You and somebody whose process you respect are looking at the same ₹600 share, with the same written rule: risk 1% of a ₹10 lakh account, and place the stop two units of volatility away. You arrive at 347 shares. They arrive at 555 shares — sixty per cent more — on the same stock, the same evening, the same rule. Neither of you has made an arithmetic error, and neither of you has been careless. You measured volatility with the range of the bars and they measured it with the Standard deviation of the closes, and the word "volatility" covered the difference so completely that neither of you noticed there was one.
Ask how far a driver travelled today and there are two true answers. The odometer says 46 kilometres. The straight line from the house to where the car is parked tonight says 4 kilometres. Neither reading is wrong and neither is a version of the other: one measures the path, the other measures the displacement. Which you want depends entirely on the question — fuel, or whether he came home.
Close-to-close standard deviation is displacement: where did it end up, relative to where it ended up yesterday. The true-range family is path: how far did it actually travel while getting there. A stop is hit by the path. A band is drawn around the displacement. Using either number for the other job is the mistake, and it has a name in neither case, which is why it survives.
What each one sees, and what it is blind to
- Uses one number per bar: the close. Everything that happened inside the bar is discarded
- A stock that travels 4% intraday every day and closes unchanged every day reads close to zero
- It does include the overnight gap automatically, because yesterday’s close and today’s close sit on either side of it
- It is the quantity a two-sigma band, a z-score and a Sharpe ratio are all built from — those tools assume this estimator, whether or not anybody says so
- Statistically the least efficient of the family: one observation per bar, so it needs a lot of bars before it settles down
- Uses the high and the low, so it sees the travel rather than the destination
- A plain high-minus-low range is blind to the gap: a stock that opens 6% down and then trades in a 1% range all day has a range of 1%
- Which is exactly why true range is defined as the greatest of three quantities — the bar’s own range, the high less the previous close, and the previous close less the low. The two extra terms exist to plug the gap hole, and nothing else
- Reported in rupees unless you divide by price yourself, which makes it useless for comparing two stocks until you do
- Several times more efficient per bar than close-to-close, because each bar contributes two extra numbers — it settles down on weeks of data where the other needs months
- σ
- Standard deviation of the bar-to-bar change, in the same units as the range
- 1.6
- The square root of 8 ÷ π, which is 1.5958 — the expected travel of a random walk relative to its displacement
- In practice
- ATR sits at that figure or above it, because true range adds the gap back in and because real bars gap
Example: This is the conversion nobody performs, and it is why the two numbers cannot be swapped. A range-based number is about sixty per cent larger than a close-to-close number for the same stock in the same week. So "two ATR" is roughly 3.2σ, and "two sigma" is roughly 1.25 ATR — and both get described as two units of volatility.
Two biases in the range estimator, pointing opposite ways
- The observed range under-states the true one. A high is the highest price that printed. Between prints the price may well have been higher, and nobody recorded it. The fewer trades in the bar, the more the recorded extremes fall short of the real ones — so range-based volatility is biased downwards, and the bias is worst in thin names and short bars.
- The bid-ask bounce over-states it. From the previous lesson: the high is likely an offer-side print and the low a bid-side print, so the range contains a full spread that is not movement at all. This bias is upwards, and it is also worst in thin names and short bars.
- There is no reason they cancel. Two errors of opposite sign, both large in the same conditions, with no arithmetic connecting their sizes. The correct conclusion is not that one wins — it is that a range measured on a thin intraday chart is not a number you should be sizing a position from at all.
- Neither estimator is annualised for you, and the conversion carries its own assumption. Multiplying a daily figure by the square root of the session count assumes days are independent. An earlier lesson in this track covers the session-count convention and the discipline of recording which count you used; the next lesson in this module is how you check the independence it rests on.
| The job | Which estimator belongs there | Why |
|---|---|---|
| Where the stop goes | True-range family | A stop is executed by the path, not by the close. The relevant question is how far the price ordinarily travels while going nowhere, and only a range-based number contains that. |
| How wide a band or a z-score is | Close-to-close σ | The arithmetic of a two-sigma band is built on the standard deviation of closes. Feeding it a range-based number widens it by about sixty per cent and quietly changes what the band is claiming. |
| Comparing two stocks | Either, divided by price | A ₹14 ATR is meaningless until you know the share price. Everything in a watchlist has to be normalised to a percentage before two rows can be compared at all. |
| Comparing your number with somebody else’s | Whichever, stated explicitly | Two people can compute "the volatility" of the same stock over the same period and differ by sixty per cent without either being wrong. The undocumented number is the unusable one. |
| Sizing a position | The same one the stop uses | Size is risk budget divided by stop distance. The estimator only enters through the stop, so the discipline is not to pick the best estimator but to use one estimator throughout a single rule. |
Hold the risk budget fixed and move the stop distance. Everything about a position other than the rupee risk — the share count, the capital committed, how often you get stopped out — is decided by that one number, which is decided by an estimator you chose.
A stock reliably travels about 4% between its high and low every day and reliably closes within a few paise of the previous close. What do the two estimators report, and which rule breaks?
Poochho ki driver aaj kitna chala — do sacche jawab hain. Odometer kehta hai 46 kilometre. Ghar se aaj raat gaadi jahan khadi hai, us seedhi lakeer ka jawab hai 4 kilometre. Koi galat nahi hai, aur ek doosre ka version bhi nahi hai: ek raasta naapta hai, doosra faasla. Stop raaste se lagta hai; band faasle ke chaaron taraf khinchta hai. ₹600 ke share ka rozana close-to-close σ 1.5% hai, yaani ₹9. Random walk ka expected high-low range uska lagbhag 1.6 guna hota hai — ₹14.40, matlab 2.4%. Ab dono aadmi wahi ek line likhte hain, "do unit volatility ka stop": ek ko ₹28.80 (4.8%) milta hai, doosre ko ₹18.00 (3.0%). ₹10 lakh ke account pe 1% risk = ₹10,000. Pehla: 10,000 ÷ 28.80 = 347 share. Doosra: 10,000 ÷ 18.00 = 555 share, yaani 60% zyada. Rupee risk dono mein ₹10,000 hi hai, isliye dono keh sakte hain ki rule follow kiya. Par dono stop ko ek hi paimane pe naapo — us ₹14.40 pe jo share aam din chalta hai: 347 waala stop poore do guna pe hai, 555 waala sirf sawa guna pe. Sawa aam din ek hi taraf chal jaaye toh chhota stop kat jaata hai, jab ki bade stop ko poore do din chahiye — aur thesis mein kuch badla bhi nahi. Kisi ka number galat nahi tha. Rule galat tha, kyunki usne "volatility" likha aur yeh nahi likha ki kaunsa naap.
- Volatility is estimated, not observed, and the estimator you choose decides the number.
- Close-to-close standard deviation measures displacement and is blind to everything inside the bar.
- True range measures the path, and its three-term definition exists solely to include the overnight gap.
- For the same stock, a range-based figure runs about 1.6 times a close-to-close figure — so "two ATR" is roughly 3.2σ.
- In thin names the range is biased down by missing prints and up by the spread, and the two do not cancel.
- The rule that matters is not which estimator, but one estimator per rule, written down beside the threshold.
Mark it done to track your progress through the curriculum.