge that the probability input does not contain.

### Probability of finishing in the money
The risk-neutral probability that the underlying is beyond the strike at expiration. It is N(d2), not N(d1), and therefore not delta.

Formula: P(S_T > K) = N(d2);  P(S_T < K) = N(-d2)

The gap between delta and N(d2) widens with sigma and with T, because it is driven by sigma*sqrt(T). On a long-dated or high-volatility contract the delta proxy is badly wrong.,This is a risk-neutral probability. It is the probability under the measure that prices the option, which is not the probability under which the world evolves, and the two differ by the risk premium.,A quoted "probability ITM" from a broker platform is normally N(d2) computed from that platform implied volatility. Changing the volatility input changes the number.

### Probability of touch
The probability that the underlying reaches a level at any time before expiration, rather than merely finishing beyond it. For a driftless underlying it is approximately twice the probability of finishing beyond the level, by the reflection principle.

Formula: P(touch K) approx 2 * P(S_T beyond K) = 2 * N(d2), capped at 1

Probability of touch is roughly double probability in the money for out-of-the-money levels. Any position that can be closed or assigned before expiration is exposed to touch, not to terminal probability.,The doubling is exact only with zero drift and a constant volatility. With a drift term the two halves of the reflection are no longer equal.,For a short option that will be managed rather than held to expiry, touch is the operative probability and terminal probability is the irrelevant one.

### Probability of profit
The probability that a position finishes on the profitable side of its breakeven, which is a different level from any strike in the structure. It is computed by evaluating the terminal distribution at the breakeven price, not at a strike.

Formula: POP for a single breakeven B, profitable above = N(d2 evaluated at K = B)

POP and probability of maximum profit are different numbers. The condor above has a 0.490270 probability of any profit and a smaller probability of the full 160.00.,A high POP with a small maximum profit and a large maximum loss carries no information on its own. Pair it with the breakeven win rate from the payoff ratio before it means anything.,The condor worked example has a risk-neutral POP of 0.490270 against a breakeven win rate of 0.68. Those two figures being inconsistent is a signal about the premiums used in the example, which are stated inputs rather than model prices, not a signal about the structure.

### Expected value of a defined-risk position
The probability-weighted average outcome. For a two-outcome simplification it reduces to one line; for the true continuum it is an integral over the terminal distribution.

Formula: EV = p * MaxProfit - (1 - p) * MaxLoss

The two-outcome form ignores every partial outcome between the breakevens and the wings, and for a condor or butterfly those are a large share of the distribution. It is an upper-bound sketch, not the expectancy.,Under the pricing model own measure a fairly priced position has an expected value of zero net of carry. Every positive EV produced by this arithmetic comes from the probability input, which is an assumption supplied from outside the model.,EV per contract is EV per share times the multiplier. EV per unit of capital is EV divided by the buying-power reduction, and it is the only version comparable across structures.

### Kelly sizing applied to a defined-risk options position
The fraction of capital that maximises the expected logarithm of wealth for a repeated bet with known probability and known payoff ratio. For a defined-risk position the payoff ratio is fixed at entry, so only the probability is an assumption.

Formula: f* = (p*b - (1 - p)) / b, where b = MaxProfit / MaxLoss

Kelly assumes the probability is known, the bet is repeatable, and outcomes are independent. Options positions on correlated underlyings violate the independence assumption, and the probability is never known.,The criterion is extremely sensitive to the probability input near the breakeven win rate. In the worked case a two-point error in p moves f* from 6.25 percent to zero.,A fractional Kelly - half of f*, for instance 3.125 percent in the worked case - reduces growth rate modestly and reduces drawdown substantially. That trade-off is arithmetic, not judgement.

### Win rate versus payoff ratio
The two figures are jointly constrained: for a fixed expected value, a change in the payoff ratio implies an exact change in the required win rate. Neither number is informative alone.

Formula: p_required = (EV_target + MaxLoss) / (MaxProfit + MaxLoss)

A high win rate and a low payoff ratio can describe exactly the same expectancy as the reverse. Any comparison quoting one without the other is incomplete by construction.,The sensitivity term is largest at small b, so structures with a small maximum profit relative to maximum loss are the ones whose required win rate moves most for a small change in the payoff ratio.

### Expected move compared to breakeven distance
The comparison that decides whether a structure needs more movement than the premium is charging for. Both sides are computable from stated inputs and neither requires a market view.

Formula: One-standard-deviation move = S * sigma * sqrt(T);  Breakeven distance = |Breakeven - S|

Breakeven distance measured in standard deviations is the only form comparable across underlyings and tenors. Measured in points or in percent it is not.,The expected absolute move is 0.7979 of the one-sigma move for a normal distribution, so a breakeven inside one standard deviation is not the same as a breakeven inside the expected move.,These figures follow from the volatility input. Changing sigma changes both sides of the comparison, which is why the comparison is a statement about the input, not a prediction.

#### Three probabilities, three different questions

| Measure | Question it answers | Closed form | Worked at K = 105 |
|---|---|---|---|
| Probability in the money | Will S be beyond K at expiration? | N(d2) for a call; N(-d2) for a put | 0.330729 |
| Probability of touch | Will S reach K at any time before expiration? | Approximately 2 * N(d2) for a driftless underlying, capped at 1 | 0.661458 |
| Probability of profit | Will the position finish above its breakeven? | N(d2) evaluated at the breakeven price rather than at the strike | Depends on the structure; see the entries |
| Delta | How much does value change per 1.00 move in S? | exp(-q*T)*N(d1) | 0.367719 |

#### Risk-neutral probabilities at fixed inputs
Reference inputs: S = 100, r = 0.04, q = 0, sigma = 0.20, T = 0.25. N(d2) is the risk-neutral probability that S exceeds the level in the left column at expiration. These are model outputs from stated inputs, not forecasts.

| Level | d2 | P(S_T &gt; level) = N(d2) | P(S_T &lt; level) = N(-d2) | P(touch) approx |
|---|---|---|---|---|
| 93.40 | 0.732788 | 0.768156 | 0.231844 | 0.463687 |
| 93.85 | 0.684724 | 0.753241 | 0.246759 | 0.493518 |
| 95 | 0.562933 | 0.713260 | 0.286740 | 0.573481 |
| 100 | 0.050000 | 0.519939 | 0.480061 | 0.960122 |
| 105 | -0.437902 | 0.330729 | 0.669271 | 0.661458 |
| 106.60 | -0.589133 | 0.277886 | 0.722114 | 0.555772 |

#### Win rate required by payoff ratio
The breakeven win rate for a two-outcome defined-risk position, computed as MaxLoss / (MaxProfit + MaxLoss), equivalently 1 / (1 + b) where b is the payoff ratio MaxProfit / MaxLoss. Arithmetic only; it says nothing about whether any win rate is achievable.

| Max profit | Max loss | Payoff ratio b | Breakeven win rate | Win rate needed for EV = +0.10 per share |
|---|---|---|---|---|
| 1.00 | 4.00 | 0.25 | 80.00 percent | 82.00 percent |
| 1.60 | 3.40 | 0.470588 | 68.00 percent | 70.00 percent |
| 2.10 | 2.90 | 0.724138 | 58.00 percent | 60.00 percent |
| 2.50 | 2.50 | 1.000000 | 50.00 percent | 52.00 percent |
| 3.10 | 1.90 | 1.631579 | 38.00 percent | 40.00 percent |
| 3.80 | 1.20 | 3.166667 | 24.00 percent | 26.00 percent |

Reference information only. Not investment advice, not a recommendation, and not a solicitation. Options involve substantial risk of loss. Contract terms, margin requirements, and exchange rules change; confirm against the current OCC and exchange rulebooks and your broker's house requirements before trading.