Implied Volatility vs Historical Volatility: What Each Needs

Historical volatility is worked out from prices that have already been recorded. Implied volatility is read back out of a premium somebody is quoting now, through a model somebody chose. One wants data and no model. The other wants a model and no data. The two are different objects wearing one word, and stacking their levels sets two unlike things against each other.

Two figures can carry the very same unit and still be answers to two different questions. Both get quoted as a distance travelled per year. The shared unit is what puts them next to each other in the same column on somebody's screen, and very nearly everything that goes wrong on this subject starts from that resemblance. One of them summarises a record that exists and can be handed over. The other restates a price that somebody is quoting this morning. On a close reading, neither definition mentions the other anywhere inside it. Neither was built to sit beside the other, and neither needs the other in order to be produced.

Think about a kilogram for a moment. The weight of a sack of rice on a shop scale and the weight a courier writes on a docket before the parcel is packed are both in kilograms. One weight was read off a physical object that was present. The other was somebody working from a form. Nobody would say the two disagree if they came out different. Same unit, different act, different provenance. The mismatch of provenance is the whole of the trouble here, transplanted into finance and given one shared word instead of two.

One habit before any arithmetic, and it costs a sentence each time it is used. Rs 2,000.00/- is a price, and it measures the exposure whoever holds the reference asset is carrying; not a paisa of it changes hands to buy either of the two contracts. Rs 180.00/- is a premium, and somebody genuinely did hand it over on day one. A contract turns into a payoff on its end date. The premium taken back out of that payoff leaves a profit. Four words, four separate jobs, and swapping any two of them wrecks the sentence they are sitting in.

What does somebody actually do to produce a measured figure?

Somebody sits down with a column of prices for the reference asset, one price per line, each line stamped with the day it belongs to. The work happens on the steps between consecutive lines rather than on the lines themselves. The level a price sits at, and the direction it has drifted in, are not what gets summarised. The summary is of how big the steps were, and the arithmetic that does the summarising is a standard deviationA summary of how spread out a set of numbers sits around its own average. Arithmetic on numbers already in hand, making no claim at all about numbers not in hand. taken over those steps. That arithmetic ends in a single figure, and once a restating rule is applied to it the figure gets quoted per year so that it can be filed next to other figures of its kind.

The property worth carrying out of this block is that a measured figure eats data and assumes nothing whatever about how prices behave. No view about the future is fed into it. No claim about what the reference asset is going to do is buried anywhere inside it. The figure is a summary, in one number, of a record that already exists, and the record is the only thing it has ever touched.

Four things have to be stated alongside it or the figure means very little. The column of prices itself is the first. A figure taken over one column and a figure taken over a different column are not the same figure even when they land on the same value. The intervalThe spacing from one observation to the next in a series: daily, weekly, monthly. Whoever takes the observations picks the spacing, and the pick changes the answer. between one observation and the next is the second. How far back the column runs is the third. And the conventionAn agreed way of restating a figure so that two people's numbers can sit in one column. Which one was used is a choice somebody made, and it belongs next to the figure. used to restate the answer per year is the fourth. How each of those four is done, and what the arithmetic looks like once they are settled, is covered separately; what matters in this guide is only that four choices exist and that all four are somebody's.

Everyday version. A shopkeeper is asked how much the price of one sack of rice bounced around last year. He can answer, but only by opening the book he wrote the prices in. He needs the book, he needs to have written prices in it often enough, he needs to know which months he is being asked about, and he needs to say whether he is answering per month or per year. Take the book away and the question is not hard, it is unanswerable. Nothing else can stand in for the book. He cannot get to the answer by asking a customer what a sack feels like it might be worth.

Notice what he never has to do. He never has to say what he thinks rice is going to do next. He never has to choose a theory of how prices move, or borrow one from somebody who has. He opens the book, he does arithmetic on what is written in it, and he reads out the result. If a second shopkeeper is handed the same book and told the same four choices, the second shopkeeper produces the same figure, and the two of them have nothing left to argue about except whether the four choices were the right ones.

What does somebody actually do to produce an implied figure?

Somebody reads a premium off a screen. Rs 180.00/- is being asked for a call on the reference asset. Then they pick a piece of machinery that turns a set of inputs into a premium, feed it everything it wants bar one input, then search until that missing input takes a value at which the machinery returns exactly Rs 180.00/-. The value they land on is the implied figure. The implied figure is not measured, not observed, and not looked up. It is the travel assumption that machinery must be given before it will return the premium being quoted.

The property worth carrying out of this block is the mirror image of the last one: an implied figure eats no history at all. A contract written on something with no past whatever could still carry one, provided a premium is being quoted for it and somebody has chosen a model. Doing without history is not a technicality and not an edge case. The absence of a series is the sharpest single difference between the two figures, and a reader who holds only this one fact is already ahead of most of what gets said about the subject.

Four things have to be present here too, and not one of them is a price from the past. The quoted premium is the first. Without a premium there is nothing to reproduce. Everything else the machinery wants is the second: the price the reference asset stands at, the level the contract is struck at, the financing rate and the stretch of time left to run. A model is the third, and somebody has to choose it. The assumptions living inside that model are the fourth, and they need saying out loud rather than being left where the manual buried them.

The provenanceWhere a figure came from and by what route: who produced it, out of what material, and on which day. of an implied figure is therefore a quotation plus a piece of machinery. Change the quotation and the figure moves. Change the machinery and the figure moves as well, with the quotation sitting exactly where it was. There is no machinery on the measured side to change, so the second sensitivity has no counterpart there.

Everyday version, and it is the same shopkeeper. A customer offers Rs 45/- for a bag of onions the shopkeeper never priced. The shopkeeper can work out what belief about the coming week would make Rs 45/- a sensible offer, but only by assuming something about how that customer reasons: what margin the customer works on, how the customer treats a week of storage, what the customer thinks the onions weigh. Change any assumption about the customer and the belief he backs out changes with it. Rs 45/- sits there unmoved. He has learned something real about the offer. He has learned nothing at all about onions.

Do the two share a single ingredient anywhere?

Set the two shopping lists next to each other and look for an item that appears twice. There is not one. The measured figure wants a series, an interval, a window and a restating rule, and it wants no model. The implied figure wants a quoted premium, the model's other inputs, a model and that model's assumptions, and it wants no series. Because two estimates of a single quantity are always built out of evidence that overlaps somewhere, and these two overlap nowhere at all, they cannot be two estimates of one thing.

The claim just made is the load-bearing one, and it is worth slowing down on. An estimateA figure worked up from evidence about a quantity nobody can read off directly. Two of them aimed at one quantity are built on evidence that overlaps. is a figure somebody produced from evidence about something they could not read off directly. Two people estimating the height of the same tree are both looking at the tree, or at a photograph of it, or at a shadow it cast; the evidence overlaps, and that overlap is precisely why it makes sense to ask whose estimate is better. Here, one person is looking at a ledger and the other is looking at a price tag and a rulebook for turning price tags into assumptions. Asking whose figure is better is asking a question with no purchase.

The plain version, and it is the sentence to remember: these are not two readings taken off different instruments. The two are readings of different things. Two thermometers in one room can disagree and one of them is wrong. A thermometer and a clock cannot disagree at all, and noticing that they show different numbers is not a discovery.

Which stretch of the calendar does each one describe?

Readers collapse this contrast most often, and it is the easiest one to see once it is drawn. A measured figure describes a window that has closed. The window runs from its first observation to its last, and every single observation inside it had already happened at the moment somebody used it. Nothing in it is provisional. Nothing in it can be revised by anything the reference asset does tomorrow.

An implied figure is fastened to a contract that has not closed. The figure is read out of a premium being quoted today, and what it restates is what that premium says about the stretch running from today to the contract's last day. Every day of that stretch is still in front of everybody. Not one of them has happened.

The two are not even describing the same calendar, so a difference between them is not a disagreement about anything. Contradiction needs a shared subject, and there is no shared subject here. Two people who each read out a figure, one covering last quarter and one covering the coming year, have not contradicted each other. If somebody says the two figures disagree, the first thing to establish is what they think both figures are about.

The household version runs like this. A household's electricity bills for the past twelve months are sitting in a drawer, and the arithmetic on them is settled and unarguable. The quote a contractor gives this morning for rewiring the house next summer is a different kind of object entirely. Both are in rupees. Both are about the same house. One of them is a record of what left the account and the other is somebody's asking figure for work not yet begun. Setting the two side by side and announcing that one is higher tells nobody anything.

Hand the same task to two people: do they land on the same answer?

The third contrast is the most practical of the three, and it settles what to ask for rather than what to think.

A measured figure is reproducible outright. Hand two people the same column of prices, tell them the same interval, the same window and the same restating rule, and both of them come back with the same figure. If they come back with different figures, one of them has made an arithmetic slip or one of the four choices was not actually shared. Arithmetic and an unshared choice are the entire space of possible disagreement, and the space is small and inspectable. Somebody can walk into it and settle the matter.

An implied figure is reproducible only inside a model. Hand two people the same premium of Rs 180.00/-, the same price, the same strike, the same financing rate and the same time left to run, but let them choose different machinery, and they come back with different figures. Neither of them has made a mistake, and that is what makes this the awkward case rather than the obvious one. Each has correctly reported what their own machinery would have had to be fed. The figures differ because the machinery differs. The difference is a fact about the two rulebooks and not about the reference asset.

People call this model riskThe chance that an answer turns out to be a property of the model somebody picked rather than of the thing the answer is about., and it lands on one of the two figures and not on the other. A measured figure carries choices, and choices can be argued about, but it carries no machinery for the choices to hide inside. The contrast is easy to overstate, so the honest version matters: a measured figure is not more trustworthy, it is more inspectable. Four visible choices is a different situation from four visible choices plus a rulebook, and it is a smaller one to check.

So what a reader should do with each is not the same. A measured figure asked for its four choices has been asked for everything there is. An implied figure asked for its inputs and nothing further has been asked for roughly half. Which rulebook produced it, and what that rulebook takes for granted, has gone unasked. The rulebook question is the one that gets left out, and it is the one that decides whether two figures in a report are even the same kind of statement.

What is really being set against what when the two are printed together?

Somebody puts the two levels in one sentence. What has actually been placed beside what? On one side, a summary of one stretch of the past, taken over a window somebody picked. On the other, a restatement, through a rulebook somebody picked, of a price about a stretch of the future. There is a third object in that sentence and it never gets named: the pair of choices holding the whole comparison up.

The comparison is not forbidden, though performing it takes more than the two numbers, and the reason is plain: the meaning of any gap depends entirely on which window was chosen for the first figure and which model was chosen for the second. A gap quoted without both of those attached is not an imprecise number. The gap is instead a number with no interpretation available for it at all. An imprecise number still points somewhere, so the missing interpretation is the worse condition and much the easier one to miss.

So there are exactly two questions to put before accepting any such comparison, and they are short enough to ask out loud in a meeting without sounding difficult. Which window? Which model? If either answer does not come back, the sentence handed over has not been made more approximate by the omission. The sentence has been left without a subject.

There is a version of this everybody has already met. Somebody says the vegetables at one shop are dearer than at another. Dearer measured how, over which basket, on which day, at what quality? Until those come back, the sentence is not slightly unreliable, it is empty; and if the two shops were weighed on different baskets, the person who said it has not exaggerated, they have simply not said anything yet. An exaggeration can be discounted and an empty sentence cannot be, so the difference between these two conditions matters.

Why can neither figure be produced from an invented pair of contracts?

The worked case carries no column of past prices for this reference asset, so the measured figure cannot be computed. No pricing model stands behind these two contracts either, so no implied figure can be run backwards out of Rs 180.00/-. Both of the figures set against each other above are therefore absent from the material to hand.

Neither of the two figures exists to be moved, so a control that moved one and watched the gap respond would have to invent both, on axes labelled with the two quantities that are precisely what is not held. The three panels below stand for the two blanks and the one figure that survives them.

So the comparison gets taught as a contrast between objects rather than a contrast between levels. The two differ in the material each is built out of, in the stretch of the calendar each is about, and in what each can honestly be said to describe. Which of the two stands higher belongs to whoever holds both figures.

Every contrast above holds without either figure in hand, whatever values the two figures might have taken. The material each one is made from, the stretch of time each one covers, and the way each one can be checked are properties of the two objects rather than of their values. A reader who has taken all three can handle both figures correctly the first time they meet one in the wild. A reader who was shown two numbers and a gap usually cannot.

Run the invented pair through both definitions: what comes out?

Everything below belongs to the same invented pair of contracts this guide runs on. A price of Rs 2,000.00/- attaches to the reference asset. The figure measures exposure, and no part of it has left anybody's account. Holding the reference asset on borrowed money is charged at 6.50 per cent a year. Both contracts reach their end date twelve months out. Nothing is thrown off by this reference asset while somebody sits on it, so no payment interrupts the year in either direction, and that matters because a payment during the year would move every figure below.

Two figures in this guide read Rs 2,000.00/- and that is not a slip of the keyboard. The first is the price the reference asset stands at. The second is the level the pair of contracts was written at. The two agree because whoever wrote the pair picked the level so that it would agree, and a pair picked that way is described as struck at the money. A premium of Rs 180.00/- attaches to the call, and one of Rs 57.93/- to the put. Working either of them out from scratch would need exactly the quantity shown absent above, so both arrive from the record rather than being worked out here.

First, the measured figure

The measured figure would want a column of prices. One price, carrying one date, is the whole of what sits here. A single observation is not a series, and there is no arithmetic that turns it into one. So the figure cannot be produced, and the space where it would go stays blank.

Second, the figure that would be read out of the premium

The implied figure would want the quoted premium, and that is here: Rs 180.00/-. Next it would want the model's other inputs, and those are here too. Rs 2,000.00/- of exposure is one of them. The level struck, also Rs 2,000.00/-, is another. A financing charge of 6.50 per cent a year is the third, and the fourth is the twelve months still on the clock. Then it would want a pricing model, along with the assumptions living inside it. Neither of those is here. So this figure cannot be produced either, and the space where it would go again stays blank.

Third, the figure that can be produced

Set the two premiums against each other. Rs 180.00/- with Rs 57.93/- taken off it leaves Rs 122.07/-. That survivor is not a travel figure of any description: it is what waiting a whole year to pay Rs 2,000.00/- is worth on day one.

The other direction supplies the check. Dividing Rs 2,000.00/- payable in a year by 1.065 gives what it is worth on day one, and Rs 1,877.9343/- comes back. The figure carries a name, the present value of the strike, and that name is needed here and nowhere else. Standing on top of it there is room for Rs 122.0657/-. The record rounds the put to two decimal places, so the two routes finish 0.43 paise apart and the relationship holds good to the paisa rather than beyond it. Anybody wanting the unrounded put can read it straight off the same line: Rs 180.00/- less Rs 122.0657/- is Rs 57.9343/-.

One more check is available and it is worth doing, because it shows what the survivor really is. Take Rs 122.0657/- forward twelve months at that same 6.50 per cent a year and Rs 130.00/- comes back exactly. Rs 130.00/- is the year's financing on Rs 2,000.00/-. Not approximately: the identity is exact on the unrounded figure. The room between a price and the sum that grows into the strike, carried forward one year, is arithmetically the same as the price multiplied by the rate.

So the one figure available to a reader here is precisely the figure that is silent on the subject of this guide, and saying that plainly carries its own weight. A reader who leaves knowing that Rs 122.07/- is financing and not travel will never mistake a premium difference for a statement about movement. The mistake is a specific one and it does get made.

With neither figure in hand, what can still be said out loud?

A reader who has followed the three contrasts already has everything needed for the practical question. The answer comes as two lists rather than as argument. The first list is longer than most readers expect, and the second is short, with a stated reason attached to each refusal in it.

Start with what can be said. A premium is what somebody actually paid, and turning it into a travel figure needs a model whose assumptions have to be named before the result means anything. A measured figure describes the window it was taken over, and the window belongs in the same sentence as the figure rather than in a footnote under it. One of the two runs on data and the other runs on a model. A gap between them cannot be interpreted at all until the window and the model are both attached to it. And when one level carries both a call and a put, whatever separates the two premiums is financing and not travel, with the arithmetic available in one line: Rs 180.00/- less Rs 57.93/- leaves Rs 122.07/-, against Rs 2,000.00/- less Rs 1,877.9343/- giving Rs 122.0657/-.

Now what cannot be said, and why not in each case. Whether either figure is high or low: high and low are comparisons, and no second figure has been set beside either one. Whether either figure implies anything about what happens next: neither one holds a claim of that kind inside it. Which of the two stands higher: neither of them is written above.

Not having a figure and not knowing what the figure would even be are two different states, and only the second leaves a reader unable to ask the right question. The distinction between those two states is the thing worth carrying away. A reader in the first state can walk into a room, be handed a number, and know within one sentence what to ask for. A reader in the second state can only nod.

Who has to keep these two apart on a working day?

One line on a screen: a figure, a per year unit after it, and nothing else. Whoever passed that line along filled a single field and left three of them blank. The blank fields are the material the figure was built out of, the stretch of the calendar it covers, and the route by which it was produced. Filling those three is the whole of the working skill on this subject, and a reader who asks for them by name gets a straight answer or gets silence, and the silence is informative too.

The field marked what it was built out of comes first. Somebody filling that field writes either a column of prices or a quoted premium and a rulebook. Nobody ever writes both. The moment that field is filled the rest of the conversation is decided. The branch it opens carries its own three questions and rules out the other three. A figure whose source field says premium and rulebook cannot be checked by asking for a longer window, and asking anyway signals that the branch was never established.

The field marked which stretch of the calendar comes next. Filling it means writing two dates, and the striking thing is how often the field comes back with only one. A figure handed over with a single date attached to it is a figure whose owner has not decided whether it describes a period or a moment, and until that is settled nobody can say whether two such figures set side by side overlap, abut or sit years apart.

Take the field marked how it was produced. On the measured branch this field is filled by naming four choices, a task both tedious and finite. On the implied branch it is filled by naming the same kind of inputs plus a rulebook and what that rulebook assumes. The field usually goes thin there. The rulebook is often institutional rather than personal, and the person handing over the figure did not choose it. The most useful sentence a working reader ever says on this subject is a request for a field rather than a judgement about a level.

None of that requires either of the two figures to be in hand. Framing the skill as filling in fields is what makes that possible: the fields exist whether or not anybody has the values, and a reader who knows the shape of the form can audit a figure they have never seen before, produced by a method they were not told about, in the first minute of the conversation.

Is telling these two apart a reason to do anything?

Whether a reader should be doing something about all this is a separate question. Answering it wants an opinion about the size of the moves the coming year might contain and about which of them are likely. Neither figure would hand that opinion over even if both were sitting in front of the reader: one summarises a window that has closed and the other restates a price. The answer wants the reader's own position too, and no written account can see that. And whatever arrangement a reader had in mind carries a cost to hold and a second cost to get back out of, neither of which is written anywhere above.

Being able to tell two figures apart is something done while reading, and nothing done while reading is an instruction to act. All of it leaves a reader in a position to receive either figure without misreading it: to know what to ask for, to know which stretch of the calendar is being described, and to know that a gap between the two is not a message until two choices are named. The capability is real and it is worth having. A capability is not the same thing as a reason to act, and the difference between the two is worth as much as anything above.

The reference asset, the level the pair is struck at, the two premiums quoted beside it and the financing rate carried through the year are invented.
Educational material. Not advice on any investment, tax, budget or market position.