How an Option Payoff Changes with the Price at Expiry

An option payoff at expiry is the amount the contract hands over on its last day, and it comes from two numbers: the price of the reference asset that day and the strike written into the contract. A call pays the price less the strike when that difference is above nil. A put pays the strike less the price on the same rule. Nothing else enters.

Everything awkward about an option comes from one feature. The buyer is allowed to choose, and until the choosing is done nobody can say what the contract will hand over. On the last day that awkwardness ends. The price has arrived and is sitting there in front of both sides, so there is nothing left to weigh and nothing left to guess at. The buyer compares one number with one other number, takes whichever is better, and the result of that single comparison is the payoffWhat a contract pays on its last day, counting nothing that was paid to hold it..

The closing of that choice is worth saying twice, and the whole shape of what follows comes from it. Every genuinely hard question in this subject lives in the days before the last one, when the price has not arrived and somebody has to form a view about where it might get to. On the last day there is no view to form. There is a price, there is a strikeThe fixed level written into the contract at which the buyer may deal if they choose to., and there is a subtraction. The last day is the one moment in this subject where the arithmetic closes completely, and that is why exact figures can be computed here and at no earlier moment.

There is an everyday version to carry alongside it. A market committee sells a trader a token in advance that lets the trader take a stall for the season at a fixed rent. On the last day of the letting, the going rent for a stall is now a known, settled, public number. There is nothing to predict any more. The trader looks at the going rent, looks at the fixed rent on the token, and pays the lower of the two. Whatever holding the token saved is what the token handed over. The saving is the payoff, and working it out needed no opinion about anything.

What is a payoff at expiry, and what is it worked out from?

A payoff at expiryThe last day of the contract, on which the right written into it is either taken up or abandoned. is what the contract pays on the last day, and it is worked out from exactly two numbers. The first is the price of the reference asset on that day. The second is the strike written into the contract when it was struck. Set those two side by side, apply the rule that belongs to the contract, and the payoff follows.

The absences from that list matter as much as the entries. No view about how far the price might travel enters a payoff at expiry, no model enters it, and no assumption of any kind enters it. Two people working carefully from the same two numbers cannot disagree about the answer. That is a stronger statement than it sounds. Almost nothing else in this subject has that property. Ask two careful people what a contract is worth with three months still to run and they can differ honestly. Each has had to supply something the record does not contain. Ask them what it pays on the last day and there is nothing left to supply.

One more distinction belongs in this block, and the rest of the account is built on it. A payoff counts nothing that was paid to hold the contract. The profitThe payoff after the premium is counted, with the premium carried forward to the same day the payoff arrives. counts it. Payoff and profit are two different quantities, they take two different values at every single price, and the habit of writing down which one is in hand is worth more than any calculation below. The payoff comes first because it is the simpler object. The profit is built on top of it later, and it needs one extra number that the payoff never touches.

What does a call pay at expiry, at any price?

A call pays the price of the reference asset less the strike wherever that difference is above nil, and it pays Rs 0.00/- everywhere else. There is no third case and no exception. At any price whatsoever, one of those two branches applies.

The floor at nil is the part worth pausing on. The whole asymmetry of an option sits there. A call payoff never goes below nil however far the price falls, and the reason is not a rule somebody added for fairness but the plain fact that the buyer is allowed to walk away, so the worst the buyer's payoff can be is nothing at all. If the price on the last day sits far below the strike, the buyer simply does not use the right. There is nothing to use it for. The contract expires having handed over nothing, and nothing is the floor.

The arithmetic runs at both ends of the range this guide uses. At a price of Rs 2,400.00/- on the last day, against a strike of Rs 2,000.00/-, the call payoff is Rs 2,400.00/- less Rs 2,000.00/-, or Rs 400.00/-. At a price of Rs 1,600.00/- the subtraction would give a negative number, so the second branch applies and the call payoff is Rs 0.00/-. Both of those figures are exact, and a reader can reproduce either of them with a single subtraction and no other information whatsoever.

What does a put pay at expiry, at any price?

A put pays the strike less the price of the reference asset wherever that difference is above nil, and it pays Rs 0.00/- everywhere else. The put rule is the same sentence as the call rule with the two numbers swapped round, and the swap is the whole of the difference between the two contracts on the last day.

Work this one at both ends too. At a price of Rs 1,600.00/- against the strike of Rs 2,000.00/-, the put payoff is Rs 2,000.00/- less Rs 1,600.00/-, or Rs 400.00/-. At a price of Rs 2,400.00/- the subtraction would give a negative number, so the floor applies and the put payoff is Rs 0.00/-. The floor is there for the identical reason it is there on the call: the buyer of a put may also walk away, and nobody can be made to use a right they do not want.

Held together, the two rules invite a natural sentence that turns out to be slightly wrong, so the statement of them needs care. Wherever one of this pair is paying anything at all the other is paying nil, and at the strike of Rs 2,000.00/- itself both of them pay nil, so the two are never paying together at any price. The tempting version of that sentence is that the put pays exactly where the call does not, and it is not quite true. At the strike, the call pays nothing and the put pays nothing as well. One price on the whole range has both of them at nil at once, and it happens to be the most important price on the diagram, so it is worth getting right rather than glossing.

Look at the picture for a moment longer than feels necessary. Each line is flat along one stretch and sloped along the other, and the flat stretch is not the line resting or the drawing running out of information. The flat stretch is the floor doing its work. Along it the right is being abandoned, and abandoning a right pays nothing. Nothing at all is a number rather than an absence.

Why does the payoff line bend, and where exactly does it bend?

Both payoff lines bend at the strike of Rs 2,000.00/-, so the answer is one number rather than a region. The bendThe point on a payoff line where the buyer's choice changes over from abandoning the right to using it. is the buyer's choice drawn as geometry. The bend sits at the strike because the strike is the price at which the choice changes over, and no other price on the whole range changes anything at all.

On one side of the bend the right is abandoned and the line runs level along nil. On the other side the right is used and the line climbs away from nil at one rupee of payoff for every rupee of price. The shape holds nothing else. Nothing else is happening in the picture. A payoff diagram looks like it might be carrying a great deal of information because it is a curve on axes, and it is not. The drawing is two straight runs joined at one point, and the point is a decision.

The bend gives a reader something to carry to any payoff diagram met afterwards, including ones drawn by somebody who labelled nothing. The position of the bend along the price axis gives the strike. The direction the line slopes away from the bend gives the contract: climbing to the right of the bend is a call, climbing to the left of the bend is a put. Two glances read two facts off a drawing that stated neither of them.

What do these two contracts pay at the two ends of the declared range?

Everything worked below runs on one invented pair of contracts, and here is the whole of what they are. The reference asset has a spot price of Rs 2,000.00/- today. Financing costs 6.50 per cent a year, and both contracts run for one year. The reference asset pays nothing at all while it is held. A payout during the holding period would change several of the figures below, and this pair carries no payout.

Three separate quantities in this guide are all Rs 2,000.00/-, and a reader is entitled to know whether one of them was simply copied into the others. Not one of the three was copied from another. The spot price of the reference asset is Rs 2,000.00/-. The strike written into both contracts is Rs 2,000.00/-, and it agrees with the spot price because the pair is struck at the money. A strike that agrees with the spot price is what at the money means. And one unit of the reference asset carries an exposure of Rs 2,000.00/-. The exposure agrees with the spot price for the plain reason that a unit of something is worth its price. Three quantities agree at Rs 2,000.00/- for three different reasons, and none of the three is the same kind of number as the other two.

The call premium is Rs 180.00/- and the put premium is Rs 57.93/-. Both are given quantities rather than results worked out here, and the distinction matters. Neither option is priced below. Pricing one would need a figure for how far the reference asset might travel, and no such figure exists anywhere in the invented record these contracts belong to. The two premiums arrive as given quantities that are consistent with each other, and every payoff and every profit below is then produced from them by subtraction that a reader can follow.

Two prices on the last day are then chosen as the ends of the range worked over here: Rs 1,600.00/- and Rs 2,400.00/-. Both prices are chosen settings, picked to give a wide and readable range, and neither is a reading taken off anything or a suggestion that either price is likely. Here is what the pair pays at each.

Taken one row at a time, the table already shows the whole point. On the top row, one contract at one price produces two different figures, Rs 400.00/- and Rs 208.30/-, and both of them are correct answers to two different questions. Neither figure is a better version of the other. The two figures answer what did the contract hand over and what is left after paying for it, and mixing them up is the single commonest error made with drawings of this kind.

How is the payoff different from the profit at the same price?

A payoff counts nothing that was paid to hold the contract. A profit counts the premium. Counting the premium sounds like a small adjustment and it is not. The reason is when the two amounts happen. The premium left the buyer's hands at the start. The payoff arrives at the end. Setting them against each other means bringing them to the same date, and the honest way to do that is to carry the premium forward at the financing rate to the day the payoff lands.

Do it for the call. The premium of Rs 180.00/- carried for one year at 6.50 per cent a year is Rs 180.00/- plus Rs 11.70/- of carry, or Rs 191.70/-. The carried premium has a name worth having: it is the financed premiumThe premium carried forward at the financing rate to the day the payoff arrives, so that the two amounts can be set against each other on one date.. At a price of Rs 2,400.00/- the call's payoff of Rs 400.00/- is a profit of Rs 400.00/- less Rs 191.70/-, or Rs 208.30/-. The whole of the difference between those two figures is the financed premium.

The same arithmetic for the put needs one detail handled with care. The put premium sits in the record as Rs 57.93/-, and that is a rounded figure. Carried at 6.50 per cent a year, the exact put of Rs 57.9343/- comes to Rs 61.70/- to the paisa. The rounded Rs 57.93/- carried the same way comes to Rs 61.69545/-. The two differ by Rs 0.00455/-, under half a paisa, and the figure used throughout is Rs 61.70/-. At a price of Rs 1,600.00/- the put's payoff of Rs 400.00/- is a profit of Rs 338.30/-, once the financed put premium of Rs 61.70/- is taken out.

The rounding is not fussiness. The relationship tying these two premiums together holds to the paisa rather than exactly, and it is worth seeing why with the numbers set out. The present value of the strike is Rs 1,877.9343/-, so the difference between the two premiums must be Rs 2,000.00/- less Rs 1,877.9343/-, or Rs 122.0657/-. The rounded put taken from the call gives Rs 180.00/- less Rs 57.93/-, or Rs 122.07/-. The two differences are Rs 0.0043/- apart. Carrying the rounded present value back the other way, Rs 1,877.9343/- multiplied by 1.065 gives Rs 2,000.0000295/-, not Rs 2,000.00/-. A claim of exact equality that the rounded figures themselves do not produce teaches a reader to stop checking, so the equality is stated to the paisa and the difference is printed.

The last row of that drawing is the one people skip. The carry of Rs 11.70/- is a small segment because it is a small amount, and it is drawn on the same scale as the rest rather than blown up to be noticeable. The carry is still the difference between calling the break even Rs 2,180.00/- and calling it Rs 2,191.70/-, and only one of those two is right.

The opening state of that control is worth stating in plain words so the finding survives with the picture stripped out. At the control's opening setting of Rs 2,400.00/-, the call's payoff reads Rs 400.00/- and the call's profit reads Rs 208.30/-, exactly the worked instance in the table above. Drag the control down and both lines fall together. The band between them is the financed premium, and the financed premium is fixed. Keep dragging and the payoff line flattens at nil at Rs 2,000.00/-. The profit line flattens at minus Rs 191.70/-, a loss of the whole financed premium and the furthest down the profit line ever goes.

One thing in particular holds still while the control moves. The dashed marker sitting at Rs 2,191.70/- does not move at all. The price marker travels across the whole range and that dashed line stays exactly where it is. Break even is a property of the contract and the premium rather than of today's price. The moving thing and the still thing on that diagram are the two halves of the point: the price is what changes, and where the profit crosses nil is what does not.

Where does the profit cross nil, and is that the same price as the bend?

No, and this is the sharpest point in the whole account. The payoff leaves nil at the strike. The profit crosses nil somewhere else entirely. Break evenThe price at which the profit line crosses nil, which is not the same price as the bend in the payoff line. for the call is the price at which the payoff has grown large enough to have covered the financed premium: Rs 2,000.00/- plus Rs 191.70/-, giving Rs 2,191.70/-. Break even for the put is the mirror: Rs 2,000.00/- less Rs 61.70/-, giving Rs 1,938.30/-.

The call's payoff line leaves nil at Rs 2,000.00/- while its profit does not reach nil until Rs 2,191.70/-, so between those two prices the contract is paying something and the position is still behind, and a reader who treats the strike as the point where a position starts making money has understated what the price has to do by exactly the financed premium. The gap is Rs 191.70/- on the call and Rs 61.70/- on the put, and the two gaps differ because the two premiums differ.

Why does the last day need nothing it does not already have?

Half a dozen exact calculations have just been worked, and the reason they were possible is not that this subject is easy. The reason is that the last day is the one moment where the difficulty has already resolved itself.

A payoff on the last day is arithmetic on a price that has already arrived, so it wants no opinion about how far anything might travel and no machinery for holding such an opinion. Every earlier day is short of one specific thing, and the last day is short of none. Every earlier day concerns a price that has not arrived yet. The moment a price has not arrived, somebody has to say something about the range of places it might get to, and the record these contracts belong to says nothing of the kind. The missing view is the gap, and the gap is not a shortcoming in the record. The gap is the reason no figure can honestly be produced for an earlier day.

The practical version is a useful test to carry. Everything set out here can be checked with a pencil. Almost nothing about an earlier day can. The difference between the two situations is one word, and the word is when. Before the last day, a number of any kind about this contract has an assumption inside it. On the last day, it does not.

Do not let that turn into the wrong lesson. The exactness set out here belongs to one day and does not extend backwards even by twenty four hours, so a reader who leaves thinking options are simple arithmetic has taken away the opposite of what the last two blocks establish. The arithmetic is simple because the hard part is over, not because the hard part was never there.

What changes one day back from the last day?

Step back a single day and the payoff stops being a number. The price it is worked out from has not happened. There is no price on the last day yet. The subtraction has nothing to subtract. Whatever anybody writes down about the contract on that day is a different sort of object with a different name.

A premium exists instead: the amount at which somebody would take the contract off a holder's hands today. A premium and a payoff are not the same quantity measured at two moments. Premium and payoff are different quantities. A premium contains something a payoff never contains: an allowance for how far the reference asset might still travel before the last day. The allowance is why a premium can be a positive number on a day when the payoff, if the contract ended right then, would be nil.

Think about what that means for a call struck at Rs 2,000.00/- with the price sitting at Rs 2,000.00/- and a year to run. If the contract ended that instant the payoff would be nil. The price and the strike agree. Yet somebody paid Rs 180.00/- for it. The buyer did not pay Rs 180.00/- for a nil payoff. The Rs 180.00/- bought the year, and specifically what might happen to the price during the year, and that is the allowance.

The allowance cannot be sized from anything set out above, and saying so plainly is more useful than producing a figure that would have to be invented. Sizing it needs a view about travel and a model to convert that view into a premium, and neither is available here. The sizing of the allowance is covered separately: what it responds to, how it is read out of a quoted premium, and what would have to be known before anyone could put a number on it.

Is being able to read this diagram a reason to act on it?

A reader who has followed this far is entitled to ask that out loud, and the question deserves a plain answer rather than a hedge. Whether a position should be taken is not a question that can be answered here, and the reason for declining is a stated one rather than caution.

Three items would have to be known before anybody could answer it, and three items can be checked one at a time.

1. A view about travel, and how likely each move isHow far the reference asset might get to before the last day, and with what likelihood attached to each place it might get to. None of that is supplied anywhere, and the record these contracts belong to holds no probability, no distribution and no outcome of any kind. The first item is missing outright.
2. The circumstances of the person askingWhat else they hold, what they owe, what they can afford to be wrong about, over what period, and under what obligations. None of that is visible from here, and an answer given without it would be answering a question it had not been asked.
3. What the arrangement costs to hold and to unwindWhat has to be placed and maintained to keep the position open to the last day, and what it would cost to get out of it earlier. Both are set by parties not named above, and both move.

All three are absent, and the first of them is the very quantity that the treatment of premiums establishes nobody here has. So the honest position is not that the answer is difficult. The answer cannot be produced from anything set out above, and producing one anyway would mean inventing the three missing items.

A drawing like this sets out what each side owes at each price on one day, and setting out what somebody would owe is not a suggestion that anybody take the position on. The record these figures come from contains no outcome, so no contract in it can be ranked, compared on how it turned out, or called attractive.

The reference asset, its price of Rs 2,000.00/-, the strike of Rs 2,000.00/-, the financing of 6.50 per cent a year, the call premium of Rs 180.00/-, the put premium of Rs 57.93/- and the two prices on the last day of Rs 1,600.00/- and Rs 2,400.00/- are invented.
Educational material. Not advice on any investment, tax, budget or market position.