Definition
Bollinger Bands are a volatility envelope drawn around a moving average: a middle band, typically a 20-period simple moving average, with an upper and a lower band set a chosen number of standard deviations away, usually two. They were developed by John Bollinger.
How it works
Because the band distance is based on the standard deviation of recent closes, the bands widen when price swings grow and narrow when they calm. Two derived readings help: %B = (Price − Lower) ÷ (Upper − Lower) shows where price sits inside the bands, and Bandwidth = (Upper − Lower) ÷ Middle shows how wide they are relative to the average.
A close near the upper band means price is high relative to its recent range, not that a reversal is due; in persistent trends price can ride along a band for many periods. Narrow bands show unusually quiet conditions, not a directional forecast.
Formula / convention
Middle = SMA(20); Upper = Middle + k × σ; Lower = Middle − k × σ, where σ is the standard deviation of the same 20 closes and k = 2 by default
Illustrative example
Illustrative: a fictional share has a 20-day average of HK$80.00 and a standard deviation of HK$2.50, so the upper band is 80.00 + 2 × 2.50 = HK$85.00 and the lower band is 80.00 − 5.00 = HK$75.00. A close of HK$84.00 gives %B = (84 − 75) ÷ (85 − 75) = 0.90, and Bandwidth = (85 − 75) ÷ 80 = 12.5%.
What to watch for
Do not read the bands as a fixed probability range: the familiar figure of about 95% inside two standard deviations comes from the normal distribution, while real prices are often fat-tailed and volatility changes. A band touch is not a standalone signal, so test any band rule on historical data, including costs, before giving it weight.