BOLLINGER BANDS

Bollinger Bands were developed by John Bollinger in the early 1980s while he was working as a capital markets analyst for the Financial News Network (FNN). Bollinger was searching for a method of defining whether prices were high or low on a relative basis — a deceptively simple question that had resisted easy answers in classical technical analysis. Bollinger Bands consist of three lines plotted directly on a price chart. Their construction requires three parameters: the period n (typically 20), the moving average type (simple, by default), and the multiplier k (typically 2).

The Middle Band

The middle band is a simple moving average (SMA) of the closing price over the chosen period:

Middle Band = SMA(Close, n)

Where n is the lookback period. This acts as a trend baseline.

Standard Deviation

The standard deviation is calculated over the same n-period window.

σ = √ [ (1/n) × Σ (Closeᵢ – SMA)² ]Upper and Lower Bands

The upper and lower bands are placed at k standard deviations above and below the middle band:

Upper Band = SMA(Close, n) + k × σLower Band = SMA(Close, n) – k × σ

With the default settings of n = 20 and k = 2, the bands reflect two standard deviations of recent price dispersion around the 20-period moving average.

Statistical Background and Caveats

The 95% Claim

It is frequently stated that with k = 2, approximately 95% of price observations will fall within the Bollinger Bands. This claim is derived from the normal distribution: in a perfectly Gaussian distribution, exactly 95.45% of observations fall within two standard deviations of the mean.

This is a useful rule of thumb, but it comes with significant statistical caveats that any serious practitioner must understand.

Why the Claim is Not Statistically Rigorous

The standard application of the normal distribution to financial prices rests on assumptions that do not hold in practice:

• Fat tails: Financial return distributions are well-documented to have heavier tails than the normal distribution. Extreme price moves occur far more frequently than a Gaussian model predicts.

• Rolling window non-stationarity: The standard deviation used to construct the bands is calculated over a rolling n-period window. The 95% property of the normal distribution applies to a stationary distribution, not a rolling one.

• Price levels, not returns: Bollinger Bands are applied to price levels, not to returns. Price levels are non-stationary by construction.

Bollinger Bands as a Continuation Signal

In trending markets, Bollinger Bands are most powerfully used as a continuation tool.

The Squeeze

A Squeeze occurs when the bands narrow significantly. This reflects a period of low volatility and typically signals consolidation or range-bound price action. Bollinger himself describes the Squeeze as “the most powerful signal Bollinger Bands give.”

The rationale is rooted in the cyclical nature of volatility: periods of low volatility tend to precede periods of high volatility.

Walking the Bands

Once a strong trend is established, price tends to “walk” along the outer band — repeatedly tagging or closing near the upper band in an uptrend, or the lower band in a downtrend. In this context, a tag of the upper band is not a sell signal but a sign of trend strength. Traders should resist the impulse to fade the band during a strong directional move.

Bollinger Bands as a Reversal Signal

In ranging markets, Bollinger Bands can be used as a contrarian reversal tool. The classic reversal patterns are the “W-Bottom” and “M-Top,”.

W-Bottom (Bullish Reversal)

A W-Bottom consists of two reaction lows. The first low tags or breaches the lower Bollinger Band. A subsequent bounce and then a re-test forms the second low, which ideally does not reach the lower band. A rally from the second low that breaks above the most recent reaction high and the middle band confirms the reversal.

M-Top (Bearish Reversal)

The M-Top is the mirror image. A first push reaches or breaches the upper band. A reaction pullback and then a second push higher that fails to reach the upper band signals weakening momentum.

Bollinger Bands with Momentum Oscillators

Volume and momentum oscillators help confirm or deny the significance of a band tag:

• RSI: An upper band tag accompanied by RSI below 70 (non-overbought) suggests trend continuation. An upper band tag with RSI above 80 and beginning to turn down raises reversal risk. Bullish and bearish divergences between RSI and price at band extremes are particularly powerful signals.

• MACD: A bullish MACD crossover as price tests the lower band strengthens the case for a long entry. Conversely, divergence between MACD and price at a band extreme warns of potential reversal.

• Stochastic Oscillator: A %K/%D crossover in oversold territory coinciding with a lower band test provides a combined timing trigger that many traders find more reliable than either signal alone.