The Call Option: The Right to Buy, and What It Costs

A call option is the right to buy the reference asset at a fixed strike, and the matching obligation to sell sits on the writer. The buyer hands over a premium on day one and decides at the end, once the price is known. Above the strike the buyer buys. Below it the buyer walks away, and the strike is never paid at all.

Everything that follows comes from one feature of that arrangement, and it is worth naming before any arithmetic starts. The buyer decides last. Not first, not halfway through, and not on the strength of any view about where the price is heading. Last, with the price already sitting on the table. A contract in which one party gets to make a choice after all the uncertainty has resolved is a strange object, and every other feature of a call is a consequence of trying to make that object fair to the party on the other side of it.

Three consequences follow. The writer is bound while the buyer is not, so the writer has to be paid something at the start, and that payment is the premium. The buyer will only ever act when acting helps, so the line describing what the contract pays has to bend at the level where acting stops helping. And a party who may walk away cannot be pursued for anything, so the buyer never owes a rupee beyond the premium, whatever the reference asset does. Three features, one cause.

Every figure worked below belongs to a single invented pair on a single invented reference asset. The reference asset has a spot price of Rs 2,000.00/-, financing costs 6.50 per cent a year, the contract runs for one year, and the reference asset pays nothing at all to whoever holds it over that year. A payout arriving during the year would change several of the numbers below, and no such payout exists on this reference asset. The condition matters more than it looks.

What is a call option actually giving its buyer?

A call optionA contract giving one side the right to buy a named reference asset at a fixed level, and giving the other side no right at all. gives its buyer the right to buy the reference asset at Rs 2,000.00/-, and it gives the writer no right whatsoever. Two words in that sentence are carrying the weight. The first is RIGHT. A right means the buyer may buy and is never required to. The second is FIXED. The level was settled the day the contract was made and does not move afterwards. Nothing the reference asset goes on to do can shift it. A call is a right, held by one side only, exercisable at a level that cannot be changed by anybody.

The fixed level has a name. The name is the strikeThe level written into the contract at the start, against which the choice at the end is measured. A strike does not move once the contract exists., and on this invented pair the strike is Rs 2,000.00/-. Notice that the spot price of the reference asset today is also Rs 2,000.00/-. The spot price and the strike agree on purpose rather than by accident. A contract struck where the two are equal is struck at the moneyWhere the strike written into the contract and the current price of the reference asset are the same number., and that is precisely what the phrase at the money means. Printing both figures without saying so would leave a reader wondering whether one number had been copied into the other by mistake.

The everyday version is worth carrying through the whole of what follows. A vegetable stall outside one office building takes a token from a customer at the start of the season. In exchange, the stall agrees that on any day the customer turns up, a crate may be bought at Rs 400/-. If the market goes to Rs 600/- the customer turns up and pays Rs 400/-. The figure was fixed when the token was left, so the stall owner cannot raise it. If the market falls to Rs 250/- the customer buys at Rs 250/- from anybody and never goes near the stall, and nobody can compel otherwise. The customer has a right and the stall has a duty, and the token is what bought the difference between the two.

Three things translate across into a call. The token is the premium, Rs 180.00/- on this pair. The stall is the writer, the side that took the payment and stayed bound. The agreed figure of Rs 400/- is the strike, Rs 2,000.00/- here. Nothing else about the arrangement needs translating, and that is a fair signal the contract is less exotic than its reputation.

How does a call option work, from the premium to the end?

How Call Options Work, the four steps a single contract runs through

Set the shapes and the diagrams aside for a moment and just watch one contract run from beginning to end. There are four steps, and each of them is a thing that physically happens rather than a thing somebody thinks.

Step one. The premium of Rs 180.00/- passes from the buyer to the writer. Handing over Rs 180.00/- is a real payment, it happens on day one, and it happens whatever the rest of the year turns out to hold. The premium never comes back. There is no clause anywhere in the contract under which the buyer receives it again, no matter what the reference asset does, and no matter which way the choice at the end goes.

Step two. Nothing happens. The contract sits there for a year. Neither party pays the other anything. The reference asset moves around, and the two sides watch it move, and watching is the whole of the activity. Step two feels like filler and it is not. Most of a contract's life is step two, and a reader who imagines money changing hands during it has misread the arrangement.

Step three. The year ends and the reference asset has a price. Whatever that price is, it is now a fact rather than a forecast. Nobody has to guess. Step three is what makes step four so simple, and step three is the one readers skip over fastest.

Step four. The buyer, and only the buyer, compares that price to the strike of Rs 2,000.00/- and either buys at the strike or does not. To use the right is to exerciseThe act of using the right the contract carries, rather than letting it run out unused. it. To leave it unused is to let it lapseWhat happens to a right nobody uses by the deadline: it simply stops existing, and nothing is owed either way.. The writer has no part in this at all and is told the outcome rather than consulted about it.

Two of those four steps move anything, and the party who acts at step four is not the party who acted at step one. The crossover is the mechanism, drawn in the picture below. The buyer pays and then waits. The writer receives and then waits. And at the end the waiting stops for one side only, the side that has been holding the choice all along.

What exactly does the buyer of a call decide, and when?

The decision is far smaller than most readers expect, and the reason is step three. By the time the buyer has to choose, the price of the reference asset is known. There is nothing to weigh, no view to form, and no forecast to make. The buyer buys if the price is above Rs 2,000.00/- and does not if it is below. The comparison is the entire test, and it has one input.

The premium is not part of that decision. The premium was paid at the start, it does not come back either way, and pushing it into the choice at the end is how people throw away money they still have. An amount already paid that no later decision can recover is sunkAlready paid, and unrecoverable by any decision taken afterwards. A sunk amount is the same whichever way the later choice goes, so it cannot tip that choice., and the premium here is sunk from the moment it leaves the buyer's hands.

Back to the stall. The token left at the start of the season is gone whether or not the customer turns up with a bag. The token is gone if the crate is bought, gone if the customer never returns, and gone if the stall burns down. So when the customer stands in the road on a Tuesday working out whether to walk over and buy at Rs 400/-, the token has nothing to say. The only thing worth looking at is what a crate costs elsewhere that morning. An amount that is identical under both branches of a choice cannot possibly help anybody pick a branch.

A lot of otherwise careful readers go wrong at exactly this point, and being blunt about why is worth the space. Those readers have just learned, correctly, that what a contract pays is not the same as what the position earned. Having learned it, they try to apply it one step too early, at the moment of exercise, where it does not belong. The full cost of that mistake is worked out further down, in exact rupees.

What does this call pay at each price of the reference asset?

The payoff is what the contract pays at the end, before anything paid to hold it is counted. For a call it is the price less the strike wherever the price is above the strike, and Rs 0.00/- wherever it is not. Take four prices and work the payoff at each, and the shape falls out on its own. The picture below confirms the arithmetic rather than carrying it.

At Rs 1,600.00/- the payoff is Rs 0.00/-. Nobody exercises a right to pay Rs 2,000.00/- for something available at Rs 1,600.00/-. At Rs 2,000.00/- the payoff is also Rs 0.00/-. Buying at the strike and buying in the market cost the same, and neither branch is worth anything over the other. At Rs 2,130.00/- the payoff is Rs 2,130.00/- less Rs 2,000.00/-, or Rs 130.00/-. And at Rs 2,400.00/- the payoff is Rs 2,400.00/- less Rs 2,000.00/-, or Rs 400.00/-.

The payoff is never below nil, and that is not a convenience of the arithmetic but the right to walk away written as a number. The floor at nil is worth sitting with. A reader either understands the floor and understands options, or spends a year understanding neither. The formula does not clip at nil because somebody chose a tidy formula. A party who may decline cannot be made to pay out, so the formula clips at nil and no price of the reference asset can push the payoff below it. Change the contract so the buyer must buy, and the flat part disappears immediately and the line runs straight down through nil. The flat part is the right, drawn.

The price of Rs 2,130.00/- in that list is not arbitrary either. Rs 2,130.00/- is the forward price of this reference asset: Rs 2,000.00/- carried for one year at 6.50 per cent a year, or Rs 2,000.00/- plus Rs 130.00/-. The forward price is where the call and the arithmetic of carrying the reference asset meet, and the cross-check further down needs it.

What does the call earn, once the premium is counted?

The payoff answers what the contract pays. The payoff counts nothing that was paid to have the position in the first place, so it does not answer what the position earned. Subtracting what was paid gives the profit, a different line with a different name and a different sign at several prices.

Doing the subtraction properly needs one small step that most treatments skip. The premium was paid at the start of the year and the payoff arrives at the end of it, so setting one directly against the other compares two amounts standing at two different dates. Carry the premium forward to the date the payoff turns up and the comparison becomes honest. At 6.50 per cent a year for one year, Rs 180.00/- becomes Rs 180.00/- plus Rs 11.70/-, or Rs 191.70/-. Rs 191.70/- is the financed premiumThe premium carried forward at the financing cost to the date the payoff arrives. Both amounts then stand at the same date., and it does the work for everything that follows.

Now run the same four prices through the second line. At Rs 1,600.00/- the payoff is Rs 0.00/- and the profit is Rs 0.00/- less Rs 191.70/-, or minus Rs 191.70/-. At Rs 2,000.00/- the payoff is again Rs 0.00/- and the profit is again minus Rs 191.70/-. At Rs 2,130.00/- the payoff is Rs 130.00/- and the profit is Rs 130.00/- less Rs 191.70/-, or minus Rs 61.70/-. And at Rs 2,400.00/- the payoff is Rs 400.00/- and the profit is Rs 400.00/- less Rs 191.70/-, or Rs 208.30/-.

The same amount came off each time, and that gives the shape of the whole thing: the profit line is the payoff line moved down by one constant, and the constant is Rs 191.70/-. Not a proportion, not something that widens as the price rises, not something that closes up as the position improves. One fixed drop, applied identically at every price on the axis. The two lines therefore run exactly parallel wherever the payoff line runs, they never converge, and they never cross each other. Anybody who has drawn them once and seen that will never again read a number off one line and describe it as the other.

At what price does the buyer of this call stop being behind?

The profit line crosses nil at Rs 2,191.70/-. Getting there is one line of arithmetic. Set the profit to nil. The payoff then has to equal the financed premium, so the price has to be the strike plus the financed premium. Rs 2,000.00/- plus Rs 191.70/- is Rs 2,191.70/-. The level has a name, and the name is the break-evenThe price at which a position's profit line crosses nil. A break-even is worked out from the contract, and it is not a statement that any price will be reached..

Most sources will print Rs 2,180.00/- instead, and the whole of the difference between the two figures is one year of financing on the premium at 6.50 per cent a year. Rs 2,180.00/- is the strike plus the premium exactly as paid, Rs 2,000.00/- plus Rs 180.00/-. Rs 2,191.70/- is the strike plus the premium carried to the date the payoff arrives. The gap is Rs 11.70/-, being Rs 180.00/- at 6.50 per cent for one year, and nobody has made a mistake. The gap is a choice about whether to compare two amounts standing at the same date or at two different ones. The smaller figure turns up elsewhere, so both appear here, each labelled for what it is. Knowing what was left out beats assuming that one of the two must be wrong.

A break-even is a level, and a level is not a forecast. Nothing in that calculation says the reference asset will reach Rs 2,191.70/-, or Rs 2,180.00/-, or any other figure. The break-even answers one question about the contract, where would this position be level. The break-even says nothing about a question concerning the world, where will the price go. The two questions are constantly confused, usually by people who have just worked out a break-even and feel they have learned something about the future. Such people have learned something about the contract instead.

The control opens at Rs 2,130.00/- and reproduces the worked example above: a payoff of Rs 130.00/-, a profit of minus Rs 61.70/-, and a flag reading buy. Rs 2,130.00/- is the uncomfortable setting. At that one price the flag says buy while the profit is still negative, and holding the two facts together is the single hardest thing about a call. Dragging the control down through Rs 2,000.00/- shows what flips and what does not. The flag flips at the strike. The profit line does not reach nil until Rs 2,191.70/-, a full Rs 191.70/- further along the axis. Between those two stationary markers the buyer exercises and is still behind, and that band of prices is where almost every misreading of a call is made.

Has the buyer of a call bought the reference asset?

No. The buyer has bought a right over the reference asset. A right is a different holding with different contents, and separating the two is the most useful distinction here for anybody who reads position reports. The buyer has paid Rs 180.00/- and holds a right written on Rs 2,000.00/- of reference asset. The Rs 2,000.00/- is exposureThe value of the reference asset a contract is written on. Exposure is a size, not a payment, and neither side has handed it over.: a size, not a payment, and an amount nobody has paid or received.

Rs 2,000.00/- has now turned up three times, and each appearance means something different. The figure is the spot price of the reference asset today. The figure is also the strike, equal to the spot because this pair is struck at the money. And the contract covers one unit at the strike, so the figure is the exposure the right is written on as well. Three quantities, one figure, and they agree for reasons rather than by coincidence. Anybody printing all three side by side has to say which is which or the reader will assume one was copied into the other.

Three things follow. Each one is something the buyer of the call does not have. The buyer receives nothing that the reference asset produces while it is held. On this invented reference asset the produce is nothing at all. The reference asset pays nothing over the year. The buyer decides nothing about the reference asset and has no say in anything concerning it. And the buyer has no duty to find Rs 2,000.00/- at any point, unless and until the buyer chooses to buy. Nobody can force that choice.

A right over a thing is not the thing. In stall terms once more: leaving a token with the vegetable stall does not make the customer the owner of a crate. The crate cannot be eaten and it cannot be sold. The figure was fixed, so the customer gets no benefit when the stall raises its display prices. The customer holds a claim that may or may not be used. The claim is worth precisely what it would cost somebody else to stand in that position, and that is a different quantity from the value of a crate.

Can the premium of this call be worked out rather than given?

Not from a model. The Rs 180.00/- is a given figure in an invented example, and no model for producing it appears here.

Working a call premium out from scratch would require two things the worked example does not supply: how far the reference asset might move before the end of the year, and how likely each of those moves is. The working example contains no figure for the first, and the word volatility is the name for that figure. For the second it carries no distribution, no probability and no record of what actually happened. Supplying one of those in order to look complete would mean inventing it, and every other figure resting on the premium would then be resting on the invention.

Something can still be checked, exactly, and it needs no view about the reference asset at all. Waiting until the end of the year to pay the strike is worth something on its own, and the amount is the strike less its value today. Bringing Rs 2,000.00/- back one year at 6.50 per cent a year gives Rs 1,877.9343/-, so the gap is Rs 122.0657/-. Rs 122.0657/- is pure financing arithmetic, and it separates this call's premium from the premium of the matching contract pointing the other way.

There is a second check, and it runs entirely on figures already given above. At the forward price of Rs 2,130.00/-, this call's payoff of Rs 130.00/- against a financed premium of Rs 191.70/- leaves a profit of minus Rs 61.70/-. The matching contract's premium of Rs 57.93/- carried forward one year at 6.50 per cent a year comes to Rs 61.695/-, or Rs 61.70/- to the paisa. Two contracts, two routes with no step in common, and the same figure at the end of both. Consistency looks exactly like that when nothing has been priced from scratch. The arithmetic that ties the two premiums together in full is covered separately.

How does anybody use this outside a classroom?

Three readers use the two lines for three different purposes, and none of them is deciding whether to take a position.

Somebody reading a position report meets two columns that both look like money and are not the same kind of thing. One column carries premiums, amounts that actually moved between two parties on a date. The other carries the value of what the contracts are written on. Exposure is that value, and nobody paid it. Adding the two together, or reporting the second where a reader is expecting the first, describes an arrangement as roughly eleven times bigger than the cash that changed hands on it. On this single contract the ratio is Rs 2,000.00/- against Rs 180.00/-, so the mistake is not subtle once somebody knows to look for it. The habit that prevents it is dull and reliable: before any figure is written down, it is named out loud as a premium, a payoff, a profit or an exposure.

An analyst handed a break-even runs one check on it before using it. Was the premium carried forward to the date of the payoff, or set against it as paid? On this contract the answer moves the figure between Rs 2,180.00/- and Rs 2,191.70/-, a difference of Rs 11.70/-. The gap sounds small until the same convention is applied across a schedule of contracts and the errors all point the same way. The check takes one line: take the quoted break-even, subtract the strike, and see whether the remainder is the premium or the premium plus its financing.

A lender looking at a borrower who has bought calls reads for one thing first, the difference between holding a right and holding the reference asset. The borrower has paid a premium and holds a right. The borrower has not bought the reference asset, receives nothing it produces, and has no duty to find the strike unless the borrower chooses to exercise. Whether that last point is a comfort or a complication depends entirely on what the borrower would do with the right, and that is a question about the borrower rather than about the contract.

And the household version is the token again, held from the other end. Anybody who has paid a booking amount to hold a price on something they were not sure they wanted has bought a call, and already knows two of the three things set out here. The booking amount is gone whether or not the purchase goes through. The price locked in cannot be moved. The third thing, the one people usually get wrong, is that on the day of the decision, the booking amount already paid should play no part in it at all.

Should a reader who now understands this buy one?

Understanding what an obligation says is a different exercise from deciding whether to take it on, and finishing the first does not begin the second. No description of a contract can settle whether somebody should hold it, and the arithmetic above never tries.

Three things would have to be known before anybody could answer, and not one of them appears above. The first is a view on how far the reference asset might move before the end of the year and how likely each move is, exactly the figure the worked example does not supply. The second is the reader's own circumstances, known to the reader and to nobody else. The third is what the position would cost to hold all the way to the end and what it would cost to close before then, neither of which is worked out.

The payoff line above describes what the two sides owe each other at each price, and a description of an obligation is not a reason to take one on. The distinction is worth carrying well beyond a call. A diagram of a contract says what happens at every price. A diagram says nothing at all about which price will arrive, and it was never built to.

What does the whole worked call look like in one place?

Everything below comes from four inputs and nothing else: a spot price of Rs 2,000.00/-, a strike of Rs 2,000.00/- struck at the money, financing at 6.50 per cent a year for one year, and a call premium of Rs 180.00/- that was given rather than computed. The reference asset pays nothing while it is held. The financed premium of Rs 191.70/- does not depend on where the price ends up, so it is the same in every row.

Read the fourth row and the fifth row together and the two break-evens stop being an argument. At Rs 2,180.00/- the payoff has caught up with the premium as paid, and the profit is still minus Rs 11.70/- because the premium has been costing something all year. At Rs 2,191.70/- the payoff has caught up with the premium including that financing, and the profit is Rs 0.00/-. Then read the second row and the third row: the buyer walks away in one and buys in the other, and the profit is negative in both. The flag and the break-even are two different markers for exactly that reason.

The reference asset, the contract, the strike and the premium are invented.
Educational material. Not advice on any investment, tax, budget or market position.