📏 Bollinger Bands
Bollinger Bands dynamically measure volatility and draw an adaptive "normality fence" around the price.
💡 Financial Meaning
When the bands are wide, the market is volatile; when they squeeze together, a breakout is imminent. A price touching the upper band signals statistical exuberance; touching the lower band signals an abnormal dip.
🔢 Mathematical Formulas
-
Middle Band (expected value):
\[ MB_t = SMA_N(C_t) \] -
Standard deviation of prices over the window:
\[ \sigma_t = \sqrt{\frac{1}{N} \sum_{i=0}^{N-1} (C_{t-i} - MB_t)^2} \] -
Upper and Lower Bands:
\[ Upper_t = MB_t + k \cdot \sigma_t, \qquad Lower_t = MB_t - k \cdot \sigma_t \]
With \(k = 2\), if returns were normally distributed the price would stay inside the bands ~95.4% of the time. In practice, financial returns have fat tails (leptokurtosis), so breaches are more frequent — but still statistically significant.
⚙️ Parameters
| Parameter | Key | Default | Description |
|---|---|---|---|
| Period (\(N\)) | period |
20 | SMA window for expected value. |
| Multiplier (\(k\)) | multiplier |
2 | Number of standard deviations. |
🎛️ Signal Processing Equivalent — Adaptive Confidence Interval Tracker
The Middle Band is a FIR (Finite Impulse Response) moving average filter — the simplest low-pass with a rectangular window of length \(N\). The bands add a time-varying envelope at \(\pm k\sigma\), which is essentially a running estimate of the signal's instantaneous variance.
In the language of adaptive filters, this is an expected-value tracker with an adaptive confidence interval. When the variance \(\sigma^2\) drops (the "Bollinger Squeeze"), the system is in a low-entropy state. In chaotic systems like financial markets, low-entropy periods are reliably followed by high-entropy (high-volatility) explosions — making the squeeze one of the most watched setups in technical analysis.
FIR vs IIR
Unlike the EMA (IIR, one pole), the SMA is a FIR filter with a perfectly flat group delay of \((N-1)/2\) samples. It trades off a wider transition band for zero-phase distortion — ideal for centring the confidence envelope.