Bollinger Bands

An SMA centerline wrapped in symmetric standard-deviation envelopes; the classical reading is that price persistently outside a band signals a volatility-driven trend, not a reversal.

Quick reference

Item	Value
Family	Volatility & Bands
Input type	f64 (typically the close price)
Output type	BollingerOutput { upper: f64, middle: f64, lower: f64, stddev: f64 }
Output range	unbounded; lower ≤ middle ≤ upper, stddev ≥ 0
Default parameters	period = 20, multiplier = 2.0
Warmup period	period (20 for defaults)
Interpretation	width tracks recent volatility; price tags band on momentum

Formula

Each step uses the trailing window of the last period inputs:

mean   = (1/n) * Σ x_i
var    = (1/n) * Σ (x_i - mean)^2     (population variance, denominator = n)
stddev = sqrt(var)
upper  = mean + multiplier * stddev
middle = mean
lower  = mean - multiplier * stddev

Wickra computes var from the streaming sums Σ x and Σ x² as Σx²/n - (Σx/n)² and clamps to 0.0 to absorb catastrophic cancellation on near-constant inputs (crates/wickra-core/src/indicators/bollinger.rs). On long-running streams the running Σ x and Σ x² are reseeded from the live window every 16 · period updates — amortised O(1), bounds the cancellation drift to roughly 16 · period · ULP · max(|x|²) (sub-picodollar on real-world price scales).

Parameters

Name	Type	Default	Constraint	Source
period	usize	20	> 0	BollingerBands::new (bollinger.rs:43)
multiplier	f64	2.0	finite and > 0.0	BollingerBands::new (bollinger.rs:47)

Python defaults come from #[pyo3(signature = (period=20, multiplier=2.0))] in bindings/python/src/lib.rs. Invalid inputs raise ValueError in Python and return Error::PeriodZero / Error::NonPositiveMultiplier in Rust.

Inputs / Outputs

Rust signature:

use wickra::{Indicator, BollingerBands, BollingerOutput};
// BollingerBands: Input = f64, Output = BollingerOutput
const _: fn(&mut BollingerBands, f64) -> Option<BollingerOutput> = <BollingerBands as Indicator>::update;

BollingerOutput fields: upper, middle, lower, stddev.

• Python streaming (update) returns the 4-tuple (upper, middle, lower, stddev) or None during warmup.
• Python batch (batch) returns a 2-D numpy.ndarray of shape (n, 4) with columns [upper, middle, lower, stddev]; warmup rows are entirely NaN.
• Node streaming (update) returns a { upper, middle, lower, stddev } object or null during warmup.
• Node batch (batch) returns a flat Array<number> of length n * 4 interleaved per row: [u0, m0, l0, s0, u1, m1, l1, s1, …]. Warmup rows are four consecutive NaNs.

Warmup

warmup_period() == period. The first period - 1 inputs return None; the period-th input emits the first BollingerOutput. Verified for period = 5: the first non-None value appears on the 5th input (index 4).

Edge cases

• Constant input. With a flat series the population stddev collapses to exactly 0.0, so upper == middle == lower == mean. The library guards against tiny negative floating-point values from catastrophic cancellation by clamping the variance with .max(0.0).
• Flat range / squeeze. Real markets never give exactly 0.0, but very low-volatility windows produce visibly narrow bands; the upper and lower bands collapse onto the middle band (the "Bollinger squeeze").
• NaN / infinity input. The implementation skips non-finite inputs: if !input.is_finite() { return self.current(); }. The window is not advanced and the previous BollingerOutput (or None) is returned.
• Multiplier validation. multiplier <= 0 or non-finite returns Error::NonPositiveMultiplier. period == 0 returns Error::PeriodZero.
• Reset. reset() clears the window and both running sums, returning the indicator to a freshly-constructed state.

Examples

Rust

use wickra::{BatchExt, BollingerBands, Indicator};

fn main() -> Result<(), Box<dyn std::error::Error>> {
    let mut bb = BollingerBands::new(5, 2.0)?;
    let out = bb.batch(&[2.0, 4.0, 4.0, 4.0, 5.0, 5.0, 7.0, 9.0]);
    for (i, v) in out.into_iter().enumerate() {
        println!("i={i} -> {:?}", v);
    }
    Ok(())
}

Output:

i=0 -> None
i=1 -> None
i=2 -> None
i=3 -> None
i=4 -> Some(BollingerOutput { upper: 5.759591794226543, middle: 3.8, lower: 1.8404082057734565, stddev: 0.9797958971132716 })
i=5 -> Some(BollingerOutput { upper: 5.379795897113269, middle: 4.4, lower: 3.420204102886732, stddev: 0.48989794855663404 })
i=6 -> Some(BollingerOutput { upper: 7.190890230020663, middle: 5.0, lower: 2.809109769979336, stddev: 1.095445115010332 })
i=7 -> Some(BollingerOutput { upper: 9.577708763999665, middle: 6.0, lower: 2.422291236000335, stddev: 1.7888543819998326 })

The first emission at i=4 uses the window [2, 4, 4, 4, 5] with mean 3.8 and population stddev sqrt(0.96) ≈ 0.9797959.

Python

import numpy as np
import wickra as ta

bb = ta.BollingerBands(5, 2.0)
prices = np.array([2.0, 4.0, 4.0, 4.0, 5.0, 5.0, 7.0, 9.0], dtype=float)
out = bb.batch(prices)
print("shape:", out.shape)
print("row 4:", out[4])
print("row 7:", out[7])

Output:

shape: (8, 4)
row 4: [5.75959179 3.8        1.84040821 0.9797959 ]
row 7: [9.57770876 6.         2.42229124 1.78885438]

Streaming variant returns a 4-tuple (upper, middle, lower, stddev) per tick or None during warmup:

import wickra as ta

bb = ta.BollingerBands(5, 2.0)
for p in [2.0, 4.0, 4.0, 4.0, 5.0, 9.0]:
    print(p, "->", bb.update(p))

Output:

2.0 -> None
4.0 -> None
4.0 -> None
4.0 -> None
5.0 -> (5.759591794226543, 3.8, 1.8404082057734565, 0.9797958971132716)
9.0 -> (9.078143885933063, 5.2, 1.321856114066938, 1.939071942966531)

Node

const w = require('wickra');

const bb = new w.BollingerBands(5, 2.0);
const flat = bb.batch([2, 4, 4, 4, 5, 5, 7, 9]);
console.log('length:', flat.length);
console.log('row 4 [upper, middle, lower, stddev]:', flat.slice(16, 20));
console.log('row 7 [upper, middle, lower, stddev]:', flat.slice(28, 32));

Output:

length: 32
row 4 [upper, middle, lower, stddev]: [ 5.759591794226543, 3.8, 1.8404082057734565, 0.9797958971132716 ]
row 7 [upper, middle, lower, stddev]: [ 9.577708763999665, 6, 2.422291236000335, 1.7888543819998326 ]

Streaming returns the named object { upper, middle, lower, stddev }:

const w = require('wickra');

const bb = new w.BollingerBands(5, 2.0);
[2, 4, 4, 4, 5].forEach(p => console.log(p, '->', bb.update(p)));

Output:

2 -> null
4 -> null
4 -> null
4 -> null
5 -> {
  upper: 5.759591794226543,
  middle: 3.8,
  lower: 1.8404082057734565,
  stddev: 0.9797958971132716
}

Interpretation

• Bandwidth as volatility. (upper - lower) / middle is the Bollinger bandwidth; a multi-month low in bandwidth is the classic "squeeze" that often precedes an expansion move.
• Tags vs breakouts. A single touch of the upper band is not a sell signal in Bollinger's own framework; persistent closes outside the band ("walking the band") signal trend continuation, not exhaustion.
• %b position. (price - lower) / (upper - lower) normalises position inside the channel and is useful as a feature for cross-asset comparison.

Common pitfalls

• Stddev convention. Wickra uses population standard deviation (denominator n, not n - 1). This matches Bollinger's original formulation and every reference implementation (TA-Lib, pandas-ta); switching to the sample variant would mis-align bands by a factor of sqrt(n / (n - 1)) and break parity with other tools.
• Partial rows. In the Python 2-D batch result, do not slice an individual column out and use it for analysis without checking for NaN — every warmup row is NaN across all four columns. Filter with mask = ~np.isnan(out[:, 0]) before reading any single column.
• Flat batch length in Node. The Node batch returns n * 4 numbers interleaved per row, not four parallel arrays. Reshape with Array.from({ length: n }, (_, i) => flat.slice(i * 4, i * 4 + 4)) if you want per-row records.

References

• John Bollinger, Bollinger on Bollinger Bands, McGraw-Hill, 2001 (the original publication of the indicator dates to the early 1980s).
• Wilder's New Concepts in Technical Trading Systems (1978) for the surrounding family of volatility envelopes.

See also

• Keltner Channels — same envelope shape but band width is driven by ATR instead of stddev.
• Donchian Channels — rolling high/low envelope with no smoothing.
• ATR — the volatility scale most commonly used to size Bollinger-style stops.