The Put Option: The Right to Sell, and What It Costs

A put option is the right to sell one unit of the reference asset at a strike fixed in advance. The obligation to buy sits on the other side. A premium is paid by the buyer at the outset, and the choice is made at the end with the price already known. Below the strike the buyer sells at the strike. Above it the buyer walks away and sells nothing.

The machinery is already familiar. A contract with a right on one side and an obligation on the other, a payment at the start that buys the choice, and a decision at the end taken by whoever made that payment. All of that is covered under the call option, and none of it is rebuilt here. One word in the middle of the sentence changes. The right points the other way.

The single reversal is enough to turn the payoff line over, and enough to make a combination that reads, at a glance, like something being made safe. The combination is not being made safe. The combination is being changed in shape, and the change is paid for. The reversal has an arithmetic and so does its cost, and both are short enough to be worked out in the open rather than quoted.

What is a put option, and what does the right to sell actually let its buyer do?

A put optionA contract giving one side the right to sell the reference asset at a level fixed at the start, and binding the other side to buy it at that level if asked. gives its buyer the right to sell the reference asset at Rs 2,000.00/-, and gives the other side no right at all. The buyer may sell at that level. The buyer may also do nothing. That side sold its own freedom to choose at the moment it took the payment, so whichever the buyer picks, the other side has to live with it.

Where a call buyer may require somebody to sell to them, a put buyer may require somebody to buy from them. Almost every mistake made here comes from mirroring the wrong limb, so the two sentences sit best side by side. The direction of the obligation reverses between the two contracts. The side the obligation sits on does not reverse. In a call and in a put alike, the buyer has the right, the writerThe side that takes the premium at the start and carries the obligation to perform if the other side asks. has the obligation, and the premiumThe amount the buyer pays the writer at the start of the contract. The premium changes hands once, and it does not come back. travels from the first to the second.

Here is the everyday version, and it is worth carrying to the end. A stallholder sells festival lamps from a stall. A wholesaler with a shop two streets away offers an arrangement in October: a small token paid now, and at the end of the season the wholesaler will take whatever stock is left off the stallholder's hands at a fixed figure for each lamp, if the stallholder wants that. If the season goes well and everything sells at a better figure, the arrangement goes unused, and the token stays with the wholesaler. If the season goes badly the stallholder takes the arrangement and the wholesaler is bound. The token is the premium. The wholesaler is the writer. The fixed figure for each lamp is the strikeThe fixed level written into the contract at the start, against which the choice at the end is made..

One thing about that arrangement surprises readers, and it matters twice further on. The buyer of a put does not need to have the reference asset in order to have the put. The two are separate holdings. A right to sell something can be held without holding the thing itself, in the same way that a ticket to a concert can be held by somebody who does not intend to attend. Whether the buyer also has the reference asset changes the shape of what they end with. The protective combination further down is entirely about that shape. The contract itself is unchanged either way.

Why do three different figures on this pair all read Rs 2,000.00/-?

Before any arithmetic, one thing needs clearing out of the way. The confusion trips readers throughout this sequence, and hardest here. The protective combination puts all three figures in one paragraph. Three separate quantities on this pair are the number Rs 2,000.00/-, and they agree on purpose rather than by accident or by copying.

The first is the spot price of the reference asset, the cost of one unit today. The second is the strike written into the contract, the level the choice at the end is made against. The two agree because this contract 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 at the money means. The third is the exposureThe value of the reference asset a contract is written on. An exposure is a size, not a payment, and nobody has handed it over., being the value of reference asset the contract is written on. For one contract against one unit that is again Rs 2,000.00/-.

Two of those three are levels and the third is a size, and not one of the three is an amount that anybody has paid. The only amount here that actually moves between the two parties is the premium, and it is worked out next. A report that puts a premium in one column and an exposure in the next has not made an error, and a reader who adds them has. So the two kinds of figure have to stay apart from this point on.

Where does the put premium of Rs 57.93/- come from?

The premium is the first number needed and the number a reader most expects to see fall out of a model, so the premium is the honest place to start. The premium does not fall out of a model here. A pricing model would need one input above all, namely how far the reference asset might travel, and the word for that input is volatility. The working record behind this sequence holds no figure of any kind for it. So the premium arrives by a different route entirely, and the route is arithmetic on a rate.

The route runs in four steps, every one of them reproducible on paper. The first step takes the strike of Rs 2,000.00/-, not payable until the end of the year. Dividing the strike by one plus the financing cost of 6.50 per cent for the year brings it back to today and gives Rs 1,877.9343/-. The difference between the strike and that present value is Rs 122.0657/-. The relationship that ties the two contracts on this pair says that the call premium less the put premium equals exactly that difference. The call premium is given at Rs 180.00/-, so the put premium is Rs 57.9343/-, carried here as Rs 57.93/-.

The check back is worth stating plainly rather than waving at. Rs 180.00/- less the rounded put premium of Rs 57.93/- is Rs 122.07/-. Exact parity is Rs 122.0657/-. The two figures differ by Rs 0.0043/-, forty-three hundredths of a paisa, and the difference is there because the put has been rounded to two places from here on. A claim of exact equality that its own printed figures do not produce teaches a reader to stop checking, so the relationship holds to the paisa rather than exactly.

The same care applies one step further on. Carried forward at 6.50 per cent for the year, the unrounded premium of Rs 57.9343/- becomes Rs 61.70/- to the paisa, and that is the figure used everywhere here. Carrying the rounded Rs 57.93/- instead gives Rs 61.69545/-, a fraction of a paisa short. The same thing happens in the other direction: Rs 1,877.9343/- carried at 6.50 per cent gives Rs 2,000.0000295/-, not Rs 2,000.00/-. None of these gaps matters to a reader. All of them matter to one habit, never calling a rounded step exact.

How do put options work, from the premium to the end?

How Put Options Work, and the four steps one contract runs through

Put the whole life of one contract in a row and there are four steps, only two of which involve anything moving. At the start, the premium of Rs 57.93/- passes from the buyer to the writer. The premium is gone from the buyer at that moment, and it does not come back under any circumstances the contract describes. In the middle, nothing happens. The contract sits. The reference asset does whatever it does, and neither party is called on to do anything about it.

At the end, the price of the reference asset is whatever it is. The price at the end is the third step, and it is worth stating separately as the only step neither party controls. Then comes the fourth. The buyer compares that price to the strike of Rs 2,000.00/- and either sells at the strike or does not. If the buyer sells, the writer is on the other side of that sale. The trade calls that event assignmentWhat happens to the writer when the buyer uses the right: the writer is called on to perform, and has no say in whether that happens..

Money moves twice at most, once at the start from the buyer to the writer and once at the end from the writer to the buyer, and the second movement happens only if the party who made the first one says so. The same structure holds for the call option, and the shared structure is what makes the two contracts one subject rather than two topics. The party who pays first is the party who decides last. Everything asymmetric about an option is contained in that arrangement.

What does the buyer of a put decide, and when?

One comparison, taken once, at the end, with nothing else in it. Is the price of the reference asset below Rs 2,000.00/-? If it is, the buyer sells at Rs 2,000.00/- and collects the difference. If it is not, the buyer walks away and sells nothing. Selling at Rs 2,000.00/- when the open price is higher would be choosing to receive less. The comparison is the entire decision, and a reader who can state it in one line has the contract.

The premium of Rs 57.93/- is not part of that comparison. A reader who learns that once for calls tends to forget it for puts, so it bears saying again. The premium was paid at the start. The premium is with the writer. The premium does not come back whether the buyer sells or walks away, so it cannot change which of the two is better at the end. Bringing it into the decision is how people talk themselves out of a payoff they still have, by reasoning that they have already lost the premium and may as well do nothing. The premium is gone either way. The only live question at the end is which of the two available outcomes is larger.

The purchase makes the buyer of a put one particular thing. Not a forecaster and not somebody who has committed to anything. The buyer has bought the ability to look at the answer before choosing, and that is the whole of what the premium purchased. Whether the ability turned out to be worth its price is a question that can only be asked once the year has run.

What does the put 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 this put it is Rs 2,000.00/- less the price where the price is below the strike, and Rs 0.00/- where it is not. Work it at four prices and the shape appears without any drawing at all. At Rs 1,600.00/- the payoff is Rs 400.00/-. At Rs 2,000.00/- it is Rs 0.00/-. At the forward price of Rs 2,130.00/- it is Rs 0.00/-. At Rs 2,400.00/- it is Rs 0.00/-.

The put payoff line rises towards the left and flattens towards the right, the call turned over, and it stops rising at Rs 2,000.00/- for a reason that has nothing to do with the contract. A price cannot fall below nil. The reference asset can become worth nothing, and at that point the buyer sells something worth nothing for Rs 2,000.00/-, and there is nowhere further for the payoff to go. The bound is a fact about the reference asset rather than a term of the contract, and it is why the two sides of this pair are not mirror images in every respect. A price has no ceiling, so a call has no matching bound.

What the payoff line does and does not say is worth separating. The line gives the shape of an obligation across every price the reference asset could reach. The line says nothing about which price will arrive, and nothing about how likely any of them is. The working record behind this sequence holds no figure of that kind at all.

What does the put earn once the premium is counted?

A payoff is not a profit and never was. The payoff counts only what the contract pays at the end. The profit counts what the contract pays at the end less what it cost to hold, and the two are dated a year apart, so the cost has to be brought to the same date before the subtraction means anything. The premium of Rs 57.93/- was paid at the start. Carried at 6.50 per cent for the year it is Rs 61.70/- by the end, and that carried figure, the financed premiumThe premium carried forward at the financing cost to the date the payoff arrives, so that the two amounts being subtracted are dated the same way., is what gets subtracted.

Subtract it at the four prices and the profits fall out. At Rs 1,600.00/- the profit is plus Rs 338.30/-. At Rs 2,000.00/-, at Rs 2,130.00/- and at Rs 2,400.00/- the payoff is Rs 0.00/- and the same financed premium comes off, so the profit is minus Rs 61.70/- in each case. The profit line therefore sits Rs 61.70/- below the payoff line at every single price, and that puts the crossing of nil at Rs 1,938.30/- rather than at the strike.

The crossing is the break-evenThe price at which a profit line crosses nil. The break-even is a level read off arithmetic rather than a forecast of anything., being Rs 2,000.00/- less Rs 61.70/-. A second figure for the same crossing appears elsewhere, Rs 1,942.07/-, being Rs 2,000.00/- less the premium of Rs 57.93/- as paid, without the financing on it. Neither is wrong and they answer slightly different questions. The first asks what price leaves the buyer level once the money parted with a year earlier is counted at the end date. The second asks the same question with the two amounts dated a year apart, the simpler convention and the more common one. The two figures sit Rs 3.77/- apart, and Rs 3.77/- is exactly the financing on the premium. Rs 1,938.30/- is the financed break-even and it is the figure carried through the arithmetic below, so meeting Rs 1,942.07/- elsewhere need not suggest either is broken.

What is a protective put, and what does the combination end with?

The Protective Put, and what two holdings add up to

A protective putHaving the reference asset and a put written on it at the same time, so that the two holdings are valued together at the end. is two separate holdings valued at the same moment. One unit of the reference asset, and one put on it struck at Rs 2,000.00/-. Nothing joins them in law. The two are added because whoever has both will add them when they count what they ended with.

Do the addition at a price of Rs 1,600.00/-. The unit is worth Rs 1,600.00/-. The put pays Rs 400.00/-. Together they come to Rs 2,000.00/-. Do it at Rs 2,400.00/-. The unit is worth Rs 2,400.00/-, the put pays Rs 0.00/- because the buyer walks away, and together they come to Rs 2,400.00/-. Whatever the price turns out to be, the two holdings together end with the larger of the price and Rs 2,000.00/-, and that sentence is arithmetic rather than a promise about anything.

Work it at the four prices and the row reads Rs 2,000.00/- at Rs 1,600.00/-, Rs 2,000.00/- at Rs 2,000.00/-, Rs 2,130.00/- at Rs 2,130.00/- and Rs 2,400.00/- at Rs 2,400.00/-. Below the strike the line is flat and above it the line rises one for one, the same bend a call payoff makes, lifted up. The resemblance is not a coincidence, and the arithmetic behind it is covered separately under the relationship that ties the two premiums together.

One point needs care, and it is the reason the combination is drawn rather than merely stated. The word protection suggests a cushion sitting underneath a position, absorbing falls and letting rises through untouched. A protective put is not that. The premium was paid whichever arm the price ends on, so both arms of the combined line are affected. And the flat arm sits at a level written into a contract, not at the level the reference asset happened to start from. A different strike moves the flat arm, and pricing a second strike would need a second premium that the working record does not supply.

The default setting reproduces the worked example exactly. At Rs 1,600.00/- the put pays Rs 400.00/-, the unit is worth Rs 1,600.00/-, the two together come to Rs 2,000.00/- before the premium and Rs 1,938.30/- after it. Sweep the control upward and watch two things happen at once. The distance between the diagonal and the flat line closes until it reaches nil at the strike, and that closing distance is the put paying less and less as the price climbs. And the shaded strip between the two upper lines does not move at all. The premium is a fixed amount, paid before any of this was known.

What does a protective put cost, and what does it not do?

The premium now enters the picture, and this is where an account of a protective put either stays honest or turns into a sales sheet. The put premium of Rs 57.93/- is a real amount that leaves whoever bought the put at the start, at the same moment as the arrangement is made. By the end of the year, counted at the same financing cost of 6.50 per cent that every other figure here uses, it is Rs 61.70/-. Taken off the combination, the two rows sit one above the other.

The combination ends at Rs 2,000.00/- before the premium is counted and at Rs 1,938.30/- after it, and at a price of Rs 2,400.00/- it ends at Rs 2,400.00/- before the premium and Rs 2,338.30/- after it. The distance between the two rows is Rs 61.70/- in every column, high price and low price alike, and that distance is the whole of what the arrangement cost. There is no column anywhere in the table where the second row equals the first.

Three plain sentences hold the line here, and they hold it better than any notice at the foot of a screen would. The first: the name protective describes the shape of a line and is not a claim about safety. The second: the arrangement does not remove a loss, it exchanges one shape of outcome for another and charges Rs 61.70/- by the end for the exchange. The third: answering whether that exchange is worth making needs a view on how far the reference asset might move and how likely each move is, and the working record behind this sequence holds no figure of either kind.

The flat arm is worth being exact about. The flat arm is a level fixed by the strike written into the contract. The pair is struck at the money, so on this pair that level happens to equal the price the reference asset started at. The flat arm is not a floor under the price the holder paid, and it does not follow the reference asset around. And once the premium is counted it sits at Rs 1,938.30/-, a figure that does not appear on any payoff diagram drawn in the usual way.

Has the buyer of a put sold the reference asset?

No, and the question is worth asking out loud because the word sell is in the name of the contract twice over. The buyer of this put has not sold the reference asset. The buyer is not shortOwing something not held, which has to be delivered or bought back later. The buyer of a put is not in this position. of it. The buyer owes nobody any of it, and at the end of the year the buyer may simply do nothing and no obligation of any kind appears. The buyer has a right that can be used or dropped, and dropping it is free.

Two consequences follow, and both are statements about the contract rather than about what is likely. The price of the reference asset cannot fall below nil, so the largest payoff this contract could ever produce is bounded by the strike of Rs 2,000.00/-, and the largest amount the buyer can be out of pocket is the premium of Rs 57.93/-, or Rs 61.70/- counted at the end date. There is no arrangement anywhere in the contract by which the buyer can be called on for more. The bound is not a comforting fact about the world. The bound is a description of a document.

Nothing in either sentence carries a likelihood. The largest and smallest number a contract can produce gives the edges of the shape. Those edges say nothing about where inside the shape anything will land.

How does anybody read this outside a classroom?

Start with a lender looking at a borrower who has bought puts. The lender is not reading the payoff diagram at all. The lender is reading two figures. The first is the premium that left the borrower at the start. The second is whether the borrower can be called on for anything more, and for a bought put that is nothing. A borrower who has bought puts has spent money and taken on no further obligation. A borrower who has written them is a different reading entirely, and that reading belongs to the collateral question, covered separately.

Next, somebody reading a position report. Three columns will look like money and only one of them is. The premium column is cash that moved. The exposure column is a size and nobody paid it. The payoff column, if there is one, is what a contract would pay at a stated price and not what anybody has received. The three sit side by side and look identical, so the single most useful habit for reading such a report is to ask of every figure whether it has already moved, might move, or will never move at all.

Then the household version, the stall arrangement from the opening turned around. Somebody has a small holding they intend to keep, and buys an arrangement that lets them hand it over at a fixed figure at the end of the season. The purchase is not certainty. The purchase is the removal of one arm of a range of outcomes, paid for with a fixed token, and the token has gone whether or not the arm removed ever came near them. The token is best thought of as the price of a choice, and the honest question to ask afterwards is not whether the arrangement paid out but whether the choice was worth its price. The second question needs figures the working record does not hold.

Finally, an analyst reading a disclosure that names this combination. The useful question is never what shape the position makes. Arithmetic settles the shape, and it can be drawn in a minute. The useful question is at what level the flat arm sits, what was paid for it, and over what period. Those three turn a shape into an amount. A disclosure that names a structure without those three has described a picture rather than a position.

Should a reader who now understands this buy one?

An answer to that needs three things, and not one of them can come out of arithmetic. The first is a view on how far the reference asset might move over the year and how likely each move is, and the working record behind this sequence carries no figure of that kind at all. The second is the reader's own circumstances. The third is what the arrangement would cost to hold and to unwind in a real market, a set of figures that cannot be stated from memory.

A shape that removes the left-hand arm of a line is not thereby a good idea, and a reassuring shape does not turn a diagram into a reason to act. Everything drawn here describes an obligation: what each side owes at each price the reference asset might reach. Not one line says which price will arrive. A description of an obligation and a prediction of a price are different kinds of statement, and only the first kind can be drawn.

The instrument is settled, and settled exactly. A right to sell at a fixed level, bought for a premium that is gone the moment it is paid, producing a payoff that is bounded by the strike and a profit that is that payoff less the premium carried to the same date. Beside the reference asset it makes a flat arm at Rs 1,938.30/- and a rising arm above the strike. The document says that much and no more. Deciding what to do with it is a separate question, and answering it would need the very figures the working record does not hold.

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

Every figure in the table below comes from four inputs and nothing else: a spot price of Rs 2,000.00/-, a strike of Rs 2,000.00/-, financing at 6.50 per cent for one year, and a call premium of Rs 180.00/- that the working record gives rather than computes. The put premium of Rs 57.93/- was derived from those in four steps above. The reference asset pays nothing at all while it is held, and that is why no payout appears anywhere in the arithmetic.

The second row read across brings two separate points together in one line. The price of Rs 1,938.30/- is where the put on its own breaks even. The payoff of Rs 61.70/- there exactly equals the financed premium. Rs 1,938.30/- is also the level of the flat arm of the combination once the premium has been taken out, in every row below the strike. The two are the same number for the same reason: both are Rs 2,000.00/- less Rs 61.70/-. One figure, Rs 61.70/-, moves every profit line and every flat arm on this pair, and it is the only figure anybody actually paid.

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