Options Greeks, pricing and put-call parity

What each Greek measures, on one BP call

An option's price depends on six inputs: the share price, the strike, the time left, the volatility the market is charging, the interest rate and any dividend before expiry. The strike is fixed when the contract is chosen. The Greeks measure how much the price moves when each of the others changes, one at a time, with everything else held still. This page works each one through in pence and pounds on a UK contract, then adds them up across positions, links calls to puts through put-call parity, and explains why an American call is so rarely exercised early.

The page follows intrinsic and time value on the basics page and the assignment and expiry page, and sits before implied volatility, which covers what the volatility input is, how it is ranked and how it varies by strike and expiry.

Model inputs. BP at 530p (a model level; BP closed at 519.6p on Monday 17 August 2026, price data: Yahoo Finance), the October 560 call, which expires on Friday 16 October 2026, implied volatility 26%, Bank Rate 3.75%, no dividend (the call expires before BP's third-quarter ex-date of 12 November 2026), ICE standard contract of 1,000 shares, so each 1p of premium is £10. Priced as it would be on Wednesday 16 September 2026, with 30 days left and BP still at 530p. An American call with no ex-date in its life is priced with Black-Scholes-Merton. BP is used as a model underlying; this is not a view on BP. The methods page lists every model input and its source.

Value is the model price; here it is all time value, because the strike is above the share price. Delta is the change in value for a 1p rise in BP, and also the number of shares the option behaves like. Gamma is the change in delta when BP moves. Theta is the value lost over one calendar day at the present rate of decay, vega the change for one volatility point (26% to 27%), and rho the change for one percentage point on the interest rate.

Each Greek is a slope: it is exact for a very small change and an estimate for a larger one. The table below sets the estimate against a full repricing of the same call. For small changes the two agree to a few pence. For a 10p move they part company, because delta itself changes on the way, and that change is gamma.

The signs follow from who holds what. A bought call or put has positive gamma and positive vega and pays theta; a written option has the opposite signs on all three. Delta is positive for a bought call and negative for a bought put. Rho is positive for calls and negative for puts. The position Greeks section shows how the signs combine when legs are added together, and why a spread can change sign as the share price moves.

Every pound figure on this page scales with the contract. BP has no 100-share mini option on ICE. On the 22 large UK names that do (listed on ICE; check your broker offers it and quotes a two-way price), every pound figure would be one tenth as large for the same prices in pence. Contract sizes and multipliers are set out on the basics page.

Option price and Greeks calculator (Black-Scholes and binomial, dividends included)

The pricer below opens on the same call: an ICE UK stock option of 1,000 shares, 530p, strike 560p, 30 days, 26% and Bank Rate 3.75%. It shows the European and American values side by side (the American value from a binomial tree of about 1,000 steps), the early-exercise premium between them, the split into intrinsic and time value, the five Greeks per contract in the units used on this page, and the model probability of finishing in the money. Type a market price into it and it solves for the implied volatility. Add a cash dividend and its ex-date and it runs the dividend check described in the early exercise section, with both stamp tax tests. It also takes the FTSE 100 and Mini FTSE 100 index options, US shares and index options in dollars with a pound conversion, and a custom contract for adjusted series.

The five charts in the sections that follow, one after each Greek, redraw from whatever is typed here. Everything stays in the browser: nothing typed is stored or sent. The figures are illustrations under the stated inputs; the pricer does not know anyone's circumstances or the live market.

Where the price comes from: intrinsic and time value

With 30 days left the 560 call is worth 5.82p although BP stands 30p below the strike. All of that is time value: the price of the chance that BP is above 560p on 16 October. The chart shows the call's value against the share price at three dates. The 30-day curve sits above the expiry line everywhere; as expiry approaches the curve sinks towards the hockey-stick shape of the payoff, and the gap between the curve and the payoff line is the time value that is left. Every Greek below describes one part of that curve: delta is its slope, gamma its bend, theta the rate at which it sinks, and vega how far it would lift if the market charged more for volatility.

Time value is largest at the strike and fades on either side. Far below the strike there is little chance of the call being worth anything; far above it the call behaves almost exactly like the shares, and what is left is mainly the interest on the strike that the holder has not yet paid. That last part is why an American call is rarely worth exercising early, which the early exercise section puts in pounds.

Delta: share-equivalents and pounds per penny

Delta converts an option into the number of shares it behaves like for a small move. The 560 call's delta of 0.2546 per share means one contract moves like 255 BP shares: £2.55 for each 1p. For a writer the sign flips, so a written 560 call behaves like being short 255 shares. A put's delta is negative; the October 500 put's is −0.1948 with 30 days left, so one bought contract behaves like a short position of 195 shares.

Delta moves with everything else. It rises towards 1 for a call as the share price climbs past the strike, falls towards 0 as time runs out on an option that is out of the money, and sits close to 0.5 for an at-the-money option. A 10p rise to 540p takes the 560 call's delta from 255 to 341 share-equivalents, which is why the delta-only estimate above undershoots the full reprice.

Delta is not a probability

Delta is often read as "the chance of finishing in the money". The two numbers are close for a short-dated option, but they are different quantities and the gap has a direction. In the model, a call's delta is N(d₁) and the risk-neutral probability of finishing in the money is N(d₂), where d₂ = d₁ − σ√T. The gap therefore grows with volatility and time. With one volatility for every strike, as the assumptions table gives BP, delta is always the higher figure for calls, and for puts the size of delta is always the lower one. Skew can reverse that order, as the FTSE 100 example in the position Greeks section shows.

The put columns are Black-Scholes (European) values, which keep the comparison clean. As American options on the binomial tree the puts' deltas are slightly larger in size, most for the deepest: −0.9698 for the 600 put.

Three things follow. First, the 560 call's delta of 0.2546 overstates its 23.13% model probability of expiring in the money; with 60 days left the pair is 0.3405 against 30.27%, and for a one-year at-the-money option on a share with 45% volatility (no dividend, a 3.625% rate) it is 0.6200 against 44.26%. Second, the probability of BP trading at 560p at some point before expiry, 46.15%, is about twice the probability of finishing above it, which matters to anyone whose plan reacts to the strike being reached rather than to the expiry price. Third, every one of these probabilities is a model probability under the risk-neutral drift and the 26% the market is charging. None is a forecast of where BP will trade. The implied volatility page turns the same inputs into an expected move, and shows why the at-the-money straddle prices a move of about 0.8 standard deviations.

Gamma: how fast the share-equivalent changes

Gamma is the change in delta for a move in the share. This page quotes it as the change in share-equivalents per 10p on an ICE contract, per 10 points on the FTSE 100 and per $1 on a US option. The 560 call's gamma of 81 per 10p is why its delta went from 255 to 341 after a 10p rise. A bought option has positive gamma: its share-equivalent grows as the share moves its way and shrinks as it moves against, so gains accelerate and losses slow down. A written option has negative gamma, and the reverse is true.

Gamma is highest at the strike and grows sharply as expiry approaches. For the at-the-money 530 call, with BP held at 530p and IV at 26%:

Between 30 days and 3 days, gamma rises 3.2 times. On the last day one contract's share-equivalent swings by about 553 shares for a 10p move, so an at-the-money option near expiry can go from behaving like almost no shares to behaving like most of the contract within one session. For a writer, that is the practical meaning of "short gamma": the position gets longer as the price falls and shorter as it rises, which is the wrong way round both times. The same pattern sits behind pin risk at expiry.

Theta: the daily cost, and the move that pays it

Theta is the value lost over one calendar day when nothing else moves. For a closed-form (European) value, such as this 560 call, it is the annual rate of decay divided by 365, the rate at that moment; for a tree-priced American option it is a one-day reprice, and the two can differ by pennies (the methods page's formulas). This page quotes it in pounds per calendar day for one contract, weekends included; a screen that divides by trading days instead shows each figure larger by about 365/252, so the convention matters when figures are compared. The 560 call's theta of −£2.25 is the cost to the holder of BP standing still for a day, and the amount a writer collects for the same day.

Theta is the rent for gamma

The last column of the gamma table answers a practical question: how far would BP have to move in a day for the gain from gamma to cover that day's theta? For a small move ΔS, gamma adds about ½ × gamma × ΔS² to the value of the option once its delta is offset. Setting that equal to the day's theta gives a break-even move of about 7.58p with 30 days left and 7.33p with 3 days left: almost the same, and close to the 7.21p that the rule of thumb S × σ ÷ √365 gives before interest. What changes as expiry nears is the size of the stake on each side: £2.89 a day at 30 days, £8.58 at 3 days and £14.66 on the last day. A holder of an at-the-money option pays a higher daily rent as expiry approaches for a bigger payout on the same-sized move; a writer collects the higher rent and takes the bigger loss when the move comes. Whether the rent turns out dear or cheap depends on how far the share really moves, which is the comparison of implied and realised volatility on the implied volatility page.

Theta is not one curve

"Time decay speeds up into expiry" is true for an at-the-money option only. An option out of the money has less and less value left to lose, so its theta in pounds rises, peaks and then falls. The table compares the at-the-money 530 put with the 500 put, about 5.7% out of the money, with BP held at 530p:

The 500 put's theta in pounds peaks at −£1.83 a day with 17 days left and then falls, to −£1.35 at 7 days. Measured against what the option is still worth, though, decay keeps accelerating: 3.85% of the value a day at 30 days, 11.66% at 14 and 34.40% at 7. The at-the-money put's theta keeps rising in pounds to the end. Both facts matter to writers of out-of-the-money options: in the final weeks the premium left to collect is small, while the loss a move can cause is not, and that trade-off, rather than any slowing of decay, is the case made for the common 21-day management convention that the methods page describes. The chart below shows the same shapes for calls at the money, 10% out of the money and 10% in the money; the in-the-money curve levels off at the interest on the strike.

Vega: what one volatility point is worth in pounds

Vega is the change in value when implied volatility rises by one point, from 26% to 27%, not by 1% of 26%. Implied volatility itself is the volatility input that makes the model price match the market price; the implied volatility page covers how it is measured, ranked and shaped across strikes and expiries. Here the question is only what a change in it does to a price.

Two patterns stand out. At the money, vega grows with the square root of the time left: the 90-day figure is 1.72 times the 30-day one, against √3 ≈ 1.73. So longer-dated options carry more volatility risk per contract in pounds, which is what a calendar spread trades on. And the out-of-the-money 560 call is the more sensitive per pound paid: one volatility point is 8.4% of its value with 30 days left against 3.6% for the at-the-money call. Most of a cheap out-of-the-money option’s value rests on the volatility assumed, so it gains or loses a large fraction of its price when that assumption changes.

Vega is also a slope, and the slope is not constant for options away from the money. A five-point rise to 31% adds £25.35 to the 560 call against the £24.38 that five times vega suggests. The sensitivity of vega to volatility is sometimes called volga (or vomma); it is small here, but it grows for options far from the strike, which is one reason sold wings can lose more than a vega table implies when volatility jumps.

Rho: Bank Rate and the interest on the strike

Rho is the change in value for a one-percentage-point rise in the interest rate. A call gains from a higher rate because its holder keeps the strike money on deposit until exercise; a put loses because its holder waits longer to receive the strike. For the 560 call with 30 days left, rho is £1.06 per point, small next to its other Greeks. Rho scales with the strike, with the time left and with the model probability of exercise, so it matters for long-dated options and for pages that compare holding options with holding cash.

With the assumptions table's Bank Rate of 3.75% (in force on the 17 August 2026 model date and held on 17 September 2026; later decisions do not re-price the examples), the BP June 2027 530 options, which expire on Friday 18 June 2027 with 305 days left and three dividends of 6.39p inside their life (12 November 2026, and assumed dates of 18 February and 13 May 2027, the amounts also assumed), give a clearer picture. Priced as American options on a binomial tree, the call is worth £485.01 a contract with rho of £17.27 per point, and the put £504.28 with rho of −£22.56. A quarter-point cut in Bank Rate, other things equal, would move the call by about −£4.32 and the put by about +£5.64. The LEAPS page applies the same effect to options with more than a year left.

Charm and vanna: when delta moves without the share price

Delta changes when the share price moves; that is gamma. It also changes when time passes and when implied volatility moves, with the share price standing still. The first effect is called charm and the second vanna. Neither appears on most broker screens, but both explain surprises that a delta-only view leaves unexplained, particularly for anyone who hedges an option with shares, or who writes calls against shares they hold.

Adding Greeks across legs: units and position Greeks

A position's Greeks are the sum of its legs' Greeks, each multiplied by the number of contracts (negative for a written leg) and by the contract size. Shares add to delta only: 1,000 BP shares add 1,000 share-equivalents and nothing to gamma, theta or vega. That sum is only meaningful if every page quotes the Greeks in the same units, so the library uses one set, in the table below.

The same-sized move is a different amount of risk on each market. For the FTSE 100 October options (ICE ESX) on 17 August 2026, 60 days before expiry, with the index at the 10,750 model level, at-the-money IV 14.0% from the model surface, Bank Rate 3.75% and the FTSE 100 dividend yield of 3.05% (FTSE Russell factsheet, 28 August 2026), the at-the-money call is worth 248.24 points, or £2,482.42 at £10 a point. Its delta is £5.17 per index point, theta −£20.97 a day, vega £172.80 per volatility point and rho £87.25 per percentage point; a 10-point rise lifts its delta by 0.0065, so each further index point is worth about 6.5p more.

The matching put is worth 235.94 points (£2,359.41), with delta −£4.78 per point and theta −£18.93 a day. At a flat 14%, the model probability (risk-neutral, lognormal) of the call finishing in the money is 49.68%, against a delta of 0.5168. Taken from the assumed volatility skew, which charges more for lower strikes, the probability is 56.15%, above the delta: skew takes probability from small falls and adds it to large falls and to small rises, as the implied volatility page shows. The call and the put at one strike share the same vega, £172.80 each, against £8.52 for the BP 530 call over the same 60 days, so a count of contracts says little about the size of a position across markets; the pound figures do.

Ten common structures on one BP chain

The table adds up the legs of ten structures from this library, all on BP October options on Monday 17 August 2026, 60 days before expiry, at 530p and a flat 26% (the assumptions table gives BP no skew). One contract of each leg; the shares legs are 1,000 BP shares. The puts are at Black-Scholes values, as in the delta table; priced as American options on the tree, the 530, 500 and 470 puts are worth £2.42, £0.88 and £0.19 a contract more. The signs tell most of the story before any number is read.

Read across the rows and each structure's character is visible in its signs. Every structure that is paid theta, the covered call, the cash-secured put, the collar, the credit spread, the strangle and the condor, is also short gamma and short vega: the daily income is the price of carrying the risk of a large move or a rise in volatility. The long call, long put and straddle are the mirror image. The collar's option legs almost cancel, so it is close to a pure share position with its tails cut off; its 406 share-equivalents are what is left of the 1,000 shares. The bull call spread is nearly flat on gamma, theta and vega at entry, which is why its price depends mostly on direction. These are the pages' own lessons in one table; each strategy page works its structure through in its own detail, and the strategy builder adds up the Greeks of any position of up to six legs.

Where a spread changes character

Position Greeks are a snapshot at one share price. For any vertical spread, the long and short strikes pull in opposite directions, and which one dominates depends on where the share is. The 530/560 bull call spread shows it clearly. With BP near the long 530 strike, the long call's gamma outweighs the short call's, so the spread is long gamma and pays theta. Near the short 560 strike the short call dominates: the same spread is short gamma, earns theta and loses from a rise in volatility. The switch happens close to the midpoint of the strikes, about 544.8p with 30 days left.

With 7 days left the switch is abrupt: −£3.88 a day at 530p becomes +£3.84 at 560p. A spread bought as a directional trade that has moved most of the way to its short strike has become a position that earns time decay and loses if the share falls back, even though nothing about its legs has changed. The strategy pages describe this switch for their own strikes and link here for the mechanism.

Following one trade: what the Greeks predicted and what happened

The Greeks are most useful for explaining a change in value after the event. Take one BP October 560 call bought on Monday 17 August 2026 at the model value of 11.97p (£119.68 a contract), 60 days before expiry. Then assume, for illustration only (these are not BP's actual prices), that by Tuesday 1 September, with 45 days left, BP has risen 15p to 545p and implied volatility has eased from 26% to 24%. The call is then worth 12.97p (£129.75), a gain of £10.06 before costs. The entry Greeks split that change like this:

The rise in BP more than paid for fifteen days of time decay and the fall in volatility. The £4.44 the four entry Greeks miss comes from their own movement along the way: with volatility two points lower the call's delta was smaller all the way up (vanna), and the passing days shaved delta (charm) while the daily theta grew as BP neared the strike. Repricing one input at a time gives an exact split instead, though its parts depend on the order chosen: the price move alone adds +£58.61, the fifteen days then take −£33.62 and the lower volatility −£14.93. On 1 September the call's delta is 411 share-equivalents, its theta −£2.20 a day and its vega £7.44: the position has become more like shares and more expensive to hold.

A US option held in pounds carries a second exposure

A UK investor holding a US option has two sensitivities where a US holder has one. Take a hypothetical US share at $150 (not a real company) and its at-the-money call with 30 days left, IV 30%, the assumptions table's US rate of 3.625% and no dividend, on a 100-share contract, at an illustrative $1.3559 per £1 on 17 August 2026 (ECB reference-rate cross); a broker's own rate will differ. The call is worth $5.36 a share, $536.38 a contract, which is £395.59.

The premium held is a dollar asset, so its sterling value falls by about 1% when the pound rises 1% against the dollar, with no change in the share. On one contract that is £3.93; on ten, £39.28. The share exposure the call gives, about 53 share-equivalents, amounts to £5,873.72 of dollar-priced stock, and a gain on it is worth fewer pounds if the dollar weakens before it is realised. For UK tax each leg is converted at the rate on its own date, never as one net dollar figure, as the tax worked examples show; trading hours, brokers and US withholding are on the US options page.

Put-call parity: the price link between a call and a put

A bought call and a written put with the same strike and expiry together pay off exactly like a forward purchase of the underlying at the strike: above the strike the call is exercised, below it the put is assigned, and either way the holder ends up buying at the strike. Two things with the same payoff must cost the same, so for European options with a dividend yield q:

C − P = S × e^−qT − K × e^−rT

The call price minus the put price equals the present value of the underlying (after the dividends it will pay) minus the present value of the strike. Volatility does not appear: whatever the market charges for volatility raises the call and the put by the same amount, which is why the FTSE call and put above have the same vega.

Checked on the FTSE 100 October options

On the model inputs above (60 days, 10,750, q 3.05%, r 3.75%), the at-the-money call is 248.24 points and the put 235.94, a difference of 12.30. The right-hand side is 10,696.24 − 10,683.94 = 12.30 points. At the 10,500 strike, where the model surface gives both options 14.94%, the call is 406.58 and the put 145.82; the difference, 260.76, again equals the right-hand side. Parity holds strike by strike, skew or no skew: at any one strike the European call and put carry the same implied volatility, which is what parity requires. Market quotes can differ from it within the bid-ask spread and costs; a wider apparent gap usually points to a stale quote or a different dividend assumption.

Synthetics, conversions and reversals

Rearranging the equation gives the synthetic positions. Buying the 10,750 call and writing the 10,750 put costs 12.30 points (£123.01) and has a delta of £9.95 per point: a synthetic long position in the index, equivalent to buying it forward at the strike. The small cost reflects the gap between the 10,750 strike and the model forward of 10,762.38. A conversion (buy the shares, buy the put, write the call) locks in a fixed amount at expiry; a reversal is the opposite. Both are arbitrage trades for firms with low costs, and the Options Industry Council sets out the full list of synthetic equivalents.

On UK shares the costs decide the matter. A conversion on BP at the model values for the October 530 strike, with 60 days left, would buy 1,000 shares at 530p, buy the put at 20.86p and write the call at 23.88p: a net cost of £5,269.85 for £5,300 at expiry whatever BP does, a gain of £30.15, about 3.48% a year, a little less than the 3.75% model rate because the American put costs more than a European one. Buying the shares carries 0.5% stamp duty or stamp duty reserve tax (SDRT) on the price paid (GOV.UK), £26.50, and two option commissions at the assumptions table's £1.40 add £2.80, leaving £0.85 before the share-dealing commission and bid-ask spreads.

Parity with American options

ICE stock options and US equity options are American, so parity becomes a range rather than an equation. With no dividend in the option's life the American call is worth the same as the European one, but the American put is worth more, because it can be exercised early. For the BP October 530 options: the call is 23.88p, the European put 20.62p and the American put 20.86p. The European pair satisfies parity exactly (3.26p = 530 − 530 × e^−rT); the American pair gives 3.01p, inside the bounds S − K ≤ C − P ≤ S − K e^−rT, which here run from 0p to 3.26p.

Box spreads and locked-in returns: when the return is income

Combining a bought synthetic at one strike with a written synthetic at another removes the underlying altogether. A FTSE 100 box on the October expiry buys the 10,500 call, writes the 11,000 call, buys the 11,000 put and writes the 10,500 put. Whatever the index does, it pays 500 points, £5,000, at expiry. At the model values it costs 496.93 points, £4,969.27, so it returns £30.73 over 60 days: exactly the 3.75% model interest rate, because a box is a loan in option form. Four commissions at £1.70 cut the return to £23.93 before bid-ask spreads.

For UK tax, a return of this kind is not necessarily a capital gain. The disguised-interest rules in ITTOIA 2005 s381A charge income tax, where no other income tax provision does, on a return from an arrangement that is economically equivalent to interest: a return that can reasonably be assumed to reflect the time value of money, at a rate reasonably comparable to a commercial rate of interest, and with no practical likelihood, when the arrangement starts, that it will fail to arrive unless the party due to produce it cannot pay. Box spreads, conversions and very tight collars can fall within that definition. The exclusion in s381E applies only where an arrangement involves only excluded shares, so an arrangement that combines shares with options is not excluded by it. HMRC's Capital Gains Manual at CG12310 still points to the chapter these rules replaced. Whether a particular position falls within the rules depends on its facts, which is a question for a tax adviser rather than a worked example.

American against European: early exercise and dividends

An American option can be exercised on any business day; a European option only at expiry. The right is worth something only when exercising now beats selling the option or holding it. The site's pricing rule follows from that: Black-Scholes-Merton for European options and for American calls with no ex-dividend date in their life, and a binomial tree (200 and 201 steps, averaged) for American puts and for calls with an ex-date. An American option is never shown below its intrinsic value.

Why an American call is rarely exercised early

On 17 August the BP October 500 call, 60 days before expiry and 30p in the money, is worth 42.03p (£420.25). Exercising it would capture only the 30p of intrinsic value and give up 12.03p, or £120.25 a contract. Put-call parity shows what that 12.03p is made of: 3.07p is the interest on the 500p strike for 60 days at 3.75%, which the holder keeps by paying for the shares later rather than now, and 8.95p is the value of the protection against BP falling below 500p that a call holder has and a shareholder does not (the 500 put, priced as a European option). Early exercise throws both away, and on a UK share the exercising holder also pays 0.5% SDRT on the strike (who pays SDRT). A holder who wants out sells the call instead; the basics page sets out exercising against selling.

A dividend is the one reason to exercise a call early, just before the ex-date, and only when the dividend is larger than the time value that would be given up. The assignment page works through BP's calls before the 12 November ex-date, including the stamp tax test. The quick version compares the dividend with the call's time value: an options intermediary with SDRT relief can gain from exercising only when the dividend exceeds it, and a private holder who does not otherwise want the shares also pays 0.5% of the strike, so needs the dividend to exceed the time value plus that 0.5%. Passing the quick test is necessary, not sufficient: the assignment page shows a call that passes it and is still worth more held through the ex-date. The pricer above runs both quick tests for calls in the money.

When an American put is worth more

For a put the logic runs the other way. Exercising a deep in-the-money put early brings the strike money in sooner, and when the interest on that money is worth more than the put's remaining time value, holding on costs money. That is why a European put can trade below its intrinsic value and an American put cannot. On the BP October puts, 60 days before expiry with no dividend in their life:

The premium grows with depth in the money. At the 640 and 700 strikes the European value is below intrinsic, and the American value equals intrinsic: in the model, immediate exercise is the rational choice for the holder, and the writer of such a put is likely to be assigned at any time. The assignment page gives the share price below which early exercise becomes rational for each number of days left.

What a dividend does to calls and puts

A share falls by roughly the dividend on its ex-date, so a known dividend inside an option's life lowers call values and raises put values. The assumptions table has BP paying 6.39p with an ex-date of Thursday 12 November 2026 (8.66 US cents, the last rate declared before the model date, at the model exchange rate; BP was due to set the third-quarter amount with its results on 30 October 2026). For the BP December options, which expire on Friday 18 December 2026 with 123 days left on 17 August, the present value of that dividend is 6.33p:

Neither change equals the full dividend, because each option only partly behaves like the shares. The 480 call shows the early-exercise right at work: as a European option it would be worth 61.16p with the dividend, and the American right to exercise before 12 November adds 2.08p, £20.76 a contract. For puts a dividend delays early exercise, because the holder gains by waiting for the ex-date fall. The FTSE 100 examples use a continuous dividend yield instead, because the index's dividends arrive spread across the year.

Pages that apply these Greeks

Each of these pages uses one part of this page on its own structure and numbers:

How these numbers are calculated

Self-check: ten questions with the working

These use this page's own numbers; the working is in each answer. Nothing is scored.