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Miha Kralj
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116 lines
5.6 KiB
Markdown
116 lines
5.6 KiB
Markdown
# ACCBANDS: Acceleration Bands
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> *Acceleration bands widen with high-low range, framing the expected reach of each bar's ambition.*
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| Property | Value |
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| ---------------- | -------------------------------- |
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| **Category** | Channel |
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| **Inputs** | OHLCV bar (TBar) |
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| **Parameters** | `period`, `factor` (default 4.0) |
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| **Outputs** | Multiple series (Upper, Lower) |
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| **Output range** | Tracks input |
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| **Warmup** | `period` bars |
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| **PineScript** | [accbands.pine](accbands.pine) |
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- Acceleration Bands construct a volatility envelope using the intra-bar high-low range rather than close-to-close standard deviation, creating chann...
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- **Similar:** [BBands](../bbands/bbands.md), [KC](../kc/kc.md) | **Complementary:** ADX for trend strength | **Trading note:** Wider than Bollinger Bands; effective for breakout trading using high-low range volatility.
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- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
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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.
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## Historical Context
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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.
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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.
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## Architecture & Physics
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### 1. Per-Bar Normalized Width
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For each bar, compute the range as a fraction of total price:
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$$w_t = \frac{H_t - L_t}{H_t + L_t}$$
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When $H_t + L_t = 0$ (price is zero), $w_t = 0$ to prevent division by zero.
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### 2. Adjusted Prices
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The high and low are expanded by the normalized width scaled by the factor:
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$$\text{AdjHigh}_t = H_t \times (1 + F \cdot w_t)$$
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$$\text{AdjLow}_t = L_t \times (1 - F \cdot w_t)$$
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### 3. Band Construction (Three SMAs)
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$$\text{Upper}_t = \text{SMA}(\text{AdjHigh}, n)$$
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$$\text{Lower}_t = \text{SMA}(\text{AdjLow}, n)$$
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$$\text{Middle}_t = \text{SMA}(\text{Close}, n)$$
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### 4. Complexity
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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.
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## Mathematical Foundation
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### Parameters
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| Parameter | Description | Default | Constraint |
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|-----------|-------------|---------|------------|
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| `period` | Lookback period for the three SMAs ($n$) | 20 | $> 0$ |
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| `factor` | Multiplier for normalized width ($F$) | 4.0 | $> 0$ |
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### Breakout Rule (Headley)
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A trend is confirmed when:
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$$\text{Close}_t > \text{Upper}_t \quad \text{AND} \quad \text{Close}_{t-1} > \text{Upper}_{t-1}$$
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(Two consecutive closes above the upper band.) Reverse logic for downside breakouts.
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### Output Interpretation
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| Output | Description |
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|--------|-------------|
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| `upper` | SMA of adjusted highs (resistance envelope) |
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| `lower` | SMA of adjusted lows (support envelope) |
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| `middle` | SMA of close (center line) |
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## Performance Profile
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### Operation Count (Streaming Mode)
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ACCBANDS computes per-bar normalized width, two adjusted prices, and three independent SMA running sums:
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| Operation | Count | Cost (cycles) | Subtotal |
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| :--- | :---: | :---: | :---: |
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| ADD (H + L for denom) | 1 | 1 | 1 |
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| SUB (H - L for range) | 1 | 1 | 1 |
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| DIV (range / denom for w) | 1 | 15 | 15 |
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| MUL (factor × w) | 1 | 3 | 3 |
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| MUL (H × (1 + F·w), L × (1 - F·w)) | 2 | 3 | 6 |
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| SUB (oldest from 3 running sums) | 3 | 1 | 3 |
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| ADD (new value to 3 running sums) | 3 | 1 | 3 |
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| DIV (sum / count, three SMAs) | 3 | 15 | 45 |
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| **Total (hot)** | **15** | — | **~77 cycles** |
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The three DIV operations dominate. When the denominator is zero ($H + L = 0$), a branch sets $w = 0$, adding one CMP.
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### Batch Mode (SIMD Analysis)
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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:
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| Optimization | Benefit |
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| :--- | :--- |
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| Width + adjusted price computation | Vectorizable with `Vector<double>` (ADD, SUB, MUL, DIV) |
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| Three SMA running sums | Sequential; cannot parallelize across bars |
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| Band output assembly | Trivial; already scalar from SMA |
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## Resources
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- **Headley, P.** *Big Trends in Trading*. Wiley, 2002. (Original Acceleration Bands specification)
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- **TA-Lib** `TA_ACCBANDS` function. (Reference implementation with factor = 4.0)
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