This is a good question. Indeed, Bollinger Bands are a remnant of the time, when indicator calculations were done by hand. Prior to the advent of PCs this was a useful simplification of the real underlying distribution of all trades.
However, in order to calculate the accurate standard deviation bands over a selected lookback period, you would need to use tick data in single-tick resolution and take into account the volume of each transaction.
A compromise between the oversimplified Bollinger Bands and the accurate standard deviation bands calculated from data in single-tick resolution is indeed a model that uses all four data points OHLC and the volume of each bar.
Approximation of the price volume distribution
The first approximation of the distribution did not take into account volume, but just used so called time price ooportunities (TPOs). A more sophisticated model introduced volume. The result are volume-weighted time price opportunities (VWTPOs). Both TPOs and VWTPOs do not use the open and the close of a bar, but only depend on high and low. A better approximation can be obtained by using all four data points.
-> The statistical mean of the underlying price volume distribution can be approximated with the volume-weighted moving average calculated from all Open, High, Low and Close.
-> The standard deviation can be best approximated by a model that uses 8 data points per bar (Open, High, Low, Close, 2 x bar center, 2 x body center).
The result are standard deviation bands, which are a better representation of the underlying trade statistics than Bollinger Bands.
As you will notice below, Bollinger Bands do not widen up correctly when price breaks out and volume increases.
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@Fat Tails have you had the time to upload this indicator somewhere? I couldn't find it in the downloads section. I'm currently doing analysis using fundamental stats like this, so this indicator would be very useful to me.