Time Decay: Why an Option Loses Value Doing Nothing

Time decay is the fall in an option premium as the time left to run shortens, with everything else standing still. A premium pays for a choice made later, and part of it is an allowance for what can still happen before then. Less time leaves less that can still happen, so the allowance shrinks. Nothing is deducted from anybody: a price changes.

The mechanism is the whole of it, and it is short enough to fit in a breath. The difficulty is not the mechanism. The difficulty is that the size of the fall turns on how quickly the reference asset can cover ground, and no such figure sits anywhere in the working papers these contracts run on. So the direction comes first and in full, then both ends of the fall exactly, then the most familiar claim in the subject set aside, and last the one quantity that genuinely decays and can be computed to the paisa at every point along the way. The absence is not a footnote here; it is the spine of the subject.

So what actually decays here?

The word decay does damage before a second sentence has been read. The word suggests something rotting quietly in a corner, and that picture is wrong in a way that costs a reader later. Nothing about the contract weakens. The strike stays at Rs 2,000.00/-. The right the buyer holds on the end date is the same right it was on the first morning. The duty sitting on the writerWhoever received the premium on day one and stays bound afterwards. A buyer may decline to use the right; this side may not decline to honour it. is the same duty, not a paisa lighter. The contract read on the last day and the contract read on the first cannot be told apart.

The premium is what falls, meaning the amount somebody else would hand over today to stand where the buyer stands. A premium is a market price for a position, not a property of the paper. The paper is fixed. The price of the paper moves.

Consider a stall allotment at a seasonal fair. A trader has paid for the right to trade from a particular pitch for the whole season, and that right is written down and does not change: same pitch, same season, same terms. Two months in, somebody offers to take the allotment off the trader's hands. Because there is less season left to trade, the offer will fall short of what was paid. Nothing in the allotment has altered. The number somebody will give for it has. The split between the allotment and its price is the entire distinction, and once it is held, most of what confuses people about time decay stops confusing them.

The two things have to be kept carefully apart. The contract is a set of promises with a date on them. The premium is what those promises fetch. Letting the two blur leaves a reader believing that a right is quietly eroding, and a reader who believes that goes looking for the erosion in the wrong place: in the terms, in the strike, in some clause. The erosion is not there. It is in the price, and only in the price.

Why does the premium fall as the time left to run shortens?

The premium splits into the two parts already settled by earlier reading. One part is intrinsic value. Intrinsic value is what the contract would hand over if today were the last day, floored at nil because nobody is forced to take up a right that pays them nothing. Everything else in the premium is time value, and time value is an allowance. The allowance is what somebody is willing to pay for the fact that the story is not over yet.

Less time left to run means less room for the price of the reference asset to travel, so the allowance for travel is smaller, so the premium is smaller. The sentence in bold is the mechanism, complete. Read it twice for what it does not say. The sentence fixes the direction and stays completely silent on the amount.

Why silent? Because how much smaller turns on the distance the reference asset covers in a given stretch, and that quantity has been missing from the outset. Two contracts could stand side by side with the same time left, the same strike and the same price, and their allowances could still differ by a wide margin. The asset each one references moves differently. Without a number for that movement, the direction is knowable and the size is not. Direction without size is the honest limit of what three figures and one rate can say.

Here is another way to feel it. A vegetable seller who leaves a crate unsold until the last hour of the market has fewer possible outcomes left than the same seller at nine in the morning: fewer customers can still walk past, fewer prices can still be asked. Somebody buying the crate off him would pay less at four o'clock than at nine for exactly that reason, and neither of them needs a number for how busy the street usually gets in order to know which way the price goes. To know how much less, they would need that number. Anybody working from three figures and a rate is in the nine-o'clock seller's position on direction and in nobody's position at all on size.

Which account did the money leave?

One misreading is worth killing outright, so bluntness beats tact here. When a premium falls, nothing is deducted. No amount leaves anybody's account. No charge is levied by any party. There is no counterpartyThe side facing the holder across one contract, whose duty is the mirror of that holder's right. Every contract has exactly two, and what one of them owes is what the other is owed. quietly collecting a daily sum, and no institution taking a slice for the passage of time.

Look at the actual cash. Rs 180.00/- moved once, from the buyer to the writer, on the day the pair was written. After that, on the arrangement described here, the next movement of money is on the end date. Between those two moments the money does nothing at all. The money sits where it went. A falling premium is a change in what somebody else would pay for the contract, and a change in a price is not a payment.

Consider a hall booked for a wedding, paid for in full, for a fixed date. As the date closes in, the booking gets harder to hand on to somebody else, and what a person would give for it drifts down. At no point in that drift has anybody been charged anything. Nobody sent a bill. The hall did not shrink. The number a would-be taker had in mind is what changed, and that number is not an account entry anywhere.

The reason this misreading is so persistent is that theta, the measure of exactly this fall, is almost always described in words that sound like a bill. Phrases such as the daily cost of holding, or what the position bleeds each day, are everywhere. Both phrases describe a price movement in the grammar of an expense, and the grammar sticks even after the reader has been told better. An ear that has accepted that a premium is being drained goes looking for a drain, and there is not one.

Where exactly are the two ends of this fall?

Both ends of the fall are exact on this pair, and neither of them needs anything unavailable here. Two exact ends is a better position than it sounds, so the setting out goes slowly.

Start with the near end. The reference asset price stands at Rs 2,000.00/-, and that figure is exposureThe size of a position measured by the thing it references, not an amount anybody has handed over. A price of Rs 2,000.00/- says what the contract is written against; it does not say that Rs 2,000.00/- has been paid. rather than money anybody has parted with. The strike is also Rs 2,000.00/-. The two figures agree deliberately, not by accident and not because one was copied into the other. The strike was placed on the price on purpose, and being at the money amounts to exactly that. So if the contracts ended this instant, the call would take Rs 2,000.00/- away from Rs 2,000.00/- and hand over Rs 0.00/-. The put arrives at the same nil from the other direction.

Intrinsic value is therefore Rs 0.00/- on both contracts, so the whole of the call premium of Rs 180.00/- and the whole of the put premium of Rs 57.93/- is time value, with no intrinsic part in either of them at all. Every paisa in both premiums is an allowance for what has not happened yet. Two premiums of pure time value are unusual, and that is the reason this particular pair is such a clean teaching case: normally the premium is a mixture that has to be separated again and again, and here the separation has already been done by where the strike sits.

Now the far end. On the last day there is no time left to make an allowance for, so time value is Rs 0.00/-, and the premium can only be the intrinsic value. Notice that this end is not a calculation. Nothing was worked out to get there. The far end is true by what the words mean: an allowance for what can still happen, on a day when nothing further can happen, is nothing. The record holds no reading for either premium on that day and none is needed for the statement to be exact.

So the position is a strange but honest one. Two ends known to the paisa, and a middle that has to be treated truthfully. The middle is treated below.

Why is the shape of the fall not drawn here?

Two points on this line are exact and everything between them is guesswork wearing the clothes of a curve. Where the pair begins, the whole of the call premium of Rs 180.00/- is an allowance for what can still happen. Where it ends, that allowance is Rs 0.00/-, and the nil is not a measurement; it follows from there being no time left to make an allowance for. The premium on a day in the middle cannot be named from three figures and a rate, so the middle stays blank. A blank is uncomfortable to look at. A curve drawn to fill it would be comfortable, and it would be invented.

Now the part that matters more, and it concerns a claim most readers have already met. Readers are routinely told that time decay accelerates as the last day approaches, often with a picture of a line bending sharply down at the right. The acceleration claim is a result produced by a pricing model from its assumptions, not an observation and not arithmetic, and without a model nobody is in a position to assert it at all. The claim therefore goes unasserted.

Precision about what is being declined and what is not is worth the trouble. The claim is not false. The claim follows perfectly well inside the frameworks that produce it, and those frameworks are covered separately. The refusal is to pass the claim on as though it fell out of the two exact ends and the direction. Nothing of the sort follows from them. The claim needs the missing input, plus a set of assumptions about how a price moves through time, and accepting a conclusion whose inputs cannot be seen is the opposite of the habit this subject rewards.

There is a habit underneath this that transfers well beyond options. When a shape is handed over, the question is what produced it. If the answer is a model, the shape carries the model's assumptions with it, and what is held is now two things rather than one: a picture, and a set of conditions under which the picture is right. Losing track of the second is how a diagram ends up being treated as a fact about the world.

What survives with no model at all?

More than might be guessed. The two ends, exactly, and both are already in hand. The direction, certainly: with everything else standing still, a shorter stretch cannot make the allowance for travel bigger, so the premium does not rise for that reason. And a name for the rate, at no cost at all. Theta is how far an option premium moves per unit of time passing, so theta is the measure of precisely this fall. Naming it is free. Putting a figure on it would need the input that is missing, so no figure appears here for it or for any of the other four sensitivities.

Three things stand after the model has been taken away, and three is not nothing. But there is a fourth, and it is the one worth the rest of this guide. There is exactly one quantity here that decays, can be computed at every point in between, and consults no model at any step. Neither premium is that quantity. It is the distance between them.

What happens to the gap between the two premiums?

With the price of the reference asset held dead still at Rs 2,000.00/- and the clock running, something exact happens to the distance separating the two premiums, and every step of it can be followed.

The tie between the two premiums is settled separately. The put premium taken away from the call premium leaves an amount that has to match the distance separating the reference asset price from the strike measured at today's date. The tie is the parity relationshipThe fixed arithmetic tie between a call premium, a put premium, a price and a strike at one expiry. Break it and two positions that must cost the same would not, which is why it is arithmetic rather than an opinion., and note which direction it runs: it says nothing about either premium on its own, only about the distance between them.

Work it at the near end. The strike of Rs 2,000.00/- with a year to run, at the financing rateThe annual rate at which an amount is moved between two dates, here 6.50 per cent a year. It carries a period because a rate without one states nothing. of 6.50 per cent a year, gives a present value of the strike of Rs 1,877.9343/-. So the gap is Rs 2,000.00/- less Rs 1,877.9343/-, or Rs 122.0657/-.

An identity given and never tested teaches a reader to accept identities, so the identity is worth checking against the premiums themselves. Rs 180.00/- less Rs 57.93/- lands on Rs 122.07/-, and the arithmetic lands on Rs 122.0657/-. Forty-three hundredths of one paisa separates them, and the cause is that the put premium is carried in the record to two places rather than four. So the tie holds to the paisa. Calling it exact would ask a reader to trust a figure instead of checking it. Trusting instead of checking is the opposite of the habit the subject rewards.

Now shorten the stretch. DiscountingRunning an amount backwards through a rate to find what it is worth on an earlier date. Fewer months to run means less to strip out, so the earlier figure sits closer to the later one. the strike over a shorter run takes less off it, so the strike measured at today's date climbs towards the strike itself, and the gap narrows. With no time left at all, there is nothing to strip out, the strike measured at today's date is simply Rs 2,000.00/-, and the gap is Rs 0.00/-.

One more result, and it is worth having because it says what the gap actually is. Carry the gap forward one year at the same 6.50 per cent a year and it lands on Rs 130.00/-, exactly the financing on Rs 2,000.00/- for a year. The match is not a coincidence and not corroboration from a second source. The algebra forces it: the price less the strike measured at today's date, multiplied back up by one plus the rate, is just the price times the rate. The gap between the two premiums is the year's financing on the price of the reference asset, brought back to today, and nothing else whatsoever. No travel, no allowance, no model. Which is exactly why it can be drawn when the premiums themselves cannot.

The near-straightness is worth stopping on. The one decay that can actually be drawn here does not accelerate towards the end. The gap runs within a rupee of a straight line for its whole life, at its widest Rs 0.9609/- above the chord on a quantity that starts at Rs 122.0657/-. Nobody should read the gap as evidence about the premiums. The gap is a different quantity behaving in a different way. The near-straightness stands instead as a reminder that the shape of a fall is a fact to be established rather than assumed, and that the intuitive shape and the arithmetic shape need not agree.

The worked instance, set out in full

Everything above rests on the same invented pair, set out here in one place, in the order the arithmetic runs. Taken slowly, the rows reproduce every figure in this guide with a calculator.

Two rows in that table deserve a second look. The premium levels appear only on the day the pair is written and nowhere else. The gap appears at every date, and that difference is the entire architecture of the argument. A level would need the missing input. The gap needs a rate and a division.

Who is on the other side of this, and is either side the good side?

A premium that falls is lower for whoever would sell the contract on and lower for whoever would buy it back to close. Selling the contract on and buying it back are two different situations, and the same movement lands on them in opposite ways. Somebody wanting to hand the contract to a third party receives less than they would have on the first day. Somebody wanting to unwindClosing out of a position before the end date by passing it to somebody else, rather than holding it through to whatever the end date brings. a written position pays less than they would have on the first day to be rid of it.

The description ends there. Writing contracts is not thereby a way of standing on the profitable side of the clock, the two sides cannot be ranked on how they turned out without knowing how far the reference asset travelled, and the fall on its own settles nothing about which side anybody should be on.

The reason is worth stating rather than leaving it to sound like a formality. The fall in the allowance is only one of the two things moving in any real week, so saying which side benefits would need to know how far the reference asset actually travels while the clock runs. The travel figure is the one thing missing from the outset. A refusal that has a reason behind it is a different object from a refusal that is there for safety, and this one has a reason.

How does anybody actually use this on a working day?

Rather than sorting this by job title, sort it by the field somebody has to fill in, because the same four fields turn up whoever is holding the pen and the interesting differences are in who can fill which.

Watch what that table does to four familiar readers. A household that has paid a premium and is watching a screen can fill the first field and the fourth without help, and the fourth is the one that stops them budgeting for a payment that is never going to arrive. Somebody assessing a position at a lender can fill the first, has the second on a terminal, and knows that the third is the field where an assumption has been made whether or not anybody said so out loud. An analyst writing about a position has the same three and the professional duty to say which model produced the third. And whoever wrote the contract is filling exactly the same four fields with exactly the same difficulty, from the mirror side.

The field that separates the careful reader from the careless one is the last: asking which account moved, before deciding what a movement means. The question costs nothing but the asking, it needs no data at all, and it removes an entire class of error at a stroke. A very good return, for one question.

Is any of this a reason to act?

Should anybody be on either side of this? No answer follows from the mechanism, and the refusal is not politeness. An answer would rest on the ground the reference asset might cover before the end date, and on how likely each of those landing points is, and that is precisely the figure missing from the outset. A reader's own position matters as much, and none of it is visible from here. So does the cost of keeping the arrangement until the end date, and the separate cost of handing it on before then, and neither figure appears anywhere above. Strip those away and an answer has nothing left to stand on.

Notice how little has actually been claimed, once the items are counted. A premium contains an allowance for what has not happened yet. A shorter time leaves a smaller allowance. Nothing is deducted from anybody while that happens. Both ends of the fall are exact on this pair. The distance between the two premiums decays by financing alone and can be checked anywhere along the way. The list is complete, and not one item on it says anything about what any price will do.

A premium falling as the time left to run shortens is a description of how a price behaves, and a description of how a price behaves is not a reason to be on either side of it. The distance between understanding a mechanism and having grounds to act on it is the whole of the missing figure, plus everything about a particular reader's situation that is not visible here.