Option Payoff: What an Assembly Pays at the End Date
An option payoff calculator takes each leg of an assembly as a sign, a call or a put, a level and an end date, takes one price for the reference asset at that date, and returns what each leg pays and what the assembly pays in total. A payoff ignores what was paid to put the assembly on. The calculator prices nothing.
Build the assembly, then settle it at a chosen price
With the legs copied off a contract note, the number of contracts and the units a contract covers set, and a price named for the reference asset at the end date, the panel works each leg on its own and adds them. The panel opens on the two-leg assembly used throughout this guide. Every figure below is an illustration of arithmetic. None of it is a quotation, a price or a prediction.
| Leg | Bought or written | Call or put | Level | Premium a unit | Contracts |
|---|---|---|---|---|---|
| One | |||||
| Two | |||||
| Three | |||||
| Four |
- Bought or written. Contract note, the buy or sell column on that row. A leg set to nil contracts is switched off and drops out of every figure below.
- Call or put, and the level. Contract note, the instrument description on that row.
- End date. Contract note, the expiry printed on that row. Every leg in this panel runs to the same end date, one year out, so no box asks for it. A payoff is drawn at one date, and legs finishing on different dates are worked separately.
- Premium a unit. Contract note, the rate column, before brokerage and every other charge. The box stays empty where no premium is available.
- Contracts. Position statement, the quantity column for that row.
The build-up, leg by leg
Each leg drawn against nil, and the three totals under it
The assembly across every price, redrawn as the inputs change
.
Educational illustration, and not a quotation of any price. On screen: one year to the end date; financing at 6.50 per cent for the year, applied only to premiums that have been supplied; the reference asset pays nothing while it is held. With the units a contract covers left at one, every figure here is a figure a unit. With a real quantity in that box, every figure becomes a figure on a position, and that quantity is set by SEBI at sebi.gov.in.
The calculator is a smaller machine than most people expect, and the smallness is the point. There is no market feed inside it and no view about where the reference asset is going. Given rows and one price, it hands back arithmetic. Everything difficult about options sits outside that boundary, and this guide is mostly about where the boundary runs and what happens to a reader who forgets it is there.
Two formulas and a sign do the whole job. A call pays the price less its level wherever that difference is above nil, and nothing otherwise. A put pays its level less the price on the same terms. A written leg pays the negative of whatever the same leg bought would have paid. Applied leg by leg and added together, those three rules give the assembly. The three rules are why the output of a calculator like this is exact, and they are also why the same calculator can say nothing whatever about what any of it is worth.
What does an option payoff calculator actually compute?
The calculator computes one number for one named price: what each legOne contract sitting inside a larger position. A leg is complete on its own and keeps its own obligation however many other contracts are held beside it. pays at that price, and what the assembly pays in total. Name a different price and it computes a different number. The calculator is not solving for anything and not deciding anything: it is evaluating an expression that already existed the moment the contracts were written.
The word for what it returns is a payoffA gross amount, not a result. It cannot be negative for a leg that was bought, and it can be nil.: what the assembly pays at the end, before anything anybody paid for it is counted. The word for the other thing, the one this tool mostly refuses, is a profitA net amount, arrived at once money leaving at the start and money arriving at the end stand on one date.: that same payoff after every premium has been counted and moved to the same date. Two words, two numbers, and the label between them is the whole reason this guide exists.
A tariff card at a metered utility works the same way. The card states exactly what three hundred units cost and exactly what four hundred cost, and the arithmetic is beyond argument in both cases. The card cannot state how many units a household will use next month. A payoff calculator is that tariff card. The calculator converts a supplied price into an obligation with complete precision, and it holds no opinion at all about which price arrives.
| πi(ST) | the payoff of leg i at the end date, in rupees a unit |
| εi | the sign of leg i: plus one where the leg was bought, minus one where it was written |
| φi | the type of leg i: plus one for a call, minus one for a put |
| ST | the price of the reference asset at the end date, the one figure the reader supplies |
| Ki | the level written into leg i, read off the contract |
What does the form take, field by field?
Four fields on each leg and one price for the whole assembly. The panel above carries two more on each row. The premium changes nothing about the payoff, and the number of contracts only scales it. A reader who has met a pricing screen before will notice at once how much is missing from all of it.
The first field is the sign. Plus for a leg that was bought, minus for one that was written, read off the contract note rather than off the name anybody has given the pair. A row carrying the wrong sign does not describe a smaller version of the same obligation, it describes the opposite one. A bought call paying Rs 200.00/- becomes, with one setting changed, a written call costing Rs 200.00/-. The arithmetic stays correct, so every figure downstream inherits the error silently.
The second field is the type: call or put. The third is the levelSome markets call it the strike. It is a term of the contract, not a price and not a forecast., the figure written into that contract which the payoff is measured against. The fourth is the end dateThe date on which a leg finishes and its payoff is fixed. Before that date a contract has a value; on it, a contract has a payoff., on which that leg finishes and its payoff stops being a question. Then, once for the whole assembly rather than once a leg, the price of the reference asset at that date.
Beside each leg there is one optional box: the premium, where one is available. The panel above opens with one of the two filled and the other empty. The working notes behind this guide leave them in that state. Most of the honesty of this tool lives in what that box does and does not do, and the premium box has a section of its own below.
Where does each number on the form come from?
A field note is not a definition. A field note says where to find the number, not what the number means, and here the answers divide into two piles.
The first pile is the contract itself. The sign, the type, the level and the end date are terms of it, printed on the note or the position statement. The four fields are not worked out; they are copied. A statement showing a level and a date but no sign has supplied three fields of four, and the fourth is fetched rather than inferred from the name of the position.
The second pile is set by an authority and is deliberately not printed here. Which levels exist and how far apart they sit, the dates contracts run to, and the quantity one contract covers are all set by SEBI at sebi.gov.in. The panel above has a box for that last one and it opens at one unit, a placeholder rather than a value: until the real quantity goes in, every figure this tool returns is a payoff a unit of the reference asset rather than a payoff on a position.
The four levels drawn with here, Rs 1,600.00/-, Rs 1,800.00/-, Rs 2,200.00/- and Rs 2,400.00/-, are declared levelsA level chosen here to draw a shape, sitting at a stated distance from the spot price. It carries no premium, and it is not a level read off any venue.: placed at ten and twenty per cent either side of the spot price of Rs 2,000.00/-, standing in for nothing real. The one level carrying a premium here is Rs 2,000.00/-. The level equals the spot price because the pair struck there is struck at the money, and that is what at the money means.
Every assembly here is made of two shapes, so both are worth holding on to. A bought call is flat at nil and turns upward at its level, rising rupee for rupee with the price. A bought put is its mirror. Written legs are those two shapes reflected in the horizontal axis. Reflecting a picture is what a minus sign does to it.
The tool returns Rs 200.00/- for an assembly. What further figure is needed before that becomes an amount of money, and where does it come from?
Where does the price at the end date come from?
From the reader. The complete answer is that short, and the field is the one readers misunderstand most often. A box that accepts a number looks as though the machine behind it knows something about which number belongs there.
It does not. A price at the end date has to be named rather than derived. The arithmetic carries no distribution, no probability and no run of past prices to draw one from. The reference asset sits at a spot price of Rs 2,000.00/- and pays nothing while it is held. A payout during the holding period would change other arithmetic in this subject area, so the second half of that sentence matters.
The question being put to it is a conditional one, and it answers exactly that and nothing wider: if the price ends here, the assembly pays this. Asked about Rs 1,600.00/- it answers. Asked about Rs 2,600.00/- it answers too, with the same confidence. The confidence belongs to the arithmetic rather than to the price.
What does the tool require before it will return anything at all about what an assembly will pay?
A leg is written rather than bought, and at the end date the price sits well above its level. Before reading on, what sign does that leg's payoff carry?
What does the tool return once it runs?
Three things, and it is worth naming them separately because readers tend to look at the third and skip the first two.
First, each leg's payoff at the price supplied, one figure a leg. Second, the assembly's payoff: those figures added with their signs and nothing more elaborate. Third, a redrawn diagram with a marker at that price, so the number can be seen against every other number it could have been.
The total is the payoff of the assembly, and a payoff ignores what was paid to put the assembly on, so it is neither money made nor money lost. It is what the contracts oblige at that price. Whether it leaves anybody better off is a different calculation, and it needs the premiums.
Notice what the addition leaves alone: it does not merge the legs. Each is evaluated against its own level, with its own sign, and only then are the results added. Averaging two levels and evaluating once invents a third contract that nobody wrote, and it disagrees with the truth at almost every price.
| Π(ST) | the payoff of the whole assembly at the end date, in rupees a unit |
| πi(ST) | the payoff of leg i, from the expression above, with its sign already applied |
| n | the number of legs in the assembly, two on the worked default here |
Why is there a premium field, and why can it be left empty?
Because the reader may have the premiums and this calculator does not.
The calculator does not price a leg, and the refusal is structural rather than a matter of taste. Producing a premium needs a figure for how far the reference asset might move over the life of the contract, and no such figure exists in the working notes this guide is built on. Exactly two are available, both at one level and one end date: a call at Rs 180.00/- and a put at Rs 57.93/-, given rather than derived. For any other level, the declared Rs 2,200.00/- included, there is no premium and no honest way to manufacture one.
So the box sits empty, waiting for a reader with their own contract note. The rule is one line: with a premium supplied for every leg the tool draws a profit line, and with any one missing it draws the payoff and stops. A blank is a fact about what is known. A plausible number in the same box is a fact about nothing at all, and the two look identical once typed.
The empty box has a visible effect on the assembly the panel opens with. Leg one sits at a level carrying a premium and leg two does not, so no profit line appears, and the panel names leg two rather than leaving the reader to wonder whether it is broken. With leg two set to nil contracts every leg still in play has a premium beside it, so the line appears at once.
Why is the premium box optional rather than required?
How is the output read without turning it into money made?
Take the simplest possible case, one leg, and watch two numbers separate that a careless reading would treat as one.
The leg is plus one call at Rs 2,000.00/- for one year. The price at the end date is Rs 1,600.00/-. The price finished below the level, so the expression returns nil rather than a negative and the tool returns a payoff of Rs 0.00/-. Nothing is owed by anybody. Rs 0.00/- is the complete payoff answer.
Now suppose the contract note shows the premium actually paid. In this invented example the premium is Rs 180.00/-. The premium left at the start of the year and the payoff arrives at the end of it, so the two cannot be compared until they stand on one date. Carry Rs 180.00/- forward one year at 6.50 per cent for the year and it becomes Rs 191.70/-, so the profit on the same leg at the same price is a loss of Rs 191.70/-. The Rs 11.70/- of difference is the financing on the premium, and nothing more.
Two numbers, one leg, one price: a payoff of Rs 0.00/- and a profit of minus Rs 191.70/-. The only thing separating them is the label, and a reader who copies the payoff into a row headed profit has made no arithmetic error that checking the arithmetic will ever catch.
Run the same leg at Rs 2,400.00/- and the separation stays the same width: payoff Rs 400.00/-, carried premium still Rs 191.70/-, profit Rs 208.30/-. At Rs 2,130.00/- the payoff is Rs 130.00/- and the profit is a loss of Rs 61.70/-. The loss of Rs 61.70/- is the put premium of Rs 57.93/- carried at the same rate, agreeing to the paisa rather than exactly because the put figure is rounded. The put premium comes from a relationship settled separately, and is quoted only as a check that can be run independently.
| P(ST) | the profit of the assembly at the end date, in rupees a unit |
| Π(ST) | the payoff of the assembly, from the sum above |
| ci | the premium of leg i, supplied by the reader, never produced here |
| εi | the sign of leg i, so a premium received on a written leg enters as a negative |
| r | the financing cost, 6.50 per cent for the year |
Plus one call at Rs 2,000.00/-, with a supplied premium of Rs 180.00/-, and a price at the end date of Rs 1,600.00/-. What are the payoff and the profit?
With every premium box cleared and the price control moved, does a profit line appear on the diagram?
Drive the tool: one price in, every leg's payoff out
One control moves: the price of the reference asset at the end date. Every leg's line, the assembly's line, the marker and the level markers redraw together. The two buttons swap the assembly between the two worked defaults in this guide, and the premium boxes decide whether a profit line can be drawn at all.
Both ends of this control are declared settings, placed at thirty per cent either side of the spot price of Rs 2,000.00/-, and neither is a limit on anything a price could do.
At a price of Rs 2,130.00/- at the end date, leg one pays a payoff of Rs 130.00/-, leg two owes a payoff of Rs 0.00/-, and the assembly pays a payoff of Rs 130.00/-. The profit is not shown, because a premium is missing for leg two.
Educational illustration. Not a quotation, not a price, and not a prediction of any price. Assumptions on screen: one year to the end date; financing at 6.50 per cent for the year, used only where a premium has been supplied; the reference asset pays nothing while it is held; what one contract covers is set by SEBI at sebi.gov.in, so every figure is a figure a unit rather than a figure on a position; the level of Rs 2,200.00/- is declared for drawing and carries no premium.
Slide it from the left and watch the order things happen in. Both legs return nil below Rs 2,000.00/-, so nothing moves until then. From there to Rs 2,200.00/- only leg one is working, so the assembly climbs as fast as the price does. Above Rs 2,200.00/- leg two owes at the rate leg one pays, the leg lines fan apart, and the assembly line between them goes flat. The flat stretch is the first place a reader learns to distrust the leg they were looking at.
What does the tool open on, and what do those readings say?
Both instruments open on the same two-leg assembly: plus one call at Rs 2,000.00/- for one year, minus one call at Rs 2,200.00/- for one year, one contract on each leg, the units a contract covers left at one, and the price at the end date set to Rs 2,130.00/-. Three readings come back. Leg one pays Rs 130.00/-, leg two pays nothing, and the assembly's payoff is Rs 130.00/-. The premium of Rs 180.00/- on leg one is shown as paid. Leg two has no premium beside it, so no net premium and no profit are struck at all.
Every one of those figures is arithmetic rather than a quotation. Leg one pays Rs 2,130.00/- less Rs 2,000.00/-, or Rs 130.00/-. Leg two is written at Rs 2,200.00/-, the price finished below that level, so the expression returns nil and the sign has nothing to flip. Add them and the assembly pays Rs 130.00/-. Why no profit appears is printed beside it: Rs 2,200.00/- is a declared level and no premium exists for it here.
Here is the same assembly at six stated prices, written out so that this table survives with the tool switched off entirely.
| Price at the end date | Leg one pays | Leg two owes | Assembly payoff |
|---|---|---|---|
| Rs 1,600.00/- | Rs 0.00/- | Rs 0.00/- | Rs 0.00/- |
| Rs 2,000.00/- | Rs 0.00/- | Rs 0.00/- | Rs 0.00/- |
| Rs 2,130.00/- | Rs 130.00/- | Rs 0.00/- | Rs 130.00/- |
| Rs 2,200.00/- | Rs 200.00/- | Rs 0.00/- | Rs 200.00/- |
| Rs 2,400.00/- | Rs 400.00/- | Rs 200.00/- | Rs 200.00/- |
| Rs 2,600.00/- | Rs 600.00/- | Rs 400.00/- | Rs 200.00/- |
The second worked case separates the two words this guide keeps insisting are different, and it is the one place a profit can honestly be shown: one leg, plus one call at Rs 2,000.00/-, with the Rs 180.00/- premium this record does carry. With leg two set to nil contracts, the panel above reproduces the table below row for row. Across the payoff and profit columns side by side, the gap never changes.
| Price at the end date | Payoff | Premium carried to the end | Profit |
|---|---|---|---|
| Rs 1,600.00/- | Rs 0.00/- | Rs 191.70/- | minus Rs 191.70/- |
| Rs 2,000.00/- | Rs 0.00/- | Rs 191.70/- | minus Rs 191.70/- |
| Rs 2,130.00/- | Rs 130.00/- | Rs 191.70/- | minus Rs 61.70/- |
| Rs 2,400.00/- | Rs 400.00/- | Rs 191.70/- | Rs 208.30/- |
One detail in that second table is worth pausing on. The spot price of Rs 2,000.00/-, the level of Rs 2,000.00/- and the exposureThe value of the reference asset a contract is written against. Nobody has paid it and nobody is holding it. It is the base the payoff is measured from, not an amount at stake. of Rs 2,000.00/- a unit are three different quantities that happen to be the same number, for two separate reasons rather than one: the level matches the spot because this pair is struck at the money, and the exposure matches the spot because the contract is written against one unit at its spot price. Nobody has paid Rs 2,000.00/-, and reading the third as an outlay is the error the coincidence invites.
On the two-leg default at a price of Rs 2,400.00/- the tool prints Rs 400.00/- against leg one and Rs 200.00/- against leg two. What is the assembly's payoff?
The error that gets made: reading a figure a unit as a figure on a position
The tool returns Rs 200.00/-, and the reader writes Rs 200.00/- into a column headed amount at stake. Rs 200.00/- is neither an amount nor at stake. The figure is what the assembly pays for a unitPer single unit of the reference asset, rather than per contract. A contract covers some quantity of units, and that quantity is a term set by an authority rather than by the reader. of the reference asset, and what one contract covers is set by SEBI at sebi.gov.in and moves.
A caterer quotes a rate a plate for a wedding. The rate is exact, and it is not the bill. The bill needs the number of plates, that number comes from somebody else entirely, and nobody has ever mistaken a rate card for an invoice at a wedding because the two arrive on different sheets of paper. On a screen they arrive on the same one.
Who makes it: anybody who met payoff diagrams before they met contract specifications. Meeting the diagrams first is the ordinary order of learning, and almost every diagram they have ever seen was drawn a unit while almost none of them said so. What it costs: a position sized off a figure that is wrong by whatever one contract covers, in whichever direction. The reader has no way of noticing. The arithmetic on the screen is entirely correct, and the check they would run confirms it.
The fix is one habit. Read every figure this tool returns as a payoff a unit, and go to SEBI at sebi.gov.in for the quantity before multiplying anything by anything.
The panel at the top has a box for that quantity, and it opens at one unit. With a real quantity in it, every figure on the screen moves by exactly that factor while the figure a unit stays where it was. The whole of the error is produced in one keystroke.
In the panel at the top, with every leg left alone and the units one contract covers raised from one to fifty, what happens to the assembly payoff?
What is set by an authority rather than written here?
Four things this tool leans on are set by an authority, and the name and the site stand in place of every value. The four are what one contract covers and in what quantity, the figure that turns a payoff a unit into a payoff on a position; the levels at which contracts are made available and their spacing; whether a contract settles in cash or by delivery; and the dates a contract runs between. All four sit with SEBI at sebi.gov.in. Where the reference is a rate or a currency rather than an asset, the equivalent arrangements sit with the Reserve Bank of India at rbi.org.in.
Each of those rows is set by the authority named inside it, and each moves, so the value belongs at the source rather than in a fixed account of it. The four levels drawn here are declared geometry, and the levels actually made available sit with that authority instead. Every one is confirmed at the source before any figure from this guide touches a real position.
What does a payoff out of this tool not establish?
A reader who has driven the controls a few times arrives at a reasonable question: is this assembly worth holding? The calculator does not answer that question.
Here is what the tool has actually produced. A conditional arithmetic statement about one price: if the reference asset finishes at Rs 2,130.00/-, this assembly pays Rs 130.00/- a unit. Turning that into a decision would need three things, and not one of them is available here.
The first is a view on how far the reference asset might move and how likely each move is; the arithmetic behind it holds no distribution, no probability and no run of past prices, so there is no honest route to one. The second is the reader's own circumstances. No calculator can see them and none would be entitled to reason about them. The third is what the assembly costs to place, to hold and to unwind. Costing that needs the premium at the second level, and no such premium is available.
A calculator returning a decision instead of a payoff would be pretending to hold all three. Pretending is a far more damaging thing to build than a tool that stops early and says why. Signed primitivesA leg written out as a plus or a minus against one call or one put at one level and one end date. The word primitive means the row cannot be broken down any further. and two formulas can be checked by anybody with a pen. A recommendation cannot be checked by anybody at all.
The tool shows a payoff of Rs 200.00/- at the price entered. What has it established about whether to hold the assembly?
How does somebody work through a payoff on a position statement?
Four habits, each of them a consequence of something above rather than general caution. The most useful thing anybody does with a payoff figure is decide which of the four words it is, price, premium, payoff or profit, before writing it into any row at all.
- Copy the four fields off the contract, and go back for any one that is missingSign, call or put, level, end date, on every leg. A statement showing a name and two levels has supplied three fields on each row and left out the sign, and reconstructing the sign from the name of the position is guessing dressed as reading.
- Work each leg on its own before adding anythingOne expression a leg, evaluated at the price in question. Averaging two levels and evaluating once invents a contract that nobody wrote, and it will disagree with the truth at almost every price while looking neat at one or two.
- Label the answer a payoff, and leave the profit row empty until every premium is in handA payoff worked off the rows is exact. A profit needs every premium in the assembly carried to the end date, and one missing premium takes the whole profit line away rather than a part of it. A household that budgets around a figure it invented is in exactly the position of a position sheet doing the same thing, and both discover it at the same moment.
- Get the quantity before multiplying, and get it from the authorityEvery figure here is a payoff a unit. What one contract covers is set by SEBI at sebi.gov.in and moves, so the multiplication happens after that lookup rather than before it. This is the step people skip, because the figure on screen already looks like money.
Notice what none of those four steps is. None is a view about the price, and none ranks one assembly against another. A person can execute all four perfectly and still be no closer to knowing what to do. The method is being honest about which question it answers.
Why can a payoff be exact while a price stays out of reach?
The two halves of this guide stand on completely different footings. A payoff is arithmetic on terms already written down: the level and the sign are in the contract, and the price at the end date is a named number, so the answer is exact for the same reason that adding up a bill is exact.
A premium is not that. A premium is a figure about what might happen between now and the end date, and producing one needs an input describing how far the reference asset might move. The input is unavailable at every level, so the two premiums available are given rather than derived and no third can be conjured out of them.
Three things follow. A payoff can be computed at every price for every assembly here. A profit can be computed only where every premium is in hand. And filling the gap with a plausible figure would produce the most convincing wrong number in the whole subject. The invented figure would sit in a column of exact ones and be indistinguishable from them.
References
| Source | Document | Where |
|---|---|---|
| Securities and Exchange Board of India | Framework for what one contract covers and in what quantity, which is the figure that turns a payoff a unit into a payoff on a position and is the row a reader most wants filled; for the levels at which contracts are made available and the spacing between them; for whether a contract settles in cash or by delivery; and for the dates a contract runs to | sebi.gov.in |
| Reserve Bank of India | The equivalent arrangements where the reference is a rate or a currency rather than an asset | rbi.org.in |
| International Organization of Securities Commissions | The place cross-border conduct principles sit | iosco.org |
| arXiv Quantitative Finance | Preprint repository for the standard statement of payoff expressions and for the separation between a payoff and a profit | arxiv.org |
| Social Science Research Network | Working paper repository for the same material, for notation and structure | ssrn.com |
The reference asset, its spot price of Rs 2,000.00/-, the financing cost of 6.50 per cent for the year, the level of Rs 2,000.00/- and the declared levels of Rs 1,600.00/-, Rs 1,800.00/-, Rs 2,200.00/- and Rs 2,400.00/- are invented.
Educational material. Not advice on any investment, tax, budget or market position.