**Bollinger Bands** and the related indicators *%b* and *BandWidth* are technical analysis tools invented by John Bollinger in the 1980s. Having evolved from the concept of trading bands, Bollinger Bands can be used to measure the highness or lowness of the price relative to previous trades.

Bollinger Bands consist of:

- a middle band being an
*N*-period simple moving average (MA)
- an upper band at
*K* times an *N*-period standard deviation above the middle band (MA + *Kσ*)
- a lower band at
*K* times an *N*-period standard deviation below the middle band (MA − *Kσ*)

Typical values for *N* and *K* are 20 and 2, respectively. The default choice for the average is a simple moving average, but other types of averages can be employed as needed. Exponential moving averages are a common second choice. Usually the same period is used for both the middle band and the calculation of standard deviation.

Purpose

The purpose of Bollinger Bands is to provide a relative definition of high and low. By definition, prices are high at the upper band and low at the lower band. This definition can aid in rigorous pattern recognition and is useful in comparing price action to the action of indicators to arrive at systematic trading decisions.

Indicators derived from Bollinger Bands

There are two indicators derived from Bollinger Bands, %*b* and *BandWidth*. %*b*, pronounced 'percent b', is derived from the formula for Stochastics and tells you where you are in relation to the bands. %*b* equals 1 at the upper band and 0 at the lower band. Writing *upperBB* for the upper Bollinger Band, *lowerBB* for the lower Bollinger Band, and *last* for the last (price) value:

- %
*b* = (last − lowerBB) / (upperBB − lowerBB)

*BandWidth* tells you how wide the Bollinger Bands are on a normalized basis. Writing the same symbols as before, and *middleBB* for the moving average, or middle Bollinger Band:

- BandWidth = (upperBB − lowerBB) / middleBB

Using the default parameters of a 20-period look back and plus/minus two standard deviations, *BandWidth* is equal to four times the 20-period coefficient of variation.

Uses for %*b* include system building and pattern recognition. Uses for *BandWidth* include identification of opportunities arising from relative extremes in volatility and trend identification.

In a series of lectures at The World Money Show in Hong Kong, Asian Traders Investment Conference in Singapore, the Italian Trading Forum in Rimini, Italy, The European Technical Analysis Conference in London, England and the Market Technicians Symposium in New York, USA, all in Spring of 2010, John Bollinger introduced three new indicators based on Bollinger Bands. They are BB Impulse, which measures price change as a function of the bands, BandWidth Percent, which normalizes the width of the bands over time, and BandWidth Delta, which quantifies the changing width of the bands.

Interpretation

The use of Bollinger Bands varies widely among traders. Some traders buy when price touches the lower Bollinger Band and exit when price touches the moving average in the center of the bands. Other traders buy when price breaks above the upper Bollinger Band or sell when price falls below the lower Bollinger Band. Moreover, the use of Bollinger Bands is not confined to stock traders; options traders, most notably implied volatility traders, often sell options when Bollinger Bands are historically far apart or buy options when the Bollinger Bands are historically close together, in both instances, expecting volatility to revert back towards the average historical volatility level for the stock.

When the bands lie close together a period of low volatility in stock price is indicated. When they are far apart a period of high volatility in price is indicated. When the bands have only a slight slope and lie approximately parallel for an extended time the price of a stock will be found to oscillate up and down between the bands as though in a channel.

Traders are often inclined to use Bollinger Bands with other indicators to see if there is confirmation. In particular, the use of an oscillator like Bollinger Bands will often be coupled with a non-oscillator indicator like chart patterns or a trendline; if these indicators confirm the recommendation of the Bollinger Bands, the trader will have greater evidence that what the bands forecast is correct.

Effectiveness

A recent study concluded that Bollinger Band trading strategies may be effective in the Chinese marketplace, stating: "Finally, we find significant positive returns on buy trades generated by the contrarian version of the moving average crossover rule, the channel breakout rule, and the Bollinger Band trading rule, after accounting for transaction costs of 0.50 percent." Nauzer J. Balsara, Gary Chen and Lin Zheng *The Chinese Stock Market: An Examination of the Random Walk Model and Technical Trading Rules*. (By "the contrarian version", they mean buying when the conventional rule mandates selling, and vice versa.)

A paper by Rostan, Pierre, Théoret, Raymond and El moussadek, Abdeljalil from 2008 at SSRN uses Bollinger Bands in forecasting the yield curve.

In his 2006 master's thesis, Oliver Douglas Williams at the University of Western Ontario studied Bollinger Bands and suggested that fundamental analysis was key to setting Bollinger Band parameters, a process John Bollinger dubbed rational analysis. Williams concluded: "Alone, Bollinger Bands do not seem to yield the extraordinary results. Fundamental analysis is required to determine the best moving average window to match the business cycle of the asset. When combined with other techniques such as fundamental analysis, Bollinger Bands can give systematic traders a method of choosing their buy and sell points."

Companies like Forbes suggest that the use of Bollinger Bands is a simple and often an effective strategy but stop-loss orders should be used to mitigate losses from market pressure.

Statistical properties

Security prices have no known statistical distribution, normal or otherwise; they are known to have fat tails, compared to the Normal. The sample size typically used, 20, is too small for conclusions derived from statistical techniques like the Central Limit Theorem to be reliable. Such techniques usually require the sample to be independent and identically distributed which is not the case for a time series like security prices.

For these three primary reasons, it is incorrect to assume that the percentage of the data outside the Bollinger Bands will always be limited to a certain amount. So, instead of finding about 95% of the data inside the bands, as would be the expectation with the default parameters if the data were normally distributed, one will typically find less; how much less is a function of the security's volatility.

Bollinger Bands outside of finance

In a paper published in 2006 by the Society of Photo-Optical Engineers, "Novel method for patterned fabric inspection using Bollinger Bands", Henry Y. T. Ngan and Grantham K. H. Pang present a method of using Bollinger Bands to detect defects in patterned fabrics. From the abstract: "In this paper, the upper band and lower band of Bollinger Bands, which are sensitive to any subtle change in the input data, have been developed for use to indicate the defective areas in patterned fabric."

The International Civil Aviation Organization is using Bollinger Bands to measure the accident rate as a safety indicator to measure efficiency of global safety initiatives. %b and BandWidth are also used in this analysis.

Notes

**^** When the average used in the calculation of Bollinger Bands is changed from a simple moving average to an exponential or weighted moving average, it must be changed for both the calculation of the middle band and the calculation of standard deviation.
**^** Bollinger Bands use the population method of calculating standard deviation, thus the proper divisor for the sigma calculation is *n*, not *n* − 1.