Bands and Ranges: What Survives Statistical Scrutiny

The final defendants are the volatility dressers: Bollinger Bands, which wrap price in a statistical costume, and the Average True Range, which hides a legitimate statistical idea inside chartist clothing. This module closes the audit — and ends, deliberately, on the one fragment worth salvaging, because knowing why it survives teaches as much as knowing why the rest do not.

Bollinger Bands: borrowed clothes

Bollinger Bands plot a rolling mean of price flanked by bands at plus and minus two rolling standard deviations:

$\text{upper/lower}_t = \text{SMA}_n(P)_t \pm 2\,\hat{\sigma}_{n,t}$

The construction looks statistical — that is its charm and its trick. Two standard deviations whispers "ninety-five percent" to anyone who remembers the normal distribution, and the folklore duly claims that price touching a band is a rare event due for reversal.

You already own every tool needed to see through this. The ninety-five-percent figure belongs to an independent, identically distributed normal variable — and you spent the foundations course establishing that financial data violates each word of that phrase. Prices (not even returns — the bands are drawn on prices) are non-stationary, so a rolling mean of them estimates nothing stable. Returns are fat-tailed, so multi-sigma excursions arrive far more often than normal theory predicts. And volatility clusters, so the rolling sigma is always a step behind: in a calm-to-storm transition the bands are too tight and "rare" touches happen constantly; in a storm-to-calm transition they are too wide and touches vanish. The band touch is not a probability statement. It is a moving target drawn with yesterday's ruler. Run the audit empirically and the costume falls off: band-touch frequencies on real assets bear no resemblance to the five percent the borrowed mathematics implies, and the mean-reversion rule built on touches fails the conditional-versus-base-rate test just as its cousins did — Bessembinder and Chan's verdict on the band rules matches the one on crossovers: nothing survives costs and honest testing.

ATR: a real estimator in chartist clothing

The Average True Range smooths the true range — each day's high-low span, widened for overnight gaps — into a running average. And here the audit returns a different finding, which is exactly why this module exists. The daily range is genuinely informative about volatility. You met this idea seriously in Course 102: the Parkinson and Garman-Klass estimators extract variance from the high-low range with substantially more efficiency than close-to-close returns, because the range observes the path, not just the endpoints. ATR is a crude, unnormalized cousin of those estimators — built from the right raw material, decades before the literature formalized why that material is rich.

So the object is salvageable; the uses are not. The chartist deploys ATR for stop-loss multiples and "breakout" thresholds — fixed multipliers, untested claims, the same base-rate sins as before. Strip those away and what remains is an honest, if inefficient, volatility proxy. The statistical upgrade path is direct: replace ATR with Parkinson or Garman-Klass if you want what the range knows about volatility, and with a GARCH model from Course 202 if you want what volatility's own dynamics — its clustering, its persistence — can tell you about tomorrow's risk. Those tools make the same kind of statement ATR gestures at, but with defined estimators, known efficiency, and out-of-sample evidence that they actually work. Volatility, unlike direction, genuinely is forecastable — which is why this institute spends entire courses on it.

The audit's closing argument

Three modules, one pattern. Each indicator is a transformation of past prices — a filter, a rescaling, a band — that contains no information the price series did not already hold. Each bolts an untested directional claim onto that transformation. Each claim, translated into a conditional probability and tested honestly — against base rates, out of sample, after costs, with multiple-testing corrections — collapses to noise. And each has a statistical counterpart that does the legitimate part of the job with estimated parameters and quantified uncertainty: the hypothesis test where RSI gestures, the time-series model where MACD gestures, the volatility estimator and GARCH where the bands gesture.

That is the institute's position on technical analysis, stated without rancor: it is not a rival school of analysis. It is statistics done without the statistics — and once you can do the statistics, you have no further use for it.