Miha Kralj
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ACCBANDS: Acceleration Bands
Acceleration bands widen with high-low range, framing the expected reach of each bar's ambition.
| Property | Value |
|---|---|
| Category | Channel |
| Inputs | OHLCV bar (TBar) |
| Parameters | period, factor (default 4.0) |
| Outputs | Multiple series (Upper, Lower) |
| Output range | Tracks input |
| Warmup | period bars |
| PineScript | accbands.pine |
- Acceleration Bands construct a volatility envelope using the intra-bar high-low range rather than close-to-close standard deviation, creating chann...
- Similar: BBands, KC | Complementary: ADX for trend strength | Trading note: Wider than Bollinger Bands; effective for breakout trading using high-low range volatility.
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
Acceleration Bands construct a volatility envelope using the intra-bar high-low range rather than close-to-close standard deviation, creating channels that accommodate the full price excursion of the underlying asset. Each bar's contribution to band width is normalized by price level (w = (H-L)/(H+L)), making the bands scale-invariant across instruments. Three independent Simple Moving Averages of the adjusted high, adjusted low, and close prices form the upper, lower, and middle bands respectively. Headley's original breakout rule declares a trend when price closes outside the bands for two consecutive bars.
Historical Context
Price Headley developed Acceleration Bands and detailed them in Big Trends in Trading (Wiley, 2002). Headley observed that standard deviation bands often lag in fast-moving breakout scenarios because they require several bars of expanded volatility before the bands visibly widen. By incorporating High and Low prices directly into the band width calculation through a per-bar normalized range, he created a system that reacts immediately to range expansion.
The normalization w = (H-L)/(H+L) is the key design choice. Dividing range by the sum of high and low produces a dimensionless ratio that is comparable across any price level. A $5 stock with a $0.50 range and a $500 stock with a $50 range both produce w = 0.05. The default factor of 4.0 was Headley's empirically determined value for equity markets on daily timeframes, matching the TA-Lib reference implementation.
Architecture & Physics
1. Per-Bar Normalized Width
For each bar, compute the range as a fraction of total price:
w_t = \frac{H_t - L_t}{H_t + L_t}
When H_t + L_t = 0 (price is zero), w_t = 0 to prevent division by zero.
2. Adjusted Prices
The high and low are expanded by the normalized width scaled by the factor:
\text{AdjHigh}_t = H_t \times (1 + F \cdot w_t)
\text{AdjLow}_t = L_t \times (1 - F \cdot w_t)
3. Band Construction (Three SMAs)
\text{Upper}_t = \text{SMA}(\text{AdjHigh}, n)
\text{Lower}_t = \text{SMA}(\text{AdjLow}, n)
\text{Middle}_t = \text{SMA}(\text{Close}, n)
4. Complexity
Three independent circular buffers maintain running sums for O(1) streaming updates. Each bar requires computing w_t, the two adjusted prices, and three buffer updates.
Mathematical Foundation
Parameters
| Parameter | Description | Default | Constraint |
|---|---|---|---|
period |
Lookback period for the three SMAs (n) |
20 | > 0 |
factor |
Multiplier for normalized width (F) |
4.0 | > 0 |
Breakout Rule (Headley)
A trend is confirmed when:
\text{Close}_t > \text{Upper}_t \quad \text{AND} \quad \text{Close}_{t-1} > \text{Upper}_{t-1}
(Two consecutive closes above the upper band.) Reverse logic for downside breakouts.
Output Interpretation
| Output | Description |
|---|---|
upper |
SMA of adjusted highs (resistance envelope) |
lower |
SMA of adjusted lows (support envelope) |
middle |
SMA of close (center line) |
Performance Profile
Operation Count (Streaming Mode)
ACCBANDS computes per-bar normalized width, two adjusted prices, and three independent SMA running sums:
| Operation | Count | Cost (cycles) | Subtotal |
|---|---|---|---|
| ADD (H + L for denom) | 1 | 1 | 1 |
| SUB (H - L for range) | 1 | 1 | 1 |
| DIV (range / denom for w) | 1 | 15 | 15 |
| MUL (factor × w) | 1 | 3 | 3 |
| MUL (H × (1 + F·w), L × (1 - F·w)) | 2 | 3 | 6 |
| SUB (oldest from 3 running sums) | 3 | 1 | 3 |
| ADD (new value to 3 running sums) | 3 | 1 | 3 |
| DIV (sum / count, three SMAs) | 3 | 15 | 45 |
| Total (hot) | 15 | — | ~77 cycles |
The three DIV operations dominate. When the denominator is zero (H + L = 0), a branch sets w = 0, adding one CMP.
Batch Mode (SIMD Analysis)
The three SMA running sums are sequential. The per-bar width computation (w, adjusted prices) is independent across bars and vectorizable in a batch pre-pass:
| Optimization | Benefit |
|---|---|
| Width + adjusted price computation | Vectorizable with Vector<double> (ADD, SUB, MUL, DIV) |
| Three SMA running sums | Sequential; cannot parallelize across bars |
| Band output assembly | Trivial; already scalar from SMA |
Resources
- Headley, P. Big Trends in Trading. Wiley, 2002. (Original Acceleration Bands specification)
- TA-Lib
TA_ACCBANDSfunction. (Reference implementation with factor = 4.0)