Call and Put Options: Two Rights, Opposite Directions

At one and the same strike, a call is a right to buy and a put is a right to sell. Both rights sit with the buyer, both cost a premium on day one, and both make the payoff line bend at that one level and nowhere else. The two point opposite ways. Hold a bought call against a written put at that strike and no choice is left in the position at all.

Every difference between the two comes from which way the right points. Everything they have in common comes from the plain fact that it is a right and not a duty. One side may walk away on both contracts, and it is the same side both times. The shared right and the opposed direction are why the two look like opposites from the outside and behave like close relatives the moment the arithmetic is written down.

Four different kinds of number run through both contracts, and it is worth pinning them down before any of them arrives. A PRICE is what the reference asset costs, Rs 2,000.00/- today. A PREMIUM is money that genuinely moves from buyer to writer at the start. A PAYOFF is what the contract hands over on the end date, counting nothing that was paid earlier. A PROFIT is that payoff once the premium has been brought forward to the same date and taken off. Confusing the last two is the mistake this entire subject turns on.

One more thing before the contracts. Spot Rs 2,000.00/-, strike Rs 2,000.00/-. The same figure twice, and not by accident. Nobody copied one into the other. The pair was struck level with what the reference asset costs today, and being struck level with what the reference asset costs today is what the phrase at the money describes. The reference asset hands its holder nothing across those twelve months, and financing costs 6.50 per cent a year.

What exactly is a call option?

Whoever buys a call pays Rs 180.00/- on day one and holds one right in return: on the end date they may BUY the reference asset for Rs 2,000.00/-. On the end date the buyer looks at what the reference asset is worth, and decides. If it is worth more than Rs 2,000.00/-, the buyer uses the right and takes it at Rs 2,000.00/-. If it is worth less, the buyer leaves the right alone and buys nothing.

For as long as this contract runs, whoever bought it holds the right and whoever wrote it carries the duty, and nothing that happens later swaps those two round. The counterpartyWhoever stands on the other side of a contract and owes the holder whatever that contract says they owe. on a call, the writer, took the Rs 180.00/- at the start and from that moment has no say at all. If the buyer wants the reference asset at Rs 2,000.00/-, the writer delivers it. The writer's whole position is that: money at the start, no choice afterwards.

A mall with ten shops in it makes the same shape. A tenant pays the mall owner a token now to hold one particular shop at a fixed rent for the whole festive season. Come the season, the tenant looks at how the street is trading. If it is busy the shop gets taken at the rent fixed earlier. If the street is dead the tenant walks away and the shop is somebody else's problem. The token is gone either way, and it was gone the day it was paid. The mall owner cannot come back in October and say the fixed rent no longer suits. The token is the premium, the fixed rent is the strike, and the mall owner is the writer.

Two habits of language matter from here on. The word price already has a job: it belongs to the reference asset. So the Rs 180.00/- is always called the premium and never the price of the option. And the Rs 2,000.00/- that the contract is written on is EXPOSURE. Exposure is the amount the contract references, the figure every payoff is measured against, and it is not money that either side has paid or received.

And what exactly is a put option?

Read this part against the last one rather than after it. Whoever buys a put pays Rs 57.93/- on day one and holds one right in return: on the end date they may SELL the reference asset for Rs 2,000.00/-. On the end date the buyer looks at what the reference asset is worth, and decides. If it is worth less than Rs 2,000.00/-, the buyer uses the right and sells at Rs 2,000.00/-. If it is worth more, the buyer leaves the right alone and sells nothing.

One word changed between those two paragraphs, from BUY to SELL, and nothing else changed at all. The right still belongs to the buyer. The obligation still belongs to the writer, who took the premium at the start and has no say afterwards. The decision on the end date still sits with the same side. With the two paragraphs laid on top of each other and held up to the light, one word shows through.

Parallel wording is what makes the reflection obvious. Keep the everyday version parallel on purpose. A stall holder is going into a season with stock to shift. She pays a bulk buyer a token now, and in return that buyer agrees to take the whole season's stock at a fixed price if she wants him to. Come the end of the season she looks at the street. If nobody is paying, she hands the stock over at the fixed price and the bulk buyer must take it. If the street is paying well, she sells to the street and the bulk buyer hears nothing from her. She decides. He does not. The token is gone either way, and it was gone the day she paid it.

Notice which person is the buyer in that second story. The buyer is the stall holder, and she is selling stock. On a put, the person who holds the right is the person who wants to sell the thing, and the person who wrote the contract is the person who may end up having to buy it. The inversion is the single commonest place a reader loses the thread, and it is worth reading twice: the buyer of a put is a would-be seller of the reference asset.

What do the two contracts actually share?

Five things, and they are listed flatly because every one of them is a difference a reader was about to imagine. On both contracts the right belongs to the buyer. On both, the obligation belongs to the writer and cannot be handed back to anybody. On both, a premium changes hands on day one and is never returned. On both, the payoff line bends at the strike and bends nowhere else. And on both, by the time the decision is made the price is already sitting there to be read, so the buyer's decision on the end date is mechanical rather than a judgement.

The mechanical decision is the least expected of the five, and it deserves a sentence of its own. There is no skill in the exercise decision. On the end date the reference asset is worth what it is worth, and the buyer compares it with Rs 2,000.00/-. A clerk with a ruler could do it. Everything interesting about these contracts happened earlier, when the premium was agreed. The two contracts are the same machine facing different directions.

Where do a call and a put genuinely differ?

Two things, then, and not five. The first is the one everybody already has: direction. A call's payoff rises as the reference asset rises and lies flat below Rs 2,000.00/-. A put's payoff does the reverse, rising as the reference asset falls and lying flat above Rs 2,000.00/-. Two mirror images through the same vertical line.

The second difference is the one worth stopping on, and most readers have never been told it. The stretch beyond the bend is not the same length on the two contracts. No level is barred to the reference asset, so above Rs 2,000.00/- a call's payoff has no arithmetic ceiling. The reference asset cannot fall below nil, so below Rs 2,000.00/- a put's payoff runs out. A put written at Rs 2,000.00/- can never hand over more than Rs 2,000.00/-, and it gets that far only if the reference asset became worthless.

The difference in length is a statement about arithmetic and about nothing else. It is not a claim that one end is more likely than the other. Nothing in the arithmetic attaches a probability to either end. The put's ceiling exists because a price has a floor at nil; the call's absence of a ceiling exists because a price has no matching roof. Neither sentence says where the reference asset is going. A payoff line states who owes what at each closing price and stops there.

What does each contract oblige its writer to do?

The writer's side is the half most comparisons leave out, and it is where the two contracts are least alike. Turn both of them round and look from the writer's side.

The writer of the call must deliver the reference asset at Rs 2,000.00/-, however far above Rs 2,000.00/- it has gone. If it stands at Rs 2,400.00/- on the end date, the writer hands it over for Rs 2,000.00/- and is Rs 400.00/- worse off on the contract. The arithmetic puts no ceiling on that number, in the same way and for the same reason that it puts no ceiling on the call buyer's payoff. The two are the same statement read from opposite ends.

The writer of the put must take delivery of the reference asset at Rs 2,000.00/-, however far below Rs 2,000.00/- it has fallen. If it stands at Rs 1,600.00/- the writer pays Rs 2,000.00/- for something worth Rs 1,600.00/- and is Rs 400.00/- worse off. Here the arithmetic does set a limit, and the limit is Rs 2,000.00/- a unit, reached if the reference asset went to nil and the writer paid Rs 2,000.00/- for nothing.

Neither of those two sentences is a ranking, and it is worth saying why in the same breath. They describe two obligations of different shapes. Which of them turns out heavier depends entirely on where the reference asset actually goes, and that is settled by the market rather than by either contract. A bounded obligation is not a smaller obligation; Rs 2,000.00/- a unit is a real number and a large one. Each obligation is defined at every level, and neither definition says which level arrives.

Where does each contract break even?

Break-even is where the buyer's profit reaches nil. The buyer has to earn the premium back on either contract. On a call that puts the level at the strike with the premium added on, and on a put at the strike with the premium taken off. Rs 2,000.00/- plus Rs 180.00/- is Rs 2,180.00/-. Rs 2,000.00/- less Rs 57.93/- is Rs 1,942.07/-. Almost every diagram prints those two figures, and both are incomplete.

Here is what it leaves out. The premium walked out of the buyer's hand on day one. The break-even is read on the end date, twelve months later. Between those two dates the money the buyer paid could have been sitting somewhere earning the financing rate, so the honest comparison brings the premium forward to the same date the payoff is read on. Financing takes the Rs 180.00/- to Rs 191.70/-, and the Rs 57.93/- stands at Rs 61.70/- by then.

So each contract has two break-even readings, and the gap between them is the same thing on both: the financing on the premium. The call reads Rs 2,180.00/- with the premium counted flat and Rs 2,191.70/- with it carried, a gap of Rs 11.70/-. The put reads Rs 1,942.07/- flat and Rs 1,938.30/- carried, a gap of Rs 3.77/-. The two gaps are different sizes only because the two premiums are different sizes; the mechanism is identical. Notice which way each one moves, too. The call buyer needs a higher price to cover a larger cost, so financing pushes the call's break-even further away. Financing pushes the put's break-even further away as well, and for the put further away means lower.

Are the two premiums independent figures?

The tie between the two premiums turns a comparison into a subject. A call at Rs 180.00/- and a put at Rs 57.93/- have both appeared above, and the natural assumption is that somebody priced the two separately and these are what came out. The two premiums cannot have been priced separately. At one strike, on one end date, on one reference asset, the call premium less the put premium is fixed by arithmetic alone, and there is no room in it for anybody's view about anything.

The tie runs as follows. The call premium less the put premium has to leave the gap between today's price of the reference asset and the present value of the strike. Dividing Rs 2,000.00/- by one plus 6.50 per cent puts that present value at Rs 1,877.9343/-, and the gap itself at Rs 122.0657/-. Set against the two premiums given above: Rs 180.00/- take away Rs 57.93/- is Rs 122.07/-. The two routes land 0.43 of a paisa apart, so the agreement is TO THE PAISA rather than exact, and the reason is simply that the put was rounded to a figure a person could use.

So what is Rs 122.0657/- actually made of? Rs 122.0657/- is discountingTaking an amount that falls due later and working out what it is worth in today's money, by dividing it by one plus the rate for the period. doing its ordinary work. The call buyer has arranged to pay Rs 2,000.00/- twelve months from now rather than today. When money costs 6.50 per cent a year, postponing a payment of Rs 2,000.00/- by twelve months is worth Rs 122.0657/-, and the gap is accounted for down to the last paisa. The gap is a financing figure, not a directional one. The tie survives even though neither premium can be priced from scratch. Pricing either one on its own needs the distance the reference asset could cover, and the working example supplies no such distance, yet the DIFFERENCE between the two needs no such input at all.

What happens when a call is bought and a put written at the same level?

Everything so far has been walking towards this. Hold two positions at once: a bought call struck at Rs 2,000.00/-, and a written put struck at Rs 2,000.00/-. Same reference asset, same end date, same level. Now walk the price of the reference asset up and down the scale, and read the pair's payoff at each point.

Below Rs 2,000.00/- the call is worthless and gets left alone. The written put is used against its writer and costs the shortfall. Above Rs 2,000.00/- the bought call is used and pays the excess. The written put is worthless and nobody troubles the writer about it. At every single price one of the two contracts is being used and the other is idle, so there is no price at which the pair does nothing, and that means the pair has no choice left in it at all.

Read the last two columns against each other. The two columns are identical. Every figure the pair produces is simply the price on the end date less Rs 2,000.00/-, with no exception anywhere on the scale and no approximation in it. And an agreement to buy the reference asset at Rs 2,000.00/- on that date pays exactly the price less Rs 2,000.00/- as well. Two positions that look nothing alike on a screen turn out to be one obligation. The word for a holding assembled this way is a syntheticA holding assembled out of other contracts, which ends up behaving like a contract nobody ever signed..

Which of these figures are payoffs, and which are profits?

The last part produced four figures that are neither, so they need naming before anybody carries them anywhere. Minus Rs 400.00/-, Rs 0.00/-, plus Rs 130.00/- and plus Rs 400.00/- are PAYOFFS. A payoff ignores every rupee that moved on day one. A payoff is the raw amount the contract hands over on the end date and no more.

Turn them into profits and the net premium has to come into the sum. Assembling the pair costs Rs 180.00/- for the call and brings in Rs 57.93/- from writing the put, a net premium of Rs 122.07/- paid at the start. Push that Rs 122.07/- through twelve months of financing at 6.50 per cent a year and it arrives at Rs 130.00/-. Now look at what that figure is. Apply 6.50 per cent a year to Rs 2,000.00/- and the financing comes to Rs 130.00/-. Set that on top of the spot and the forward price lands at Rs 2,130.00/-. The financed net premium on the pair is precisely the amount by which the forward price sits above the strike, and that is arithmetic being forced rather than a lucky landing. It could not come out any other way: take the parity line and carry every term in it forward twelve months, and today's price becomes the forward price while today's worth of the strike becomes the strike itself.

Take Rs 130.00/- off each of the four payoffs and the profit line reads the price less Rs 2,130.00/- at every price: minus Rs 530.00/-, minus Rs 130.00/-, Rs 0.00/- and plus Rs 270.00/-. Drawn together, the two make two separate lines rather than one, with the financed net premium filling the whole of the space between them and nothing else in there at all. Notice where the profit line crosses nil: at Rs 2,130.00/-, the forward price on this reference asset. Options and forwards meet at exactly that price.

The whole pair, worked from three inputs

Three numbers produce every figure on both contracts and no others are needed. The reference asset costs Rs 2,000.00/-. Money is financed at 6.50 per cent a year, and both contracts run twelve months. And the call was quoted here at Rs 180.00/- rather than produced. The gaps in those three matter as much: nothing among them says a word about the distance the reference asset could cover, and the asset itself produces no cash for its holder along the way.

Two entries in that sheet are worth a second look because they land on the same figure. The call's profit at Rs 2,130.00/- is minus Rs 61.70/-, and the put's profit at Rs 2,000.00/- is also minus Rs 61.70/-. The match is not a coincidence. The parity line forces it. Financing the gap between the two premiums over twelve months produces Rs 130.00/-, and Rs 130.00/- is precisely what the call pays at the forward price. So the call's payoff there minus its financed premium has to come to the negative of the put's financed premium. The two land within the paisa, and they would sit on top of each other if the put had never been rounded for use.

And one line on what each of these figures IS. The Rs 180.00/- and the Rs 57.93/- are amounts that genuinely move between the two parties. The Rs 2,000.00/- that both contracts are written on is exposure, the value the contracts reference, and no payment of any kind arrives during the year to change it.

How does anybody use this away from a textbook?

Three uses, and none of them is a reason to hold either contract.

An analyst handed a sheet of premiums runs the parity line as a data check before anything else. Somebody sends over a call and a put on one reference asset, at one strike, on one end date. Subtract them. Hold that difference up against today's price of the reference asset with today's worth of the strike taken off it. If those two do not land within the paisa, then either a figure was typed wrong, or the two premiums came from different moments, or one of them is stale. The check costs ten seconds and catches errors that would otherwise travel silently through everything built on top. A relationship that looks academic on first meeting earns its keep in those ten seconds.

Second, a risk officer reads the two obligations as different SHAPES rather than as different sizes. The call writer's obligation has no arithmetic ceiling; the put writer's stops at the strike per unit. The shape difference drives how each position gets collateralised, and it is the reason the two are never treated as mirror images by anybody responsible for the collateral. The actual collateral figures are not the sort of thing to carry in the head: they are set under the framework of the Securities and Exchange Board of India (SEBI) at sebi.gov.in and they move.

Third, and this is the one that touches ordinary households. Somebody is shown a packaged arrangement and told it does something clever. The question to ask is simple: which of these two contracts is inside it, and who is the writer? If the arrangement contains a written put, somebody has taken a premium and accepted an obligation that runs to the strike per unit. Knowing that settles nothing about whether the arrangement is any good; it settles what to ask. A small business owner offered a fixed price for next season's output has been offered a written call in ordinary clothes. The same discipline applies.

So should the call be held, or the put?

No answer to that follows from anything above, and the reason has nothing to do with caution. Answering would take a view on where the reference asset is headed and how wide the spread of possibilities around that view is. Answering would also take the holder's own situation, what is already held and what could be afforded to be wrong about. And it would take the cost of getting into each contract and out of it again. None of those three follows from the arithmetic above. The first of them is what no premium can be produced without, and that is why both premiums were given rather than worked out.

Everything except that choice is settled, and the settled part is the more useful part to hold. Each contract's obligations, on both sides of it, are fixed. So is where each one breaks even, on two readings rather than one. So is the fact that the two premiums cannot be set independently, together with a one-line check for whether somebody's figures respect that. And a bought call against a written put at one level is not two positions but one, so anybody describing that pair as a clever combination is describing an agreement to buy at the strike, in longer words.

A payoff line is a ledger of obligations read across every closing price imaginable, and it stays silent on which of those prices actually turns up. Ranking the two contracts, saying which side of either did better, or calling any position safe, cheap or limited would each need an outcome and a way of weighing it. A payoff line supplies neither, and the ranking has nothing to stand on.

The reference asset and both premiums are invented.
Educational material. Not advice on any investment, tax, budget or market position.