Historical Volatility: Measuring What Already Happened

Historical volatility is a measure worked out from prices that have already happened. Producing one takes four things: the run of prices itself, the gap between consecutive observations, the stretch of time covered, and a stated rule for restating the result per year. The finished figure describes what the price did across that stretch. The figure carries no claim at all about what comes next.

Everything awkward about this measure falls out of one fact that sounds far too small to matter. Movement is a relation between two observations, so it cannot be read off one. Once that is accepted, four questions arrive unbidden: which observations were taken, how far apart they sat, how many of them there were, and by what rule the answer was restated to a shared period afterwards. Somebody decided all four. The finished figure carries every one of those decisions inside it and announces none of them.

Behind the contracts used in this guide sits a single price on a single date. There is no second date. A measurement of movement therefore has nowhere to start, and the absence shows up as an empty slot at the exact point where the measurement would otherwise happen.

What is this measure actually looking at?

Most readers meeting the word for the first time reach for the level of the price, or for the direction it went. Neither is what is being measured. Historical volatility looks at the size of the movements between one observation and the next, and it looks at nothing else. Where the price finished is not an input to it. Whether the price rose or fell over the stretch is not an input to it either.

The arithmetic is built out of the steps rather than out of the destination, so a price that climbed steadily and a price that finished on the level it started from can give the same reading. Everything that follows is a consequence of that one fact.

The point stands away from prices. Ten shops sit along one mall corridor, and a year of daily takings exists for each of them. Two completely different questions can be asked of that stack of paper. One is which shop took the most over the year, a question about totals and about where each shop ended up. The other is how much each shop jumped about from one day to the next, a question about the daily differences with nothing to do with the total. The sweet shop and the phone repair counter could finish the year on the same annual takings while one of them was steady as a metronome and the other swung wildly on wedding weekends. Historical volatility asks the second question. The first question it never asks.

The shape of a movement, and not its endpoint, is therefore what a reading describes. Two paths can travel the same total distance in small pieces and land in completely different places, purely by the order in which the pieces went up and down. Reorder the pieces and the destination moves; the pieces themselves do not change at all. Since the arithmetic only ever sees the pieces, reordering them leaves the reading exactly where it was.

What has to be in hand before a measurement exists?

Four things. Set out as four separate items, each one can be ticked or left blank; buried inside a block of prose, the same four slide past a reader who has no place to put a tick.

One, a run of prices. Two observations before there is anything at all to measure between, and a great many more before the answer is worth quoting. The run of prices is the raw material, and it is the only one of the four that somebody has to be handed rather than decide.

Two, a stated gap between the observations. Whoever produced the figure looked at the price at intervals. Once a day, once a week, once every fifteen minutes. Until that gap is written down beside the figure, nobody reading it knows what the word consecutive means in the sentence that produced it.

Three, a stated stretch. The window the run of prices covers, from its first observation to its last. Six weeks and six years are both legitimate answers to a question nobody asked out loud.

Four, a stated rule for quoting the result per year. The measure is a distance per unit of time, so the unit has to be declared, and turning a per-day result into a per-year one takes a convention somebody adopted rather than a fact somebody discovered.

A figure quoted without all four is not a usable figure, and asking for the four is the whole of what checking one amounts to. Asking for the four is not a counsel of perfection. Four is the minimum, and the reason becomes visible the moment the four appear written into the arithmetic as symbols.

Now the same four things, written as arithmetic. Seeing them as symbols is what stops them feeling like housekeeping. Start with what a single step is.

Read the subscripts rather than the logarithm. Every single symbol in that line needs a neighbour. There is no version of it that can be evaluated at a lone price, and no amount of care with the rest of the method rescues that. Then the steps get gathered up over the window and restated per year.

Look at where each of the four requirements landed. The run of prices supplies every P. The gap decides what the subscript i counts. The window fixes n. The convention fixes m. Four requirements, four places in the arithmetic, and not one of them optional. The divisor of n minus one and the scaling by the square root of m are both parts of the convention somebody adopted rather than facts about the world. How many observations is enough, and how far a finished figure could have landed from a different draw of the same stretch, come under sampling errorThe wobble in a figure that comes from having measured a limited amount of data rather than all of it. Sampling error is a property of the estimate, not of the thing being estimated., and is covered separately.

Why does how often somebody looked change the answer?

The gap between observations is the requirement readers dismiss fastest, filed under method rather than under meaning. The gap belongs under meaning.

A single stretch of price history can be sampled once a day, or that identical stretch sampled once a week. The daily sampling sees a long chain of small movements. The weekly sampling sees a short chain of larger ones, and it never sees the wobbles that happened inside a week and cancelled out before Friday. The two samplings yield different collections of steps. Different collections of steps fed into the same arithmetic give different numbers. The consequence follows directly, and it is not what most people expect.

Two measures of the same asset over the same stretch can differ purely because of how often somebody chose to look, so comparing them without first matching the gap is comparing the answers to two different questions. Neither reading is the mistake. The mistake is putting them side by side.

The mall corridor again. Counting the sweet shop's jumps day by day picks up the difference between a Tuesday and a Wednesday. Counting them month by month instead, every one of those Tuesday-to-Wednesday differences vanishes inside a monthly total that only ever gets compared with another monthly total. The shop did not become calmer. The looking happened less often, and looking less often is a different measurement rather than a rougher version of the same one.

So the habit worth forming is short: ask the gap before comparing anything. If two figures are being set against each other and nobody in the room can say what gap each was built on, the comparison has not been made yet, whatever the two figures look like next to one another.

Who decided how far back to go?

Somebody did, and nothing in the arithmetic helped them. There is no term in either of the lines above that fixes where the window starts. The window has to be handed in from outside, by a person, before the sum can begin.

A short window describes a recent stretch. A short window moves quickly as new observations arrive. Each new step is a large share of the small pile it joins. A long window describes a longer stretch, and it moves slowly for the mirror reason: a new step joins a large pile and shifts it very little. The two are answers to different questions, so neither is the right one, and the question got chosen at the moment the window did.

The practical shape of the window choice surprises people who assume arithmetic settles arguments, so it is worth stating flatly. Two people can measure the same asset on the same day, both do every step of the arithmetic correctly, check each other's work, find nothing wrong with it, and report different figures. The two of them have not made an error between them. Their disagreement was about which stretch of the past anybody wanted described, and that disagreement survives every check either of them could run.

Household version. How much an electricity bill jumps about across the last three bills, and how much it jumps about across the last three years of bills, are two different questions. The short answer will move sharply the next time one bill lands high; the long answer will barely notice it. Both are honest descriptions of what was asked. Two things were asked.

A figure that arrives with no window attached is therefore not merely incomplete, it is unattached to any question. Somebody has done arithmetic on a stretch of the past and has not said which stretch, leaving the reader holding a description of something unidentified.

What does the finished figure describe, and where does it go silent?

The positive comes first, and it is genuinely a strength. The finished figure is a summary of a record, and it is checkableAnybody handed the same inputs arrives at the same answer, with nothing left to take on trust. Checkability is a property of the procedure, not a claim about how good the answer is.: hand two people the same run of prices, the same gap, the same window and the same convention, and they will land on the same answer. Reproducibility is rarer than it sounds. Very little else in this part of the subject has that property, and it is worth naming rather than assuming.

Now the silence. Nothing in that arithmetic reaches forward. Every observation used had already happened at the moment it was used, and no step of the calculation consulted anything that had not. Neither of the two lines of arithmetic contains a term that looks ahead. The result is a description of a stretch of the past, and it contains no claim whatever about the next day, the next week or the next year.

The word historical in the name is not decoration. The word is the scope of the claim. Carry that one sentence away.

The silence is worth pinning down. The silence is not the measure being unreliable, and it is not a caveat about accuracy. A figure can be measured perfectly, from a flawless run of prices, with every choice stated and every step checked, and remain completely silent about tomorrow. Perfection in the measuring does not buy a single word about what has not happened. The silence is structural, and it is a property of what was fed in rather than of how carefully anybody worked.

Why can no measurement be made from a single price?

Because a measurement of movement needs two observations before it has anything at all to measure between, and what is on hand here is one price on one date. The first line of arithmetic wants a P at position i and a P at position i minus one. There is no position i minus one here. Not a rough one, not an approximate one. There is no second observation to be had.

So a single price on a single date supports no measurement at all, and no care taken over the other three requirements supplies one. Every ingredient a measurement would take is named above, and the slot where the result would otherwise sit stays empty. Being general about an absence is how a reader ends up assuming the absence is smaller than it is, so the shape of the shortfall is worth being exact about.

Here is the exact shape. The gap between observations could be chosen and written down in one line. So could the stretch. So could the rule for restating per year. Three of the four requirements are decisions, and a decision costs nothing but the deciding. The fourth is data, and a second price cannot be conjured out of a first one. Three choices and one blank, and the blank is the one that matters.

The blank rules out one tempting move. A run of prices could be made up. The made-up run would look entirely convincing, the arithmetic would run on it without complaint, and a figure would come out of the far end with four decimal places and every appearance of having been measured. The figure would be a measurement of an invention. A reader has no way of telling from the figure itself which kind it is, and that makes such a figure worse than no figure at all.

What the audit finds here, and the one check it can still show

An audit, not a calculation, is what the record here supports. The shortfall is not arrived at after a stretch of arithmetic; it is stated first and then examined.

Start with what the record actually prints. One unit of the reference asset stood at Rs 2,000.00/- on a single named date, and that Rs 2,000.00/- measures exposure rather than an outlay: it is what a unit puts at stake, and nobody handed it across. Financing runs at 6.50 per cent a year. Twelve months is the life of every contract mentioned anywhere here. Across those twelve months nobody holding the reference asset collects anything whatever from it. A payment arriving mid-year would shift every financed figure printed further down. Both options sit at a strike of Rs 2,000.00/-. The strike matches the price exactly, and one was not lifted off the other: struck level with the price of the day is precisely the arrangement the phrase at the money names, and this pair is constructed that way. The two premiums come from the record and from nowhere else: Rs 180.00/- for the call, Rs 57.93/- for the put, with no model standing behind either.

Now the audit, requirement by requirement.

Three of the four requirements are choices anybody can make and the fourth is data nobody here has, so the measurement cannot be made.

One arithmetic check can still be shown in full, and the account then ends with something reproducible rather than only with an absence. The check sits directly beside the audit, putting a figure that can be produced and a figure that cannot on one screen.

Divide the Rs 2,000.00/- strike through by one plus the financing of 6.50 per cent a year. The division pulls the strike back to today and lands on Rs 1,877.9343/-. Set that against the reference asset price and the distance between them is Rs 122.0657/-. Now reach the very same quantity down a second road, through the premiums instead: Rs 57.93/- comes off Rs 180.00/-, and Rs 122.07/- is what stays behind. Not one step is shared between the two roads, and they finish 0.43 paise apart.

The 0.43 paise is not a fault in the arithmetic and it is not something to smooth over. The put premium has been rounded to two places by the record it comes from. Carried to four it would read Rs 57.9343/-, and at four places the two routes agree. To the paisa, yes. Exactly, no. Announcing a clean equality that the premiums as printed cannot deliver would quietly train a reader out of the one habit worth taking away from a check like this. The gap also has to be handled at its full precision for any further check. The unrounded gap carried forward a year at 6.50 per cent lands on Rs 130.00/- to the last decimal, one year of financing on Rs 2,000.00/-. Carrying the printed Rs 122.0657/- forward instead gives Rs 129.99997/- and would look like a break that is not there.

Set beside the audit above, the contrast is the teaching. Rs 1,877.9343/- was produced here, in the open, from figures printed in this guide, and it can be reproduced on a phone. The measured figure this guide is named for was not produced and cannot be produced. The material it consumes is absent rather than imperfect, so no amount of care would change that.

What can be done with the requirements when the figure is somebody else's?

The four questions below are the practical output of the account, and they are the reason the four requirements were worth naming as a list. Whenever somebody hands over one of these figures, the four requirements turn into four questions. Which run of prices. Sampled how often. Over what stretch. Restated per year by what rule.

Asking all four produces one of exactly two outcomes. Either four answers come back, in which case an unusable figure has just become a usable one and what it describes is known. Or they do not, and the silence establishes that the person quoting it cannot say where it came from. Both of those outcomes are worth more than the figure was, and that is why the questions are worth asking even when the answers are expected.

The questions do not test whether the figure is right. The questions test whether it means anything. Meaning and correctness are different tests, and the second one has to pass before the first is even sensible to run. A figure that nobody can attach to a stretch of time and a sampling gap is not a wrong answer to the question asked; it is not an answer to any question.

Asking for provenance is not a habit peculiar to this measure. The same discipline covers every figure anywhere. Ask for the source and the as-of dateThe date a figure describes, which is not the same as the date somebody sent it. Two figures from different as-of dates are not comparable however similar they look. alongside the number, for the same reason every time. A figure without its provenance is a rumour with decimal places.

What does one of these figures look like written down and passed along?

The failure at issue happens in transit rather than in the arithmetic, so the written line is worth taking apart. A measured figure almost never travels as a measurement. The figure travels as a line of text in a note, a cell in a working file, or a sentence in a meeting, and what survives that journey is decided by which parts of the line somebody bothered to type.

So take the line itself apart. A properly written one has five parts, and each part is a box somebody has to fill.

Four of those five boxes are usually empty, and the one everybody fills is the only one that means nothing on its own. The emptiness of those four boxes is the whole practical lesson, and it applies whether the person filling the boxes is writing a note for a lender, building a working file for a review, or repeating something they heard.

Think about who is downstream of each empty box. Somebody reading the note next quarter needs the stretch, or they will compare this figure with a later one that covered a different period and read a change that never happened. Somebody rebuilding the working file needs the gap and the rule, or their rebuild will disagree with the original and both of them will spend an afternoon hunting for an error neither made. Somebody being asked to rely on it needs to know which published prices went in, and that box in particular is the one where the reference priceThe single price for a given day that a market or an authority arrives at and publishes, and an observation in a run of prices is exactly that. How it is arrived at is set by the authority, not chosen by whoever is measuring. for each day, and which days count as a dealing dayA day on which dealing takes place, so that a gap between two consecutive observations spans one of them. Which days these are is decided by the authority and can change. at all, stop being a private choice and become somebody else's published arrangement.

The household version is a bill again. An envelope carrying nothing but the note that the electricity jumps about by such and such tells the next person, six months on, only that somebody once did a sum. Written with the two dates and how many bills went in, it stays useful for years.

Why does this one need no model behind it?

Here is the one respect in which this guide is easier than everything around it, and it is worth having stated cleanly.

The arithmetic here consumes data and produces a summary. The arithmetic assumes nothing about how prices behave, fits nothing to anything, and contains no free parameterA quantity inside a method that is not given by the data and has to be set by somebody before the method will produce anything. The more of these a method has, the more of its output belongs to whoever set them. that somebody had to settle beyond the three choices already named. The arithmetic is not fittedAdjusted until the method reproduces something that was observed. A fitted quantity depends on what it was fitted to and on the shape it was forced into, both of which are decisions. to anything, and it rests on no stochastic processA mathematical description of how a quantity moves through time at random. Choosing one is an assumption about the world, and the mathematics of that choice is covered separately and is not needed here. of any kind.

The other figure that shares this word is not produced from data at all, and how the two differ as objects is covered separately rather than argued here. The consequence for checking is what matters, and the consequence is sharp.

A measured figure can be disagreed with in exactly three places: the gap somebody chose, the stretch somebody chose, and the rule somebody chose. Agree on those three and two people cannot disagree about the answer. The arithmetic between the inputs and the output has no room in it for an opinion. A figure that comes out of a model can be disagreed with in all three of those places and in one more: the model itself. The fourth place is a different kind of argument from the other three. The first three are about what question was asked. The fourth is about whether the machinery answering it describes the world.

None of that makes the measured figure better. The measured figure is merely easier to interrogate, a different and smaller virtue, and one worth keeping in proportion. An easily checked description of something irrelevant is still irrelevant.

Does knowing how it is built tell anybody to do anything?

No answer to that comes from here. Nervousness has nothing to do with it either, and the actual grounds can be set out in full, so they are set out in full below.

Turned round, the question explains itself. To say yes or no, somebody would first need a view about the ground this reference asset could cover between now and the end date, and about the odds on each place it might stop. A finished stretch of dates hands nobody that, however carefully every step inside it was measured, and this guide has spent its length on why. Somebody would also need to know the reader's position, what has already been committed, and what a bad run would do to both, and a written account sees none of that and should not pretend to. And somebody would need the cost of keeping an arrangement alive to the end and the cost of getting out of one early, neither of which is worked anywhere above.

Knowing how a measurement is assembled shows how to interrogate one that lands on a desk, and interrogating is the skill this guide teaches. Interrogation is a real skill and it holds its value on its own. A reason to go and do something is another matter altogether, and the distance separating the two is no technicality.