Exercise and assignment

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.

Field	Value
Common threshold	In the money by 0.01 or more
Escape hatch	Contrary exercise instruction submitted to the broker before its cutoff, which is earlier than the clearinghouse 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.

Source: Options Clearing Corporation rules

Also described at: OCC By-Laws and Rules · SEC investor bulletin: An Introduction to Options

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.

Field	Value
Condition	Dividend per share exceeds the remaining time value of the corresponding put, on the day before the ex-dividend date
Practical screen	Deep in the money, short remaining life, corresponding put trading near zero

• 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.

Also described at: Wikipedia · Wikidata

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.

Source: Options Clearing Corporation adjustment memoranda

Also described at: OCC By-Laws and Rules · OCC equity options contract specifications

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.

Field	Value
Formula	Exercise immediately before ex-dividend if D > P + K*(1 - exp(-r*tau))
D	Dividend per share going ex
P	Value of the put at the same strike and expiry
tau	Time from the ex-dividend date to expiration, in years
Interpretation	The right-hand side is the cost of exercising early: forgone interest on K plus the put protection surrendered
Worked	K = 100, r = 0.04, tau = 30/365 = 0.0821918, corresponding put value 0.15, dividend 0.88. Interest term = 100 * (1 - exp(-0.04 * 0.0821918)) = 100 * 0.00328227 = 0.328227. Right-hand side = 0.15 + 0.328227 = 0.478227. Since 0.88 exceeds 0.478227, exercise is preferable at these inputs. At a dividend of 0.40 the inequality reverses and holding is preferable

• 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.

Source: Merton 1973; Options Clearing Corporation exercise rules

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.

Field	Value
Formula	Exercise if K*(1 - exp(-r*tau)) > (remaining time value of the put) + PV(dividends over tau)
Left-hand side	Interest earned on the strike proceeds over the remaining life
Right-hand side	Time value surrendered plus any dividend the short stock position would owe
Worked	K = 100, r = 0.04, tau = 30/365. Interest term = 0.328227. If the remaining time value of the put is 0.20 and no dividend is expected, the condition holds and early exercise is preferable. At a remaining time value of 0.40 it does not

• 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.

Field	Value
AM settlement	Settlement value derived from opening prices on the expiration date; last trading day is the preceding business day
PM settlement	Settlement value derived from closing prices on the expiration date; trades through the close
Special opening quotation	A settlement value computed from the opening price of each component, which need not equal any traded index level

• 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.

Source: CBOE contract specifications

Also described at: Cboe SPX options product specifications

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.

Field	Value
Formula	Cash settlement amount = max(SettlementValue - K, 0) * multiplier for a call; max(K - SettlementValue, 0) * multiplier for a put
Physical	Deliverable transfers; the holder ends with a stock position and a cash movement of K * multiplier
Cash	Only the in-the-money amount moves; no position results
Worked	A cash-settled call with K = 100, settlement value 103.40, multiplier 100: settlement amount = 3.40 * 100 = 340.00. A physically settled call at the same strike delivers 100 shares against payment of 10,000.00

• 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.

Also described at: Cboe SPX options product specifications · OCC equity options contract specifications

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.

Field	Value
Issuer	Every listed option is issued by the clearinghouse, not by the writer
Novation	The original trade is replaced by two contracts, each facing the clearinghouse
Exercise processing	Exercise notices are submitted to the clearinghouse, which allocates assignment among clearing members
Adjustments	Contract adjustments for corporate actions are determined and published by the clearinghouse

• 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.

Source: OCC

Also described at: Wikipedia · Wikidata · OCC: What Is OCC?

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.

Field	Value
American	Exercisable any business day through expiration. Standard for listed US equity options
European	Exercisable only at expiration. Common for broad-based index options
Consequence for parity	Put-call parity holds exactly only for European exercise
Consequence for pricing	An American option is worth at least as much as the otherwise identical European option

• 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.

Also described at: Wikipedia · Wikidata · OCC equity options contract specifications

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.

Field	Value
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
Escrowed-dividend model	Subtract the present value of every dividend inside the life from spot, then price with q = 0
Exact equivalent yield	q = minus ln(S_adj/S)/T reproduces the escrowed price identically at every strike
Naive yield	q = AnnualDividend/S, or the dividend divided by spot divided by T. Not equivalent, because it ignores the ex-date
What a dividend does to value	Lowers a call and raises a put by roughly the present value of the dividend times the respective delta
Worked	S = 100, K = 100, T = 0.25, r = 0.04, sigma = 0.20, one 0.50 dividend with an ex-date at t = 0.10. PV = 0.50*exp(minus 0.004) = 0.49800399, so S_adj = 99.50199601. Escrowed call 4.21145563 and put 3.71444300, against 4.48523641 and 3.49021978 with no dividend: the call loses 0.27378078 and the put gains 0.22422322 on a dividend worth 0.498004. The exact equivalent yield is q = minus ln(0.995020)/0.25 = 0.01996993, and pricing with S = 100 at that q returns 4.21145563 and 3.71444300 - identical to eight decimals, as it must be. The naive yield of 0.02000000 returns 4.21105178, an error of minus 0.00040385 on the call

• 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.

Field	Value
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
Rebate	The rate the short seller earns on cash collateral. A high rebate means a cheap borrow, a negative rebate an expensive one
Effect on options	Enters exactly as a dividend yield does. A hard-to-borrow name prices with an effective q above its dividend yield
Where it shows up	As an apparent put-call parity violation on a name with an expensive borrow
Worked	S = 100, r = 0.04, T = 0.25, one 0.50 dividend at t = 0.10 with PV 0.498004, so S minus PV is 99.501996. At a zero rebate F = 99.501996*exp(0.01) = 100.502008. At a 0.005 rebate F = 100.376459. At 0.020, F = 100.000752. At 0.050, so a 0.010 negative net carry, F = 99.253552. Read the other way, a market quoting a 0.25-year forward at 99.000000 implies rebate = 0.04 minus ln(99.000000/99.501996)/0.25 = 0.060231, i.e. a 6.02 percent borrow cost. Pricing a 100-strike call off that forward instead of off spot changes it materially: the effective yield is 0.060231 plus the dividend yield, not zero

• 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.

Field	Value
Formula	Fair box = (K2 - K1)*exp(-r*T); implied continuous rate = ln((K2 - K1)/Price)/T; implied simple rate = ((K2 - K1)/Price - 1)/T
Long box	Pay the price now, receive the width at expiration. Economically a deposit
Short box	Receive the price now, pay the width at expiration. Economically a loan
Impossible prices	A price at or above the width implies a non-positive rate; a price at or below zero is not quotable
American risk	On American-style legs the short box carries early-assignment risk on two of its four legs, which is a real cost the rate arithmetic does not contain
Worked	A 100/110 box, width 10.00, T = 0.25. At the reference r = 0.04 the fair value is 10*exp(minus 0.01) = 9.90049834, and the implied rate read back out is ln(10/9.90049834)/0.25 = 0.040000 continuous, 0.040201 simple - the round trip closes. A box traded at 9.80 implies ln(10/9.80)/0.25 = 0.080811 continuous, 8.0811 percent, or 0.081633 simple. A box traded at 9.95 implies 0.020050 continuous. A box at 10.00 implies a zero rate and above 10.00 implies a negative one. The 9.80 example used in the payoff sections of this site therefore embeds an 8.08 percent financing rate rather than the 4 percent reference rate, which is why its 20.00 fixed payoff is larger than the fair 9.90 box would deliver

• 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.

Field	Value
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
r in the formula	The rate at which the strike is discounted. A curve rate at the option's tenor, not an overnight rate
Funding rate for a long option	The premium is paid up front, so the cost is the account's own opportunity or borrowing rate on that premium
Funding rate for a short option	The credit is received but the margin is tied up, so the relevant figure is the rate on the buying power consumed, not on the credit
Worked	A cash-secured put at K = 95 sold for 2.40 ties up 9,500.00 for 0.25 years. Priced with r = 0.04, the discounting inside the option value is worth 9,500.00*(1 minus exp(minus 0.01)) = 94.52 over the quarter. If the account actually earns 0.01 on that cash instead of 0.04, the shortfall is 9,500.00*(exp(0.0025) minus 1) against 9,500.00*(exp(0.01) minus 1), i.e. 23.78 against 95.48, a gap of 71.70 - which is 29.9 percent of the 240.00 maximum profit on the position. The option value used none of that; it assumed the 0.04

• 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.

Field	Value
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)
Cost of carry b	r minus q minus the borrow rebate. Positive b means the forward is above spot
Equivalence	Any two of the three - spot, forward, carry - determine the third, so a price can be quoted off any of them
Why it matters for exercise	Early exercise depends on b, not on r alone. b below zero is the condition under which an American call can have exercise value
Worked	Reference inputs: S = 100, K = 100, T = 0.25, r = 0.04, q = 0, sigma = 0.20, at which the call is 4.485236 and the put is 3.490220. These are inputs chosen to make the arithmetic checkable, not observations of any market. b = 0.04 and F = 100*exp(0.01) = 101.005017. Black-76 on that forward at K = 100 returns 4.48523641, identical to the spot-form value to 1e-14, and the put returns 3.49021978. Adding a 0.06 dividend yield makes b = minus 0.02, F = 99.501248, and the European call falls to 3.696260 while an American call at the same inputs is worth 3.741151 by a converged lattice - the 0.044891 difference exists only because b turned negative

• 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.

Field	Value
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)
Value effect	Negative for a call debit spread, since the long leg loses more than the short leg
Assignment effect	The short call becomes an early-exercise candidate the day before the ex-date once its remaining extrinsic value falls below the dividend
The test	Exercise the short leg early if Dividend exceeds the remaining extrinsic value of that call after the ex-date
Worked	A 100/110 call vertical at S = 100, T = 0.25, r = 0.04, sigma = 0.20, with a 0.50 dividend at t = 0.10 whose present value is 0.498004. Without the dividend: 100 call 4.485236, 110 call 1.140397, spread 3.344839. With the escrowed dividend at S_adj = 99.501996: 4.211456 and 1.038880, spread 3.172575. The 100 leg lost 0.273781 and the 110 leg lost 0.101517, so the spread lost 0.172264 - it absorbed 62.9 percent of the effect on the long leg. Separately, the assignment test on a short call after the ex-date with 0.15 years remaining: an at-the-money 100 call has 3.388514 of extrinsic value, far above the 0.50 dividend, so no exercise. A 90-strike call has 10.785382 minus 10.00 = 0.785382 of extrinsic, still above 0.50. An 80-strike call has 20.481618 minus 20.00 = 0.481618 of extrinsic, below the 0.50 dividend, so it is an exercise candidate

• 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.