- **Similar:** [KChannel](../kchannel/kchannel.md), [SDChannel](../sdchannel/sdchannel.md), [ABERR](../aberr/aberr.md) | **Complementary:** %B and BandWidth for squeeze detection; RSI for momentum confirmation | **Trading note:** Most widely used band indicator; 2s captures ~95% of normally-distributed data but markets exhibit fat tails.
Bollinger Bands construct a volatility-adaptive envelope around a Simple Moving Average using population standard deviation as the width measure. The bands expand during high-volatility periods and contract during consolidation, dynamically adapting to changing market conditions. Under Gaussian assumptions, $\pm 2\sigma$ contains approximately 95.4% of price action, but financial returns exhibit fat tails and volatility clustering, so the bands function more as a volatility-normalized reference frame than a strict probability envelope. The derived metrics %B (price position as a fraction of band width) and BandWidth (normalized band spread) extend the raw bands into a complete analytical toolkit.
John Bollinger developed Bollinger Bands in the early 1980s while working as a market technician. He registered the name as a trademark in 1996 and published *Bollinger on Bollinger Bands* (McGraw-Hill, 2001). The indicator emerged from Bollinger's observation that fixed-percentage envelopes fail to account for changing volatility: a 5% envelope that works during quiet markets becomes useless during volatile phases, and vice versa.
Bollinger drew on statistical probability theory. Under the normal distribution, approximately 68% of observations fall within $\pm 1\sigma$, 95% within $\pm 2\sigma$, and 99.7% within $\pm 3\sigma$. By defaulting the multiplier to 2.0, he created bands targeting the 95% containment level. Chebyshev's inequality guarantees at least 75% containment at $\pm 2\sigma$ regardless of distribution shape. In practice, with fat-tailed market returns (typical kurtosis 4-6), expect 92-95% containment rather than 95.4%.
The "Squeeze" pattern (BandWidth at multi-period lows) became one of the most recognized technical analysis signals, predating and influencing the TTM Squeeze indicator. Walking the bands (price hugging the upper or lower band during trends) is a momentum signal, not a reversal signal. Bollinger Bands became one of the most widely adopted technical analysis tools, available in virtually every charting platform.
Note: this is population standard deviation (divide by $n$), not sample standard deviation (divide by $n-1$). Bollinger specified population $\sigma$, and most reference implementations (TA-Lib, TradingView) use this convention.
The circular buffer maintains running sums of $x$ and $x^2$, enabling $O(1)$ computation of both mean and variance per bar. The square root for $\sigma$ is the most expensive operation.
