# Bollinger Bands

BOLLINGER BANDS
Bollinger bands measure the height of the price relative to previous trades.

This is a technical analysis tool invented by John Bollinger in the 1980s. evolved from the concept of trading bands. Bollinger Bands are typically used to measure the highness or lowness of the price relative to the trades previously made.

Composition of Bollinger Bands:
1. A middle band being an N-period simple moving average (MA)
2. An upper band at K times an N-period standard deviation above the middle band (MA+K*sigma)
3. A lower band at K times an N-period standard deviation below the middle band (MA-K*sigma)

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. A common second choice is exponential moving averages. Usually the same period is used for both the middle band and the calculation of standard deviation.

Purpose
Bollinger Bands are there to provide a relative definition of high and low. Prices are high at the upper band and low at the lower band. This definition can help in pattern recognition and is also 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.

Interpretation
The use of Bollinger Bands varies much among traders. Some buy when price touches the lower Bollinger Band and some exit when price touches the moving average in the center of the bands. Others buy when price goes above the upper Bollinger Band or sell when price falls below. 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, they are expecting volatility to go back towards the average historical volatility level for the stock.

When the bands lie close together this indicates a period of low volatility in stock price. On the other hand, if they are far apart this indicates a period of high volatility in price. The price of a stock is predictably found to go up and down between the bands, as though in a channel, if the bands have only a slight slope and lie approximately parallel for an extended time.

Together with other indicators, many traders often use Bollinger Bands to see if there is confirmation. It is common practice to use a non-oscillator indicator like chart patterns or a trendline, with an oscillator like Bollinger Bands; through these indicators the recommendation of the Bollinger Bands is confirmed and the trader will have greater evidence that what the Bands forecast is true.

Bollinger Bands and Statistics
Security prices usually have no known statistical distribution bet ir normal or not. They are characterized to have fat tails versus the Normal. The sample size typically used, which is 20, is too small to be reliable for conclusions to be derived from statistical techniques such as the Central Limit Theorem. For this reason, these techniques usually require the sample to be independent and identically distributed. This is not the case for a time series like security prices.

Because of these reasons, it is wrong to assume that the percentage of the data outside the Bollinger Bands will always be limited to a certain amount. Therefore, instead of finding about 95% of the data inside the bands, as would be the expectation with the default parameters and a normal distribution, one should find less. How much less is a function of the volatility of the security.