What an Implied Volatility Does to an Option Premium

An implied volatility does not move an option premium. The premium is struck first, between two people who are free to agree anything, and the implied number is what that premium becomes once a chosen pricing model restates it. Read the arrow that way and the direction follows: hold the price, the strike, the time left and the financing rate still, and a higher quoted premium restates as a larger implied number.

Almost every reader arrives at this subject with the question phrased the wrong way round. The phrasing is not the reader's fault. The error is built into the way the two words are spoken together, and into the way the subject is usually introduced, so it takes a firm correction rather than a gentle hint. Because everything downstream reads differently depending on which way the arrow is taken to run, the correction comes before the machinery.

Here is the setting, and it is the same invented pair of contracts the rest of this sequence runs on. There is one reference asset, invented, standing for nothing real, with a price of Rs 2,000.00/- for one unit. The reference asset pays nothing at all while it is held. A payout during the holding period would change the financing arithmetic below, and that fact matters enough to repeat wherever it comes up. Financing costs 6.50 per cent a year. A call and a put are written on that reference asset, both struck at Rs 2,000.00/-, both running one year, and the call premiumThe amount that actually moves from the buyer to the writer at the start. It is paid once, at the beginning, and it is not the payoff. is Rs 180.00/-, a figure given here rather than worked out.

Which way does the arrow actually point between the two?

Start with what happens in the room. Two people want to deal in a contract. One of them will pay for the right to buy the reference asset at Rs 2,000.00/- a year from now; the other will take the money and carry the obligation on the far side. The two of them talk, disagree, meet somewhere, and a number is written down. Nothing forces that number to any particular level. No model is consulted, no formula is opened, and no committee approves it. The premium is a price, in the ordinary sense that a price is whatever two willing sides settle on.

Only afterwards does anybody produce an implied number. Somebody takes the premium that was just agreed, opens a pricing model, feeds in everything the model asks for except one input, and hunts for the value of that last input which makes the model hand back the premium already on the screen. The value they stop at is the implied volatilityThe travel assumption a chosen model would have had to be fed in order to return the premium actually being quoted.. The implied volatility is not an input that moved the premium but an output the premium produced, and reversing those two roles is the error that spoils every later sentence about the subject.

The everyday version makes the direction obvious in a way that the finance version does not, so take that one first. A household lives on one salary and spends a certain amount every month. A relative looks at the spending, does some sums at the kitchen table, and announces that the household must be assuming next year's income will hold up. The relative's announcement is an inference about an assumption, worked backwards out of the spending. The spending came first and was not caused by the announcement, and if the household spends more next month the relative will announce a bigger assumption. The assumption tracks the spending. The spending does not track the assumption.

The same sentence in the finance version stops sounding strange. The premium came first and was not caused by the implied number. If the premium is quoted higher tomorrow, the implied number worked out of it will be larger. The implied number tracks the premium. The premium does not track the implied number. Everything that follows is a consequence of that one sentence.

There is a reason the phrasing goes wrong so reliably, and it is worth naming rather than tutting at. The two words are a compound noun, and the second half of a compound noun usually names the category the thing belongs to. A steel bridge is a bridge. A monthly rate is a rate. So an implied volatility sounds like a volatility, meaning a property of the reference asset, with the first word doing nothing more than saying where the figure came from. The compound reading is wrong on both halves. The first word is doing the heavy lifting, and what the phrase actually names is a restatement of a price.

What does the inversion actually do, step by step?

The word implied hides a procedure, and the procedure is not complicated. The procedure only sounds mysterious while it stays a word. So take the word apart and put the machine in its place.

A pricing model is a box that eats several inputs and hands back one premium. The inputs it wants are the price of the reference asset, the strike written into the contract, how long the contract has left to run, the financing rate, and one more input that says how far the reference asset might travel between now and the end. The travel input is the one everything here turns on. Every one of the others can be looked up or read off the contract. Nobody publishes how far a price is going to move, so the travel input cannot be looked up at all.

Now what happens when the premium is already in hand. The price of the reference asset is known from the screen. The strike is known from the contract. The time left is known from the calendar. The financing rate is known because rates are quoted. And the premium is known because somebody is dealing at it. So of everything the box wants and everything it produces, exactly one item is unknown, and it happens to be an input rather than the output. An unknown input beside a known output is an unusual position to be in, and it is what makes the whole procedure possible.

So the value is guessed. A value for the travel input is chosen, put in with the rest, and the box hands back a premium. If the premium the box returns is smaller than the quoted premiumThe level at which somebody is presently willing to deal. It is the figure an inversion is performed on, and a stale screen is not one., the guess was too small, and a bigger one follows. If it comes back larger, the guess was too big, and the next one comes down. The narrowing continues until the premium the box returns and the premium being quoted sit on top of each other. The value in hand when the two met is the implied volatility, and that search is the entire content of the word implied.

1. Take the premium that is actually being dealt at. Not a premium for a similar contract, not an average of several, and not yesterday's. That one, on that strike, with that much time left to run.
2. Fill in every other input the model asks for. The price of the reference asset, the strike written into the contract, the time left, and the financing rate. All four are facts about the arrangement rather than opinions about it.
3. Choose a value for the travel input and run the model forwards. This is a guess, and it is meant to be. The first one is almost never right and it does not need to be.
4. Compare the premium the model returned with the premium being quoted. Too small means the guess was too small. Too large means the guess was too large. There is no third case to handle.
5. Adjust and run it again, and keep going until the two premiums meet. The value the search stops at is the answer. Nothing has been solved in a single stroke, and no closed expression has been evaluated; a search has been run until it converged.

The search just described is the whole inversionRunning a pricing model backwards: adjusting one input until the premium the model returns matches the premium being quoted.. The procedure is worth noticing for how ordinary it is. There is no forecasting step anywhere in that list, no view is formed about anything, and nothing is predicted. A number that was already agreed between two people goes in at the top, and a differently expressed version of the same number comes out at the bottom.

Why is there exactly one value that reproduces a quoted premium?

A careful reader asks this question second, and most explanations of the subject skip straight past it. The search only makes sense if it has somewhere to stop. Two things could go wrong. Two different travel values could both return the quoted premium, in which case the answer would not be a single number and which of the two was meant would have to be explained. Or no travel value at all could return it, in which case the search would run forever and never converge on anything.

The models used for this are built so that feeding in more travel returns a larger premium, every time and without exception, so the premium the model hands back climbs steadily as the input climbs and can meet any given level once and once only. A line that only ever rises cannot come back to a height it has already passed, so it cannot meet the quoted premium twice. And because it starts below any level worth quoting and can be driven above it, it cannot miss it either. One crossing, always.

Now the honest half of the same paragraph, and it is not an afterthought. The steady rise is a property of the model, put there deliberately by whoever built it, and it is not an observation about how prices behave in the world. Nobody went out and measured that premiums rise with travel. Somebody wrote a model in which they do, and the reason the search behaves so well is that it is searching inside a machine that was designed to make it behave well. Keep those two things separate. A well behaved procedure is a fact about the procedure.

The distinction sounds pedantic until somebody who has forgotten it turns up. Once the steadiness is believed to be a fact about the world, the number that falls out of it is believed to be a fact about the world too, and from there it is a very short walk to reading it as a forecast. Reading an implied number as a forecast is the failure set out below.

What happens to that search on the very last day?

There is one moment where all of this stops, and it is worth setting out because it is the only thing here that can be stated with no qualification attached to it at all. On the last day, when there is no time left to run, the contract pays what the contract pays: the buyer of the call takes the difference if the price of the reference asset finishes above the strike of Rs 2,000.00/-, and takes nothing if it does not. The payoff expression contains a price, a strike and nothing else. There is no travel left to have, so no travel term appears in it anywhere.

So on the last day the premium is simply the payoff, and a travel assumption has nothing left to do. The input has stopped mattering, so any travel value fed to the model at that moment returns the same figure. The steadily rising line of the previous drawing has flattened into a horizontal line. And a horizontal line, held against a level, either sits exactly on it or misses it entirely. There is no single crossing to be found, so the search that worked so cleanly the day before has no answer on the last day.

Notice something about the boundary itself. Careless statements about the boundary go wrong at exactly one point. At the strike of Rs 2,000.00/- the call pays nothing and the put pays nothing too. Both are nil together at that one price. So a sentence of the form "one of them is worth something precisely where the other is worth nothing" is false at the strike itself, and the honest statement excludes that point: above the strike the call pays and the put does not, below the strike the put pays and the call does not, and at the strike neither pays anything at all.

What is the resulting number conditional on?

Longer than most readers expect, and the length is the point rather than a caveat. An implied number is conditionalTrue only given the assumptions used to produce it, so those assumptions have to travel with the number wherever it goes. on four separate things at once, and dropping any of them leaves a figure that cannot be interpreted.

An implied number is conditional on the model chosen. A different model is a different machine, and a different machine returns a different value. The number is conditional on every assumption inside that model, and most of those assumptions are never said out loud by the person quoting it. The number is conditional on the other inputs being right, so a wrong time to expiry or a stale financing rate quietly poisons the result. And it is conditional on the quoted premium being a level somebody would actually deal at, rather than a figure sitting on a screen because nobody has updated it since the morning.

Because they were not running the same machinery, two people can invert the same premium and report different numbers without either of them having made an arithmetic mistake. That is a strange thing to say about a number, and it is worth sitting with. If two people add the same column of figures and disagree, one of them is wrong. If two people invert the same premium and disagree, both can be entirely right, and the disagreement is about what was assumed rather than about what was calculated.

An implied number therefore travels with its model in exactly the way a rate travels with its period, and a number handed over without the model behind it is missing part of itself. A rate that arrives without a period is already refused. Somebody says six and a half per cent, and the question back is: per what. The habit here is identical. When somebody hands over an implied number, the questions back are out of which premium and through which model, and until both answers are in hand what is being held is a figure rather than a fact.

Why does a premium move when the obligation has not changed at all?

Most readers actually arrive with this question, whatever they typed into the search box. Something feels wrong about a price that moves when the thing being priced has not. So start from the thing being priced, and be precise about how fixed it is.

The obligationWhat the contract binds each side to do. It is fixed in writing on the day the contract is written and does not change afterwards. is settled in writing on the day the contract is written. The strike of Rs 2,000.00/-, the last day it runs to, the quantity one contract covers, the side that may walk away and the side that may not: all of it is fixed on day one and not one item moves afterwards. Nothing gets renegotiated. Nobody amends the strike because the price of the reference asset went somewhere unexpected. Nothing written on the contract has moved, so a premium that moves after the contract is written has not been moved by anything written on it.

The one thing that moves is what somebody will pay today to stand in that position. Exactly four things can change that price, and four is a short enough list to hold in mind.

The price of the reference asset can move, and the response of a premium to that is called delta. Time can pass, and the response to that is called theta. The financing rate can move, and the response to that is called rho. And what people will pay for the travel allowanceThe part of a premium that exists because the price of the reference asset can still move before the contract ends. can change, and the response to that is called vega. Delta, theta, rho and vega are four of the five standard sensitivities, written delta, gamma, vega, theta and rho, in lower case, in that order, every time they are listed together. Each sensitivity is covered separately under its own name.

Three of those four have something that can be looked up, and the fourth does not. A premium moving for that fourth reason is precisely the move that feels as though it had no reason at all. The price of the reference asset sits on a screen. A calendar shows that a day has gone. The financing rate is quoted. The travel allowance is not published, not observed and not written down, so no screen anywhere shows what people are currently willing to pay for it. The level is inferred, and it is inferred by exactly the inversion described above. So the reader who says the premium moved for no reason is not being careless. Such a reader is noticing, correctly, that the fourth cause is the one with nothing visible behind it.

Now put the contract itself beside that card. Seeing the two together settles the matter faster than any further argument. Everything on the written contract is fixed. Not one of the four things that move the premium is written on it anywhere.

Which way does the implied number go when the premium goes up?

Now the relationship itself, stated as a direction and never as a size. Hold four things absolutely still: the price of the reference asset, the strike written into the contract, the time left to run and the financing rate. Change only the level somebody is quoting the premium at. Which way does the number an inversion returns then move?

A higher quoted premium restates as a larger implied number, and a lower one restates as a smaller one, and that follows directly from the steadily rising line rather than from anything about the reference asset. The derivation runs as follows. The model returns a larger premium for more travel. So if the level the search runs towards has been raised, the point at which the search meets it has moved along, and the search stops later. The reverse works identically: lower the level and the search meets it sooner and stops earlier. The derivation is three sentences long, and there is no more to it.

Notice hard what that does not license. Nothing in those sentences says how much larger. Nothing says whether the change is small or dramatic. Working out how far the number moves for a given move in the premium needs a specific model with specific assumptions, and neither is available here. The direction is what holds, and the size does not follow from it. A size stated without a model would be manufactured, and a reader would carry that manufactured size away as though it had been measured. A manufactured size is worse than no size at all.

The everyday version is a street vendor and a queue. If somebody watches the queue outside a stall get longer and concludes the vendor must be assuming higher demand tomorrow, the direction of that inference is sound. Longer queue, bigger assumption. But nobody watching the queue can say how much bigger, and anybody who announces a figure has invented it. The direction survives the lack of a model. The size does not.

Why can no implied number be produced from these figures?

The call premium of Rs 180.00/- is stated here. Every other input a model would want is stated here too: the price of Rs 2,000.00/-, the strike of Rs 2,000.00/-, one year to run, and financing of 6.50 per cent a year. Four inputs and an output, all present. The one thing missing is the machine that connects them. A premium cannot become an implied number without a pricing model to push it back through, and naming that model means naming its assumptions out loud. No model has been named, so no model stands behind the premium, and the account stops exactly where the premium stops.

There is a sharper version of that, and it is the version worth carrying away. A number can always be produced by picking a model and stating its assumptions. And the moment one is picked, the number belongs to that pick rather than to the contract, and the confusion at issue is exactly that. A reader who saw such a figure printed would remember the figure and forget the pick. The risk is not hypothetical; it is the ordinary way people read.

What can the worked pair actually settle?

Take the invented pair and push it as far as it goes, then stop at the exact point where it runs out. The stopping point is the most useful part of the whole worked pair.

Three figures here are all Rs 2,000.00/-, and a reader is entitled to wonder whether one of them was copied into another by accident. The three figures were not copied, and here is why they agree. The price of the reference asset is Rs 2,000.00/- for one unit. The strike written into both contracts is Rs 2,000.00/- because the pair is struck at the money, and being struck at the money is precisely what it means for the strike to equal the price of the day. And the exposure a one unit position refers to is Rs 2,000.00/- because one unit of something priced at Rs 2,000.00/- is Rs 2,000.00/- of exposure. Three different quantities, one shared figure, three separate reasons.

Now the arithmetic, and every line of it is recomputed here rather than carried across. The present value of the strike is Rs 2,000.00/- divided by one plus 6.50 per cent, or Rs 1,877.9343/- at four decimals. The parity difference between the two premiums is Rs 2,000.00/- less that present value, or Rs 122.0657/-. With the call premium given at Rs 180.00/-, the put premium is Rs 180.00/- less Rs 122.0657/-, or Rs 57.9343/-, carried below as Rs 57.93/-.

Work that last row yourself rather than taking it. Rs 180.00/- less Rs 57.93/- is Rs 122.07/-, and the parity difference is Rs 122.0657/-. The two are not the same figure. The gap is Rs 0.0042723/-, a shade over four tenths of one paisa, and the whole of it is the rounding applied to the put. So the relationship holds to the paisa and it does not hold exactly, and a claim of exact equality that the rounded figures themselves do not produce teaches the reader to stop checking.

Pushing the same rounding one step further gets sharper still, because the gap changes size depending on how much of the put is kept. Carrying the put forward one year at 6.50 per cent produces three different answers, one for each level of rounding. Only the unrounded value lands on a clean figure. The middle column is one premium moved to the end date rather than a second premium: there is one put here and this is it, standing at a later moment.

So the sentence that the put premium carried a year at 6.50 per cent is Rs 61.70/- is only true of the unrounded put, and any version of it built on Rs 57.93/- falls short by Rs 0.00455/-. This is not a pedantic flourish. The difference is between figures a reader can check and figures a reader learns to wave through.

Now the whole worked pair in one line. The parity difference of Rs 122.0657/- is financing and nothing else, so it would be exactly the same difference under any travel assumption anybody cared to make. It was built from a strike, a rate and a period. No travel input entered it anywhere. A different view about how far the reference asset might move leaves the difference between the two premiums unchanged to the paisa.

So one item is left carrying travel: the level of the call premium at Rs 180.00/-, and only the level. Getting from that level to a number is exactly the machine described earlier, and no such machine is available here. So the worked instance ends the way a careful worked instance should: with a premium written on one side of an arrow and an empty box on the other.

Where has the reader met this discipline before?

The habit this subject needs is already carried over from a subject that looks nothing like this one. The habit comes from credit spreads in fixed income, and the transfer is exact rather than a loose analogy.

There, a default rate is backed outDerived from a price by reversing a calculation, rather than observed directly or forecast. of a credit spread. Somebody takes the spread a bond is dealing at, assumes a recovery, assumes a model, and works out what default rate would account for that spread. The resulting figure sounds like a statement about the borrower. It is not. The backed out rate is a statement about the price of the bond, restated in a unit that sounds like a probability, and the fixed income treatment insists on that distinction.

The structure here is identical: a market price goes in, a model and its assumptions do the work, and out comes a number that sounds like a claim about the future when it is a claim about the price. Same shape, different subject. A reader who already refuses to read a backed out default rate as a forecast about a borrower already knows how to read an implied volatility, and nothing new has to be learned.

One habit carries across both, and it is a habit about words rather than about arithmetic: when a number has been backed out of a price, say backed out and never say forecast, every single time. Those two words send a reader to completely different places. Backed out points at the price it came from and invites a reader to ask which price. Forecast points at the future and invites nobody to ask anything at all. The word does most of the work of the discipline.

How does somebody working with these contracts actually use any of this?

Is any of this a reason to act?

The mechanism ends there, and how an implied number is manufactured can now be described. The natural next question is whether that is a reason to do anything, and the answer to it lies outside the mechanism, for the reasons set out below.

Three things would have to be known first, and none of them appears here. The first is a view on how far the reference asset might travel and how likely each move is, and that view is precisely the input everything above has been explaining is absent. The second is the reader's own circumstances, which no written account can see. The third is what the arrangement would cost to hold all the way to the end and what it would cost to unwind before then, neither of which appears anywhere here.

The first of those three lands in a particular way. The absence of that view is not a missing detail that a longer treatment could supply. The absence is the one the whole subject is built around, arriving from the other direction. An account that cannot produce an implied number is not in a position to say whether an implied number makes a contract worth holding.

And there is a plainer point underneath all three: knowing how a number is manufactured is not a view about whether the number is right. An inversion ranks nothing, compares nothing on how it turned out, and calls no contract attractive or cheap or dear. An inversion turns one quoted premium into one restated number, and a restated number carries no outcome, no track record, no probability and no distribution to rank with. A description of an obligation is a description of an obligation.

The reference asset, its price of Rs 2,000.00/-, the strike of Rs 2,000.00/-, the financing of 6.50 per cent a year and the call premium of Rs 180.00/- are invented.
Educational material. Not advice on any investment, tax, budget or market position.