Options: Rights on One Side, Obligations on the Other

An option puts the right on one side of the contract and the obligation on the other. The buyer pays a premium at the start and may walk away at the end. The writer takes that premium and may not walk away. If the buyer asks, the writer performs. Everything else about an option follows from that one asymmetry.

The asymmetry has to be paid for, and the premium is that payment. Think about why. Neither side of a contract that binds both sides equally has been handed anything the other has not, so no payment passes at the start. Both are promising the same kind of thing. Once one side may change its mind at the end and the other may not, the two sides are no longer holding the same object, and the difference between them is exactly what the payment buys.

An optionA contract that gives one side a choice at the end and binds the other side to whatever that choice turns out to be. is therefore not a bet placed with a bookmaker and it is not a fee for advice. An option is a contract with two named parties, and what makes it unusual is that the two parties are not doing the same thing. 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. is the price of that difference. Every figure in this guide belongs to an invented pair of contracts on an invented reference asset, and every one of them is worked out rather than quoted.

What does one side of an option have that the other side does not?

One side may walk away and the other may not. The asymmetry is the whole of it, and it is worth saying slowly. Every other kind of contract binds both parties in the same way. In an option the right belongs to one side and the obligation belongs to the other, permanently, for the life of the contract, and no event flips them. A price move does not flip them. Time passing does not flip them. The side holding the right holds it until the end and then either uses it or lets it go; the side holding the obligation waits.

Here is the everyday version, and it is worth carrying through everything that follows. A stall stands outside one office building, and its owner will let it for the coming season. A prospective tenant hands over a token to hold that stall. If the tenant turns up, the stall is theirs at the level agreed. If the tenant never turns up, the token is gone and nobody comes after them for anything more. Meanwhile the owner cannot let the stall to anybody else while that token is with them, whatever better offer arrives at the door. One side may change its mind and one side may not, and the token is what the first side paid for that.

The token is the premium, the stall owner is the writer, and the level agreed in advance is the strike. Translated, the three make an option. The writerThe side that takes the premium at the start and carries the obligation to perform if the other side asks. is the party who took the payment and stayed bound. The strikeThe fixed level written into the contract at the start, against which the choice at the end is made. is the fixed level written into the contract at the start, and on the pair worked here it is Rs 2,000.00/-. The spot price of the reference asset is also Rs 2,000.00/-, and those two are the same number on purpose rather than by accident, for a reason set out under the split of the premium further down.

What does the buyer of an option pay, and what does that payment buy?

The Option Buyer, and the three things that follow from paying first

The buyer pays the premium at the start, and what the buyer receives in return is a choice, made at the end. Not an asset, not a claim on anybody's earnings, not a promise that anything will happen. A choice. On the pair worked here the buyer of the call pays Rs 180.00/- at the start and gets to decide, one year later, whether to take the reference asset at Rs 2,000.00/- or to do nothing at all.

Three things follow, and they are worth holding together. The payment at the start was the whole of the buyer's obligation, so the buyer never owes anything further, whatever the reference asset does. The buyer's decision is taken with the price already known, which is why the buyer chooses last and why the choice is easy when it arrives. And the choice is mechanical rather than a matter of judgement: the buyer takes whichever side of it pays more. At the end there is no future left to have a view about, so nothing is left to weigh.

Those three properties do something particular to the shape of the buyer's position. The largest amount the buyer can ever hand over is fixed at the premium and was handed over on day one. The largest amount the buyer can ever receive is not fixed by anything here. The two limits are not mirror images of each other. Neither limit says which end of the shape is more likely to arrive. A payoff shape describes an obligation, and how likely each end of it is comes from a probability, which is a separate input somebody has to supply.

What is the premium, and what is it not?

The Option Premium, the one amount that actually moves

The premium is an amount of money that actually moves, from the buyer to the writer, at the start of the contract. Three negatives are worth stating plainly. Each of them is a mistake somebody makes on first meeting. The premium is not the strike. The strike is a level written into the contract rather than a payment. The premium is not the value of the reference asset, and nobody has paid that value. And the premium is not a deposit that comes back at the end, the way collateral does. The premium is a payment for the right to change one's mind, and it is gone the moment it is paid, whatever happens next.

On this pair, the premium of Rs 180.00/- is what moves between the parties, while the Rs 2,000.00/- of reference asset the contract is written on 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., and no part of the exposure has been paid or received by either side. Those two figures sit next to each other in every report a reader will ever see, and adding them together or swapping one for the other is the fastest way to describe an arrangement as roughly eleven times larger than the cash that changed hands. The premium is cash. The exposure is a size.

What does the writer receive, and what does the writer take on?

The Option Writer, the side that cannot change its mind

The writer takes the premium at the start and carries the obligation all the way to the end. The writer is the side most explanations skip, and skipping it is why the asymmetry never quite lands. Three things follow for the writer, and each one is the exact mirror of one of the buyer's three.

The premium was set on day one and there is no second payment anywhere in the contract, so the writer's largest possible receipt is fixed at Rs 180.00/- and cannot be improved by anything the reference asset does. The writer's obligation is decided by somebody else's choice, taken later, with better information than the writer had when the premium was agreed. And the writer places collateral against that obligation with the party standing between the two sides, at a level set by the Securities and Exchange Board of India (SEBI) at sebi.gov.in. The writer is paid for accepting that the other side chooses last, and that single sentence is what the whole subject turns on.

Why does the payoff line bend at the strike?

At the end, the buyer compares one number to one other number and takes the better of the two. The comparison between those two numbers is the bend. Both sides of a contract that binds them equally do the same thing at every price, so that contract draws a straight line. A contract where one side chooses draws a line that is flat on one side of the strike and sloping on the other. On the flat side the buyer walks away, so nothing moves however far the price falls; on the sloping side the buyer takes the contract, so every rupee of price shows up. The bend is the choice, drawn.

The payoffWhat the contract pays at the end, before anything paid to hold it is counted. A payoff is never below nil for the buyer. of the call worked here is nil at any price at or below Rs 2,000.00/-, and it is the price less Rs 2,000.00/- at any price above it. At Rs 1,600.00/- it is Rs 0.00/-. At Rs 2,000.00/- it is Rs 0.00/-. At Rs 2,130.00/- it is Rs 130.00/-, and at Rs 2,400.00/- it is Rs 400.00/-. The same four prices worked for a contract binding both sides at Rs 2,000.00/- give minus Rs 400.00/-, Rs 0.00/-, Rs 130.00/- and Rs 400.00/-. The two agree above the strike and part company below it, and where they part company is exactly where the buyer walked away.

What is a payoff, and how is it different from a profit?

The commonest error in the whole subject is made here, and it is made because two different numbers wear labels that sound interchangeable. A payoff is what the contract pays at the end, counting nothing that was paid to hold it. A profitThe payoff after the premium is counted, with the premium carried forward to the same date so the two figures are being compared at one moment. is that payoff after the premium is counted. Payoff and profit are two different lines on the same picture, and the reader has to know which one is in front of them.

The premium was paid at the start and the payoff arrives at the end, so counting them together means carrying the premium forward at the financing cost: Rs 180.00/- carried at 6.50 per cent for the year is Rs 191.70/-. That is Rs 180.00/- plus Rs 11.70/- of financing, and it is the same arithmetic the reader already used to turn a spot price of Rs 2,000.00/- into a forward price of Rs 2,130.00/-. Both figures then sit at the same moment and may be subtracted. The premium does not depend on where the price lands, so the distance between the payoff line and the profit line is Rs 191.70/- at every price on the axis.

There it is, and the default setting is the worked example. At a price of Rs 2,130.00/- the payoff is Rs 130.00/- and the profit is minus Rs 61.70/-. Push the control down to Rs 1,600.00/- and the payoff sits at Rs 0.00/- while the profit sits at minus Rs 191.70/-. Push it up to Rs 2,400.00/- and the payoff is Rs 400.00/- while the profit is Rs 208.30/-. Rs 191.70/- does not depend on the price at the end, so the green connector between the two dots is the same length at every setting of the control. The marker where the profit line reaches nil stays still at Rs 2,191.70/- while the price marker slides past it.

What is the premium made of once the contract exists?

Split the premium in two and the second half is where the rest of the subject lives. Intrinsic valueWhat the contract would pay if it ended right now. The buyer may always walk away instead, so intrinsic value is never below nil. is what the contract would pay if it ended right now, and it is never below nil, because the buyer may walk away rather than pay out. Everything else in the premium is time valueWhatever is left of the premium once intrinsic value is taken out of it. At the end there is none of it left.. Two parts, and the split is defined so that they add to the premium exactly.

On this pair the strike is Rs 2,000.00/- and the spot price of the reference asset is Rs 2,000.00/-, so the intrinsic value of both the call and the put is exactly Rs 0.00/-. The whole of the call's Rs 180.00/- and the whole of the put's Rs 57.93/- is time value. Those two figures being the same number is not a transcription slip and it is not a coincidence: this pair is struck at the moneyWhere the strike written into the contract and the current price of the reference asset are the same number., and being at the money is exactly what that means. A call struck at the level the reference asset is already trading at would pay nothing if it ended this instant, and neither would the put.

What moves the part of the premium that is not intrinsic value?

How Option Time Value Changes, and which way each change pushes

Three things move it. Two of them can be stated exactly and the third cannot. The first is the end date itself. At the end there is no time left and the premium can only be the intrinsic value, so time value falls to nil by definition rather than by any model. The direction is settled: whatever time value is on any day before the end, it is on its way to nothing. The second is the reference asset. Intrinsic value is defined against the strike and the strike does not move, so as the price of the reference asset moves, the split between intrinsic value and time value moves with it.

The third is how far the reference asset might travel before the end, and that input is not available here. There is no figure for it anywhere in the working record behind this guide, and inventing one would break every premium in it. The part that can be settled without that figure turns out to be more than nothing.

Here is the part that can be computed here and computed to the paisa: the call's time value exceeds the put's time value by Rs 122.0657/-, and that figure is nothing more mysterious than the financing saved by not paying the strike until the end. The arithmetic runs like this. Divide Rs 2,000.00/- by one plus 6.50 per cent, and the present value of the strike is Rs 1,877.9343/-. Subtracted from Rs 2,000.00/- that leaves Rs 122.0657/-. The two premiums give the same thing: Rs 180.00/- less Rs 57.93/- is Rs 122.07/-. Against exact parity of Rs 122.0657/- that is a difference of 0.43 of a paisa, and the difference is there because the put has been rounded to the paisa rather than because anything is off. The relationship holds to the paisa. The relationship does not hold exactly, and treating it as exact would be a reason to stop checking.

The rest of the call's Rs 180.00/-, meaning the part of the time value that financing does not explain, cannot be worked out from these four inputs. Working it out needs a figure for how far the reference asset might move before the end, and no such figure exists in the record these premiums come from. The premiums here were given rather than worked out, and they are consistent with each other and with the financing arithmetic. Consistency is the whole of what the two premiums support. Supplying the missing input would mean inventing it, and every other figure leaning on these two premiums would then be resting on something made up.

How does anybody read this outside a classroom?

Somebody reading a position report meets three columns that look like money and are not the same kind of money at all. The premium is cash that has already moved and will not come back. The exposure is a size, being the value of reference asset the contract is written on, and nobody has paid it. The collateral is cash that has moved but is expected back, at a level the authority sets rather than the parties. An analyst who adds the premium to the exposure has described an arrangement as roughly eleven times larger than the cash that changed hands, and an analyst who reports only the premium has described it as far smaller than the obligation behind it. Both numbers get reported, separately, with their labels attached, and that is not pedantry: it is the only way the report says anything true.

A lender looking at a borrower who has written contracts is reading for one thing in particular: which side of the asymmetry the borrower sits on. A borrower who has bought contracts has a known maximum outlay that has already left the account. A borrower who has written them has taken money in and carries an obligation decided by somebody else, later. The two borrowers have put different things on their books, and one question tells them apart: who may walk away?

The household version is the stall token again, held from the other end. For the stall owner who took the token, income from that arrangement is settled, it is the token, and it cannot improve however good the season turns out. The owner has given up the ability to let the stall to anybody else, for the whole season, on somebody else's timetable. Nobody sensible describes that as free income and nobody sensible describes it as a disaster either. The arrangement is a position with a shape, and the payoff line is that shape drawn.

Should a reader who now understands this use one?

The question is not answered here, and the reason is stated rather than left to caution: the three things anybody would need in order to answer it are not available. The first is a view on how far the reference asset might move before the end and how likely each move is. Forming that view takes a probability, a distribution or a record of what the reference asset has actually done, and none of the three comes out of arithmetic on a premium. The second is the reader's own circumstances, which differ from one reader to the next and change the answer. The third is what the arrangement would cost to hold to the end and to unwind before it. Those figures belong to the parties and the authority rather than to arithmetic.

A description can set out what would be signed. A payoff diagram describes an obligation, and understanding an obligation is not a reason to take one on. The distinction between describing and recommending is doing real work rather than covering anybody: a shape drawn on a screen is one short step from a suggestion, and a description is not that step.

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

Every figure below is worked from three inputs and nothing else: a spot price of Rs 2,000.00/-, financing of 6.50 per cent a year for one year, and a call premium of Rs 180.00/- that is given rather than computed. The reference asset pays nothing at all while it is held. A payout during the year would change every figure in the table. The put premium is not a second given: it comes out of the financing arithmetic as Rs 57.9343/- and is carried here as Rs 57.93/-.

The third row across carries the point in one line. Same contract, same price, two different numbers, and the label is the only thing that tells them apart: a payoff of Rs 130.00/- and a profit of minus Rs 61.70/-. In the last row, Rs 2,130.00/- is the forward price the reader already knows how to build, and arriving exactly at the forward price still leaves the buyer of this call behind. The call's profit turns positive only above Rs 2,191.70/-, and the put's only below Rs 1,938.30/-, both of which are the strike moved by a premium carried at 6.50 per cent for the year.

The reference asset, both contracts, the strike, the premiums and every figure shown here are invented.
Educational material. Not advice on any investment, tax, budget or market position.