# Lognormal Volatility Band

When John Bollinger introduced his Bollinger bands in the 1980’s, he gave us an indispensable tool for trading. But Bollinger bands are based on normal distribution, while markets show a lognormal distribution. This article will show the differences between Bollinger bands and lognormal volatility bands.

## The Width of Volatility Bands

Bollinger Bands are calculating the volatility of prices around a moving average. Therefore Bollinger is using the absolute distance between the current average to the last e.g. 20 daily closing prices of the market. Using the distance of the historic prices vs. the current moving average is a great idea, but using the absolute distance can bring some problems with specific instruments.

The problem is, this calculation will also show you a band which width is based on absolute levels. This makes it as likely for a stock to fall 50 points than it is to rise by 50 points. Not possible if the stock already trades at low levels.

To overcome this one could use percent levels instead of absolute levels. This would create a band which is symmetrical on a percentage scale. But also this would not be sufficient, as a stock can not fall 200%, but it can easily rise by 200%. The solution to this problem lies with logarithmic distribution.

## Lognormal Distribution vs. Normal Distribution

The lognormal distribution, in opposition to the normal distribution, takes the mentioned effects of absolute and % returns into account. Have a look at wikipedia to see the mathematical details.

On the chart below you see the difference between  a lognormal distribution based a normal distribution based calculation.

As it can be seen in February 2018, Bollinger bands exaggerate the volatility to the downside and underestimates it to the upside. This is due to the extreme movements at this time.

On the other side, if the market is moving quietly, the difference between a lognormal band and Bollinger bands is hardly visible. That’s also why Bollinger bands are still in use. On “normal” stocks and under “normal” conditions, the difference between lognormal and normal distribution is hardly worth mentioning.

In high volatility phases, the lognormal bands clearly give a better estimation on what is high and what would be regarded as a low level.

## Volatility Bands Mid Line

Bollinger bands use an arithmetic average as the mid line. But why not use an exponentially smoothed average for a better fit to the data or use a least-square regression line in the middle? The given indicator code also includes these ways of smoothing.

The chart below shows the volatility band using the median price over the last 250 trading days as a reference point. The median price will give you the level, where 50% of days in history have been above and 50% of the days have been below it.

Around this price a band with two standard deviations to the upside and one standard deviation to the downside is displayed. As it can be seen, this gives a good estimation of high and low volatility levels for this specific instrument

## Indicator Equilla Code

The lognormal volatility band indicator uses a mathematical trick to calculate the lognormal volatility. Instead of using the absolute distance to the average, like Bollinger bands do, it uses the logarithm of the distance and then the exponential function to convert it back to the levels displayed on the chart. This transformation of returns turns a standard deviation to a lognormal deviation.

Beside this logarithmic transformation and the possibility of selecting a different mid line, the programming follows the original layout given by John Bollinger. 