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1996issue C031-3

Constructing Bollinger bands, percent-b, and stochastics

A 20-day simple moving average of closes centers a two-standard-deviation band pair that can be rescaled into percent-b. A separate 14-period high-low ratio becomes percent-k and can be smoothed into percent-d. The note keeps each window, spread, and extra average explicit.

  • A Bollinger band pair is a 20-day simple moving average of closes plus and minus twice the 20-day standard deviation of those closes, with the same window used for the average and the spread.
  • Percent-b places the close on a 0-to-100 scale between the lower and upper bands, using a four-standard-deviation width, and can be withheld when twice the scaled standard deviation is zero.
  • A 14-period stochastic scales the close between the window low and high by 100; percent-k is that ratio, and percent-d is a smoothed percent-k, often a three-day simple moving average.
  • Editorial: keep the window, deviation multiple, price field, and extra averaging explicit so the band pair, percent-b, and stochastic are assembled, not treated as a black box.
Entries in this reading3 entries
Current trackBollinger Bands84 readingsStochastic oscillator170 readingsMoving average449 readings

A shared moving-average kit

Editorial reading: a lookback average can center a two-standard-deviation close envelope, that envelope can be rescaled into a 0-to-100 band oscillator, and a high-low range ratio can be smoothed into a stochastic. This editorial lesson is to keep the window, the deviation multiple, the price field, and any extra averaging explicit so each plotted object is assembled, not treated as a black box.

A lookback average of closes is used as the band centerline and as a smoother on a stochastic ratio. The archive workflow below states those averages and spreads in turn.

The 20-day band pair

A Bollinger band pair is drawn two standard deviations above and below a 20-day simple moving average, and the same 20-day window is used for the standard deviation. The channel lines are placed a stated number of standard deviations above and below a simple moving average of closes.

The standard deviation uses each day's close versus the 20-day average: square those differences, sum them, divide by 20, then take the square root. That lookback spread of closes around the average is formed by squaring deviations, averaging them, and taking the square root.

The upper band equals a 20-period simple moving average of closes plus twice the 20-period standard deviation of closes. The lower band equals a 20-period simple moving average of closes minus twice the 20-period standard deviation of closes.

Percent-b on a 0-to-100 scale

Percent-b maps the close onto a 0-to-100 scale on which 100 is the upper band and 0 is the lower band, with a four-standard-deviation width in the denominator. The close is rescaled so the upper band sits at 100 and the lower band sits at 0.

A percent-b ratio can be guarded so it is not formed when twice the scaled standard deviation is zero.

The stochastic ratio and percent-d

A 14-period stochastic locates the close between the lowest low and the highest high in that window and scales the ratio by 100. The oscillator is that ratio: it places the close between the highest high and lowest low over a lookback window.

Percent-k is the lookback-period ratio, the unsmoothed stochastic lookback ratio, and is often slowed by averaging highs and lows. Percent-d is a smoothed percent-k, commonly a three-day simple moving average. Percent-d can also be defined as a moving average of the percent-k series over a separate smoothing length.

Educational research material, not investment advice. Historical source context does not establish present-day performance.
5 of 45 in the Bollinger Bands track
19961-6 pp.Next on Bollinger BandsConstructing mechanical rules from Bollinger Bands and stochasticsFrame the work as one mechanical trading system: indicators enter only through explicit rules so entry, hold, reverse, exit, and standing aside can be tested instead of used informally.
All readings on this track · 45 readings
  1. 1992Constructing volatility-scaled bands with relative strength index confirmation
  2. 1994Implied volatility as a band-defined regime filter for index options
  3. 1995Constructing projection bands from least-squares slopes
  4. 1995Constructing regression projection bands and range oscillators
  5. 1996Constructing Bollinger bands, percent-b, and stochastics
  6. 1996Constructing mechanical rules from Bollinger Bands and stochastics
  7. 1996Constructing a standard-error envelope around a linear regression
  8. 1996Dual-horizon ratio envelopes and regression error channels
  9. 1997Rational group structure with a trend screen, RSI, and bands
  10. 1997Asymmetric volatility band construction
  11. 1998Constructing three-state filters from Bollinger band envelopes
  12. 1999Combination filters with Bollinger Bands and the relative strength index
  13. 1999Constructing stochastic timed exits and band-RSI reversals
  14. 1999Evaluating Bollinger Bands against fixed-width and range-based envelopes
  15. 2000Constructing a Bollinger Band target as a forward price
  16. 2001Numeric candlestick encoding with local size bands
  17. 2001Ranked candlestick sentiment to band-cross entries
  18. 2002Combining Bollinger Bands, RSI, and a stop-loss
  19. 2002Bollinger Bands remain filters, not forecasts
  20. 2002Constructing a stochastic RSI with Bollinger bands
  21. 2002Constructing a StochRSI and Bollinger mechanical system
  22. 2003Constructing volatility-scaled Bollinger envelopes
  23. 2003Why tick breadth fails as a market personality
  24. 2005Constructing Bollinger bands versus fixed trading bands
  25. 2006Squared versus absolute deviation in envelope construction
  26. 2006Confirming yen crossovers with implied volatility and bands
  27. 2006A daily candle reversal is a hypothesis until shorter sessions fail at the same zone
  28. 2008Rebuild the Relative Strength Index as price-scale bands
  29. 2008Reading Relative Strength Index extremes on one price axis with Bollinger Bands and moving averages
  30. 2011Three-filter confirmation for short-swing futures
  31. 2011Constructing an inverse Fisher stochastic with bands and averages
  32. 2012Constructing a Bollinger Band indicator suite
  33. 2012Stacking price extremes, crossovers, bands, and MACD
  34. 2012Adaptive Bollinger band impulse, trend, and momentum filters
  35. 2013Rescaling stochastic, percent-B, and wave-count parameters
  36. 2014Industry-group quartile pivots as a Bollinger Bands case study
  37. 2014Bollinger Bands as adaptive price envelopes: a 2014 classroom case
  38. 2016Trend-channel entry rules from stacked moving averages
  39. 2016A permission stack for Bollinger, RSI, and the 50-period average
  40. 2017Constructing weighted Bollinger bands and volume averages
  41. 2017Four swing-entry rules that share a timed exit
  42. 2017Two-wave monthly cycles as a regime filter
  43. 2019Constructing exponential-deviation-bands from a midline-average
  44. 2020Critiquing exponential variants of Bollinger Bands
  45. 2020Constructing selectable volatility and moving-average bands
By fieldIndicators and filters682 readingsBy year199680 readings
All 84 readings tagged Bollinger Bands
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2011Constructing an inverse Fisher stochastic with bands and averages1995Constructing projection bands from least-squares slopes1995Constructing regression projection bands and range oscillators1996Constructing mechanical rules from Bollinger Bands and stochastics1999Constructing stochastic timed exits and band-RSI reversals