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Miha Kralj
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- Sar.Quantower.Tests.cs: add missing opening quote on string literal (line 48) - Exports.cs: rename Correlation.Batch → Correl.Batch (CS0103) - Ad.Validation.Tests.cs: fix Ooples OutputValues key "Ad" → "Adl"
143 lines
6.7 KiB
Markdown
143 lines
6.7 KiB
Markdown
# APZ: Adaptive Price Zone
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> *The adaptive price zone contracts in calm and expands in chaos, mapping volatility into a living boundary.*
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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`, `multiplier` (default 2.0) |
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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** | [apz.pine](apz.pine) |
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- APZ constructs a volatility-adaptive envelope using double-smoothed exponential moving averages with an aggressive smoothing factor derived from $\...
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- **Similar:** [BBands](../bbands/bbands.md), [KC](../kc/kc.md) | **Complementary:** RSI for overbought/oversold confirmation | **Trading note:** EMA-based deviation adapts faster than standard deviation bands; responsive to recent volatility changes.
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- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
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APZ constructs a volatility-adaptive envelope using double-smoothed exponential moving averages with an aggressive smoothing factor derived from $\sqrt{\text{period}}$, making it significantly faster than standard EMA-based channels. The center line is a double-EMA of price; the band width is a double-EMA of the high-low range, scaled by a multiplier. Designed specifically for mean-reversion trading in non-trending markets, APZ identifies overbought/oversold extremes where price is likely to reverse rather than continue. A closing price outside the zone signals an immediate overshoot, not a breakout.
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## Historical Context
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Lee Leibfarth created the Adaptive Price Zone and published it in *Technical Analysis of Stocks & Commodities* (September 2006) under the article "Trading With An Adaptive Price Zone." Leibfarth recognized that most indicators fail in choppy, range-bound markets: trend followers get whipsawed, and oscillators saturate at extremes. APZ fills this gap by adapting its bandwidth dynamically to statistical noise, allowing traders to fade extremes in consolidation phases.
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The critical design decision is the square-root smoothing factor: $\alpha = 2 / (\sqrt{P} + 1)$. For a period of 20, $\sqrt{20} \approx 4.47$, producing $\alpha \approx 0.365$, which behaves like an EMA of period $\sim$3.5. This makes APZ extremely responsive compared to a standard 20-period EMA ($\alpha = 0.095$). The double-smoothing (EMA of EMA) adds some lag back, but the net result is still far faster than conventional approaches. The compound warmup compensator $e = \beta^{2t}$ (where $\beta = 1 - \alpha$) ensures accurate values from bar 1 without the typical EMA initialization bias.
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## Architecture & Physics
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### 1. Aggressive Smoothing Factor
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$$\alpha = \frac{2}{\sqrt{P} + 1}, \quad \beta = 1 - \alpha$$
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### 2. Center Line (Double-Smoothed EMA of Price)
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First EMA:
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$$\text{EMA1}_t = \alpha \cdot x_t + \beta \cdot \text{EMA1}_{t-1}$$
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Second EMA (double-smoothing):
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$$\text{Center}_t = \alpha \cdot \text{EMA1}_t + \beta \cdot \text{Center}_{t-1}$$
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### 3. Adaptive Range (Double-Smoothed EMA of High-Low)
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$$R_t = H_t - L_t$$
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$$\text{EMA1R}_t = \alpha \cdot R_t + \beta \cdot \text{EMA1R}_{t-1}$$
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$$\text{SmoothRange}_t = \alpha \cdot \text{EMA1R}_t + \beta \cdot \text{SmoothRange}_{t-1}$$
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### 4. Band Construction
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$$\text{Width}_t = F \cdot \text{SmoothRange}_t$$
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$$\text{Upper}_t = \text{Center}_t + \text{Width}_t$$
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$$\text{Lower}_t = \text{Center}_t - \text{Width}_t$$
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### 5. Warmup Compensation
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To eliminate EMA initialization bias, a compound compensator tracks the accumulated decay:
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$$e_t = \beta^2 \cdot e_{t-1}, \quad e_0 = 1$$
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During warmup ($e > 10^{-10}$):
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$$\text{Center}_t^* = \frac{\text{Center}_t}{1 - e_t}, \quad \text{SmoothRange}_t^* = \frac{\text{SmoothRange}_t}{1 - e_t}$$
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### 6. Complexity
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$O(1)$ per bar: 4 EMA updates (2 for price, 2 for range), plus band arithmetic. The square root is computed once at initialization. No buffers required.
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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` | Input period ($P$); $\sqrt{P}$ used for smoothing | 20 | $> 0$ |
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| `multiplier` | Band width factor ($F$) | 2.0 | $> 0$ |
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| `source` | Input price series | close | |
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### Effective EMA Period
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The actual smoothing period experienced by the double-EMA is much shorter than the input period:
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$$P_{\text{effective}} = \sqrt{P} \approx \frac{2}{\alpha} - 1$$
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For $P = 20$: $P_{\text{eff}} \approx 4.47$. For $P = 100$: $P_{\text{eff}} \approx 10$.
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### Output Interpretation
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| Output | Description |
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|--------|-------------|
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| `center` | Double-smoothed EMA of price (fast center line) |
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| `upper` | Center + scaled adaptive range (overbought zone) |
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| `lower` | Center - scaled adaptive range (oversold zone) |
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## Performance Profile
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### Operation Count (Streaming Mode)
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APZ runs four EMA updates (double-smoothed price + double-smoothed range) plus warmup compensation and band arithmetic:
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| Operation | Count | Cost (cycles) | Subtotal |
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| :--- | :---: | :---: | :---: |
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| SUB (H - L for range) | 1 | 1 | 1 |
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| FMA (EMA1 price) | 1 | 4 | 4 |
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| FMA (EMA2 price → center) | 1 | 4 | 4 |
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| FMA (EMA1 range) | 1 | 4 | 4 |
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| FMA (EMA2 range → smoothRange) | 1 | 4 | 4 |
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| MUL (multiplier × smoothRange) | 1 | 3 | 3 |
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| ADD/SUB (center ± width) | 2 | 1 | 2 |
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| **Total (hot)** | **8** | — | **~22 cycles** |
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During warmup (compensator active):
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| Operation | Count | Cost (cycles) | Subtotal |
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| :--- | :---: | :---: | :---: |
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| MUL (e × β²) | 1 | 3 | 3 |
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| SUB (1 - e) | 1 | 1 | 1 |
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| DIV (center / compensator) | 1 | 15 | 15 |
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| DIV (smoothRange / compensator) | 1 | 15 | 15 |
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| CMP (e > threshold) | 1 | 1 | 1 |
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| **Warmup overhead** | **5** | — | **~35 cycles** |
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**Total during warmup:** ~57 cycles/bar; **Post-warmup:** ~22 cycles/bar.
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### Batch Mode (SIMD Analysis)
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All four EMA recursions are state-dependent, preventing SIMD parallelization across bars:
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| Optimization | Benefit |
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| :--- | :--- |
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| FMA instructions | 4 hardware FMAs per bar; no software emulation |
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| State locality | 4 EMA states + compensator fit in registers |
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| Band arithmetic | Vectorizable in a post-pass across output arrays |
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## Resources
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- **Leibfarth, L.** "Trading With An Adaptive Price Zone." *Technical Analysis of Stocks & Commodities*, September 2006. (Original APZ specification)
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- **Mulloy, P.** "Smoothing Data with Less Lag." *Technical Analysis of Stocks & Commodities*, February 1994. (Double-smoothed EMA theory)
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