Volatility as a Quantity
The same sixty bars carry a per-bar standard deviation of 0.32 per cent and one of 3.43 per cent, and nothing between those two readings is a change in the market. It is a change in how many bars you measured over. Volatility is a number you compute rather than a mood you sense, and the computation is four steps long: turn the closes into returns, take the standard deviation, count how many of those periods fit in a year, multiply by the square root of that count. The square root of 252 is 15.87, and that single line is the entire origin of the rule that divides a VIX reading by sixteen. But one word is covering three different numbers. Over the whole run the figure is 1.50 per cent, nearly twice the median twenty-bar reading. Scaling one bar up to twenty by the square root of twenty overstates this series’ actual twenty-bar spread by more than half again, because its bars lean against each other. And the number an option chain quotes is a price rather than a measurement: the straddle at 6.00 that lesson 43 priced is quoting a standard deviation of 7.52 per cent, not 6, because a straddle costs the expected absolute move and that is 0.798 standard deviations. One piece of arithmetic survives all three complications untouched, and it is the one worth having. Share count is inversely proportional to volatility, exactly.
Prerequisites: Lesson 38, which established that a measurement belongs to the window it was made over by counting 15 swings in one series and 1 in the same series read wider, lesson 43, whose straddle at 6.00 this lesson takes apart one layer further, and lesson 20, whose sizing arithmetic is what a volatility number finally feeds.
Four steps, and the assumption inside each one
Volatility is the standard deviation of returns, and getting it out of a series of closes takes four steps and no judgement at all. Turn each pair of adjacent closes into a return, using the natural logarithm of the ratio rather than the plain percentage change: at the sizes traders deal in the two barely differ, since a 1 per cent move is a 0.995 per cent logarithm and a 5 per cent move a 4.879 per cent one, but logarithms add across periods where percentages do not, and every scaling rule below depends on that addition. Take the standard deviation of those returns, which is the volatility of one bar and for most purposes is the answer. Decide how many bars fit in the period you want quoted — 252 sessions in a trading year, 60 five-minute bars in a session, whatever your series is made of. Multiply by the square root of that count.
Each step carries an assumption worth naming before it starts doing damage. The first assumes returns rather than prices are the stable quantity, which is why a 1 dollar move on a 20 dollar stock and a 5 dollar move on a 100 dollar stock count as the same event. The second assumes the returns came from one distribution, that the market was doing the same thing throughout the window. The fourth is the square-root-of-time law, and it assumes each period’s return is independent of the last. That fourth assumption is the one that breaks, and this lesson measures how badly it breaks on the module’s own tape.
Where sixteen comes from
The square root of 252 is 15.8745. Rounded to 16 it is wrong by 0.79 per cent, which is smaller than every other error in the exercise. That is the whole derivation, and it is worth having done once rather than accepting as folklore, because knowing where the number came from tells you when it stops applying: change the period count and the factor changes with it. Weekdays instead of trading days gives 16.12; calendar days gives 19.11.
A VIX reading is an annualised standard deviation quoted in percentage points, so dividing by 15.87 runs the fourth step backwards and returns a one-session number. A reading of 16 implies 1.01 per cent a session, 30 implies 1.89 per cent, and 80, roughly the most extreme the index has reached, implies 5.04 per cent. Note what has just happened, because nobody says it out loud: that conversion is itself an application of the square-root law, over a period during which the law is being assumed rather than checked. The rule that makes the index usable inherits the assumption the index cannot verify.
A standard deviation is not a typical day
Here is the first place the common version misleads, and it misleads in the direction of making everything sound larger than it is. A standard deviation is not the size of an average move. For a normal distribution the average absolute move is 0.798 standard deviations and the median absolute move is 0.674 of one. So the reading that implies 1.89 per cent is describing a day exceeded on about 32 per cent of days: the average day under that same reading moves 1.51 per cent and the median day moves 1.27 per cent.
The gap is not academic. Told the market expects 1.89 per cent, you place a stop 1.89 per cent away and think you have bought a quiet day’s worth of room. You have bought a stop that ordinary noise reaches about a third of the time, before your idea has been tested at all. Saying that a reading of 30 means about 1.9 per cent a day overstates the average day by a factor of 1.25 and the median day by 1.48. Both numbers describe the same distribution correctly. They answer different questions, and the question a stop asks is the second one.
Which window? The lesson 38 problem, twice over
Lesson 38 established that a measurement belongs to the window it was made over. Volatility hands you two windows rather than one, and they are independent of each other. There is the estimation window, how many bars of history you feed the standard deviation, and there is the return horizon, how many bars the move you care about spans. Confusing the two is how people end up quoting a number that is right about nothing in particular.
Take the estimation window first, on the sixty closes this module has been using since lesson 38. Every figure below is a per-bar standard deviation in per cent, computed over every window of that length the series allows.
| Bars in the estimate | Windows available | Lowest | Median | Highest | Highest over lowest |
|---|---|---|---|---|---|
| 5 | 55 | 0.32% | 0.82% | 3.43% | 10.7x |
| 10 | 50 | 0.43% | 0.77% | 2.97% | 6.9x |
| 20 | 40 | 0.61% | 0.78% | 2.41% | 4.0x |
| 30 | 30 | 0.65% | 1.40% | 1.99% | 3.0x |
| 59 | 1 | 1.50% | 1.50% | 1.50% | 1.0x |
The last column is the part to look at first. A five-bar estimate of this series can come back eleven times larger than another five-bar estimate of the same series, and the ratio falls steadily as the window lengthens, because a longer window averages more of the variation away. That much is ordinary statistics. The median column is the interesting one, because it does something that looks like a contradiction: it sits near 0.8 at five, ten and twenty bars and then jumps to 1.40 at thirty.
Nothing happened to the market. What changed is the arithmetic of overlapping windows. The violent stretch in this series is the first twenty bars, and a window is contaminated by it whenever the window starts within the first nineteen returns. At twenty bars there are forty windows and nineteen are contaminated, so the clean ones are the majority at twenty-one and the median lands among them, at 0.78. At thirty bars there are only thirty windows and the same nineteen are contaminated, so the clean ones are now the minority at eleven, and the median lands among the contaminated ones instead, at 1.40. The median of the clean windows barely moves across that change — 0.69 at twenty bars and 0.69 at thirty. The headline median jumps by four fifths because of how many windows of each kind exist, which is a fact about window length and not a fact about volatility.
Lesson 41 split these same sixty bars into three consecutive twenties and found them to be radically different stretches. The volatility figures say the same thing without any interpretation: 2.46 per cent a bar in the first twenty, 0.73 in the middle twenty, 0.73 in the last. A single instrument, a single continuous run, and a 3.4-fold difference in width across it. Any number quoted for the whole sixty is an average over three regimes, and the whole-sixty figure of 1.50 per cent describes none of them.
The horizon, and where the square root fails
Now the second window. The square-root law says that if one bar has a standard deviation of 1 per cent, twenty bars have one of 4.47 per cent, because the square root of twenty is 4.47. That follows if and only if the bars are independent, and there is a standard way to check: the variance ratio. Take the twenty-bar returns — every one the series contains, starting at each bar in turn, which is forty of them from sixty closes rather than the two you get by refusing to overlap them — take their variance, divide by twenty times the variance of the one-bar returns, and the answer is exactly 1 when the law holds.
On this series it is not 1. It is 0.34 at two bars, 0.40 at five and 0.40 at twenty. Below 1 means the longer moves are smaller than the scaling predicts, which is what a series that keeps reversing does: each bar spends part of its move undoing the last one, so the moves cancel instead of accumulating. Scaling the one-bar figure up to twenty bars by 4.47 therefore overstates this series’ actual twenty-bar spread, whose true scaling factor is 2.84 — an overstatement by a factor of 1.58.
That result has a check sitting beside it, worth performing because a variance ratio is easy to compute wrongly. For two periods the algebra is short: the variance of a two-bar return is twice the one-bar variance times one plus the correlation between adjacent bars, so the two-bar variance ratio should equal one plus that correlation and nothing else. The lag-one correlation here is minus 0.658, which predicts 0.342. The direct computation returned 0.344. The variance ratio is not a separate diagnostic; it is that autocorrelation wearing different clothes.
What transfers off this page is the test rather than the figure. A lag-one correlation of minus 0.658 is enormous, an artefact of a series built by hand to be legible on a small chart; nothing anybody trades reverses that reliably, because a reversal that dependable would be arbitraged into flatness within a day. Real series usually sit near 1 and drift below it at long horizons. What matters is the direction of the error. Below 1, scaling up makes your risk estimate too large, which costs you size. Above 1, which is what a trending series produces, it makes the estimate too small, and that is the direction that empties accounts.
The chain’s number and the tape’s number
Everything so far was computed from a tape that already happened. An option chain quotes a volatility too, and it is a different animal: a price, set by people taking positions, for a period that has not occurred yet. Two things follow, and the first is a correction to this module’s own arithmetic.
Lesson 43 priced a stock at 100 with a straddle at 6.00 and called the implied move six per cent. As a break-even that is exactly right: the buyer needs the stock outside the band from 94 to 106. But six per cent is not the standard deviation the chain is quoting, and treating it as one propagates into every stop derived from it. An at-the-money straddle is priced at the expected absolute move rather than at one standard deviation, and the expected absolute move is 0.798 of a standard deviation. Divide, and the implied standard deviation is 7.52 per cent. Solving the option formula exactly rather than through the approximation gives 7.5217, so the shortcut is good to four figures. The break-even is six per cent, the volatility is 7.52 per cent, and they are a quarter apart.
That correction settles an old argument about selling premium. At an implied standard deviation of 7.52 per cent, the chance the move exceeds the six per cent break-even is 42.5 per cent. A straddle priced perfectly fairly, with no edge to anybody, therefore loses money for its buyer 57.5 per cent of the time. The observation that premium sellers win most of their trades is true, is expected, and is evidence of nothing. It describes the shape of the payoff, and lesson 43’s table showed the other half of that shape.
The second consequence is the gap. The volatility a chain quotes sits above the volatility the tape subsequently delivers more often than not, and this is neither a mystery nor a free lunch. Whoever sold that straddle is short a risk that hurts most at the moment everything else in a portfolio is also hurting, and a risk with that timing gets charged for. The asymmetry is countable on lesson 43’s own numbers: the best outcome available to the seller is the entire 6.00 collected, while a twenty per cent move costs 14.00, more than twice that best case. A hedged seller earns the straddle’s sensitivity to volatility multiplied by the difference between the two numbers, and that sensitivity is the same 0.798 once more: a straddle is the expected absolute move and nothing else, so on a 100 dollar stock its price moves 0.798 dollars for every point of volatility. An implied 7.52 against a delivered 6.00 therefore pays 1.21 on 6.00 collected, about 20 per cent, and a delivered 9.00 costs 1.18. The premium is the price of holding the wrong end of that asymmetry, not a discount somebody left lying around.
What a volatility number buys you
An account of 100,000 dollars risking 0.5 per cent per trade, which is 500 dollars. A stock at 100 dollars. A stop placed two daily standard deviations away, derived from the index reading by the conversion above. The share count is the risk budget divided by the stop distance, and that is the whole calculation.
| VIX reading | Implied daily move | Stop distance | Shares | Position value | Share of the top row |
|---|---|---|---|---|---|
| 12 | 0.76% | $1.51 | 331 | $33,100 | 100% |
| 16 | 1.01% | $2.02 | 248 | $24,800 | 75% |
| 20 | 1.26% | $2.52 | 198 | $19,800 | 60% |
| 25 | 1.57% | $3.15 | 159 | $15,900 | 48% |
| 30 | 1.89% | $3.78 | 132 | $13,200 | 40% |
| 40 | 2.52% | $5.04 | 99 | $9,900 | 30% |
The last column is not an estimate and not a rule of thumb. It is 12 divided by the reading, exactly. The share counts are 500 dollars divided by the stop distance beside them, which is that same ratio carried on an unrounded 330.72 rather than on the printed 331, and the difference surfaces once: at a reading of 20 the arithmetic gives 198 shares where scaling the printed top row would give 199. Share count is inversely proportional to volatility because the stop is proportional to volatility and the dollar risk is held fixed, and there is no judgement anywhere in that chain. The amount at risk is 500 dollars on every row. What changes is how much stock those 500 dollars buy, and it changes by a factor of 3.3 down the table without anybody deciding anything.
Two qualifications before the number is used. The index is a quote for one specific market, so sizing a position in something else off it assumes the two move together, which is an assumption with a number attached and lesson 45 measures it. And lesson 40 established that a stop does not fill where it is written — on that lesson’s own gap series it filled at a median of one and a half times its promised loss. The 500 dollars in this table is the written risk, not the paid risk, and the gap between them widens with exactly the quantity this lesson is computing.
What it does not buy you
Volatility tables that hand you a position-size multiplier by band — full size here, half there, a quarter above that — are doing two separate jobs and admitting to one. The first job is the arithmetic above, and the table has already done it. Take a normal-conditions reading of 17.5 and an elevated one of 22.5: sizing off the stop alone takes the share count to 77.8 per cent of normal, with nobody making a decision. At a high-fear reading of 27.5 it goes to 63.6 per cent. Those cuts are free, in the sense that they are implied by the stop and cost nothing to be right about.
A band table asking for 50 per cent and 25 per cent at those same readings is asking for something else on top: a further cut of 36 per cent and then of 61 per cent, beyond what the volatility already did. That extra is not sizing. It is a claim that the edge itself decays when markets are fearful, that your setups work worse rather than merely needing more room.
The claim may well be true, and for many traders it is. But it is a different kind of claim and it costs a different amount to establish. Telling a 1.0 per cent volatility regime apart from a 1.5 per cent one, at the confidence and power lesson 41 used, takes 50 bars in each condition. Telling a five-point difference in hit rate apart takes 1,560 trades in each condition. The edge claim needs roughly 31 times as much evidence as the volatility claim, and almost nobody who quotes a band table has collected it. So the order matters. Size off volatility always, because it is arithmetic and nearly free to verify. Cut further for edge decay only once you have counted your own trades, and lesson 41 tells you how long that count takes.
What this does not settle
That any one of these numbers describes the series. A standard deviation computed from a limited run of observations is itself an estimate with a spread on it, and the spread is wide. From 20 bars the 95 per cent interval runs from 0.76 to 1.46 times the figure you calculated; from 60 bars, 0.85 to 1.22; from a full year of 252, 0.92 to 1.10. Every boundary in every regime table — 20, 25, 30 — sits comfortably inside the noise of a month’s data. That is before the separate problem this lesson spent a table on, which is that the quantity was moving while you measured it.
That the variance ratio settled anything at twenty bars. Sixty closes give fifty-nine one-bar returns and forty overlapping twenty-bar returns, but they hold fewer than three independent ones, and the sampling theory in the Lo and MacKinlay paper cited below prices that honestly. The standard error on the twenty-bar ratio is 0.65, so the 0.40 printed above sits 0.9 standard errors from the 1 the law predicts, which is no distance at all. At two bars the standard error is 0.13 and the same 0.34 sits five standard errors out. The reversal in this series is therefore established at two bars and merely suggested at twenty, which makes the 1.58 overstatement the figure on this page a reader is least entitled to. Run the test at the short horizon on whatever history you have. Run it at the long horizon only on a great deal more.
That the distribution is normal. The 0.798, the 0.674 and the 32 per cent are properties of a normal distribution and of nothing else. Returns are not normal: they carry fatter tails than the formula assumes, so events far from the centre happen more often than these figures imply. Near the middle the approximation is good and the conclusions above are safe. Out at the tail — where a stop gets gapped through and where a short straddle is decided — it is optimistic, and optimistic in the direction that costs money.
That this series’ figures are anybody’s. A per-bar standard deviation of 1.50 per cent, scaled to a session of sixty bars and then to a year, annualises to 184 per cent — roughly twice the most extreme reading the index has ever reached. The module’s sixty bars are a teaching series, built to make swings visible on a small chart, and they are far more violent than anything you will trade. Every ratio and every mechanism on this page transfers. Not one of the levels does.
That closes are the best data you have. This lesson used closing prices because closing prices are what everybody has, and a close-to-close standard deviation throws away everything that happened inside the bar. An estimator built from the high and the low of each bar extracts several times more information from the same data, which means a usable number from a fraction of the history. The module has highs and lows on all sixty bars and this lesson did not use them. If you are estimating volatility seriously rather than illustratively, that is the first upgrade to make.
That measuring it is forecasting it. Volatility is unusual among market quantities in that it persists — quiet stretches follow quiet stretches, and a number computed from last week is genuinely informative about next week in a way no directional measurement is. That is why the exercise is worth doing at all. But persistence over days is not persistence over months, the regime changes are exactly what the table above caught this series doing three times in sixty bars, and a chain is quoting a number for a future the tape has not agreed to deliver. A measured volatility tells you how wide the recent past was. It is a good default for tomorrow and a poor one for the quarter.
Problems
- Compute it three ways on your own instrument. Take 60 closes. Compute the per-bar standard deviation of the log returns over the last 20, the last 40, and all 60, and write the three numbers down side by side. If they differ by more than the 20-bar noise interval quoted above allows, your instrument was not doing one thing throughout, and you have learned something no single number would have told you. It is four columns in a spreadsheet and it takes ten minutes once.
- Turn it into a share count and compare it with what you actually trade. Convert your per-bar figure into a daily standard deviation, set a stop at two of them, divide your usual dollar risk by that stop distance, and read off the share count. Now compare it with the size you have been trading. If the two disagree by more than a factor of two, one of them is wrong, and which one is answerable: count how often your stop is hit before the idea you entered on has had a chance to be right or wrong. A stop inside one standard deviation is reached by noise about a third of the time.
- Test the square root before you rely on it. Compute the variance ratio on your own series at horizons of 2, 5 and 20 bars, and bring far more than sixty closes to the twenty-bar one, because at that horizon sixty closes hold fewer than three independent observations. Below 1 and scaling a one-bar number up overstates your longer-horizon risk, so your stops are wider than they need to be and you are trading smaller than you could. Above 1 and it understates, which is the dangerous direction and the one a trending instrument produces. This is the check almost nobody runs, and it is the difference between using the square-root law and assuming it.
Sources. Louis Bachelier, “Théorie de la spéculation” (Annales scientifiques de l’École Normale Supérieure, 1900), for the origin of the square-root-of-time law itself, derived for prices five years before the same mathematics was published for pollen grains in water. Michael Parkinson, “The Extreme Value Method for Estimating the Variance of the Rate of Return” (The Journal of Business, 1980), for the range-based estimator that gets several times more out of the same bars than the close-to-close method used here. Andrew W. Lo and A. Craig MacKinlay, “Stock Market Prices Do Not Follow Random Walks: Evidence from a Simple Specification Test” (The Review of Financial Studies, 1988), for the variance ratio, which is where the test performed above comes from and where its sampling behaviour is worked out. Peter Carr and Liuren Wu, “Variance Risk Premiums” (The Review of Financial Studies, 2009), for the measured gap between what chains quote and what tapes deliver, and for the argument that it is a risk price rather than a mispricing.
Every number on this page belongs to one instrument. The moment there are two positions open, a second quantity decides how much risk is really in the book, and it is not the sum of the two volatilities. Lesson 45 takes up correlation: how it is computed, why its number moves when the window moves in exactly the way this lesson’s did, why it climbs toward one precisely in the conditions where you needed it not to, and what it does to a book of positions each of which was sized impeccably on its own.
What a Timeframe Is
The window problem in its general form, before volatility gave it two windows.
Read Lesson →Scheduled Events
The straddle whose quoted move this lesson separates from its standard deviation.
Read Lesson →Correlation
What decides whether two correctly sized positions are really one position.
Read Lesson →Educational only. Trading involves substantial risk of loss. Not financial advice. Past performance does not guarantee future results.
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