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more volatilty
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Miha Kralj
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# NATR: Normalized Average True Range
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> "The same volatility reads different on different price scales. NATR speaks the universal language of percentages."
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NATR normalizes the Average True Range (ATR) as a percentage of the closing price. This is mathematically identical to ATRP (Average True Range Percent)—both compute `(ATR / Close) × 100`. The difference is purely nomenclature: NATR is the term used in TA-Lib and many charting platforms.
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## Historical Context
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NATR derives from J. Welles Wilder Jr.'s ATR, introduced in his 1978 *New Concepts in Technical Trading Systems*. While Wilder's original ATR provided absolute volatility in price units, traders and quantitative analysts quickly recognized the need for percentage-based normalization.
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The "Normalized" moniker became standard in the TA-Lib open-source library, which formalized the calculation as `NATR = (ATR / Close) × 100`. This naming convention spread through the algorithmic trading community, creating the parallel terminology alongside "ATRP" (Average True Range Percent) used in other contexts.
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Both names describe the same mathematical transformation: making volatility comparable across instruments with different price levels.
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## Architecture & Physics
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NATR consists of three cascaded components:
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### 1. True Range (TR)
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Captures the actual price movement including gaps:
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$$
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TR_t = \max(H_t - L_t, |H_t - C_{t-1}|, |L_t - C_{t-1}|)
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$$
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Where:
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- $H_t$: Current high
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- $L_t$: Current low
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- $C_{t-1}$: Previous close
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First bar uses simple range: $TR_0 = H_0 - L_0$
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### 2. RMA Smoothing (Wilder's Method)
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ATR smooths TR using Wilder's RMA with $\alpha = 1/N$:
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$$
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ATR_t = \alpha \cdot TR_t + (1 - \alpha) \cdot ATR_{t-1}
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$$
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With warmup compensation to eliminate initialization bias:
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$$
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e_t = e_{t-1} \cdot (1 - \alpha), \quad e_0 = 1
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$$
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$$
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ATR_{compensated} = \frac{ATR_{raw}}{1 - e_t} \quad \text{when } e_t > \epsilon
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$$
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### 3. Percentage Normalization
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$$
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NATR_t = \frac{ATR_t}{C_t} \times 100
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$$
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This transforms absolute volatility into relative volatility, enabling cross-asset comparison.
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## Mathematical Foundation
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### Complete Formula Chain
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Given period $N$:
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1. **Parameters**: $\alpha = \frac{1}{N}$, $\text{decay} = 1 - \alpha$
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2. **True Range**:
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$$
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TR_t = \begin{cases}
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H_t - L_t & \text{if } t = 0 \\
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\max(H_t - L_t, |H_t - C_{t-1}|, |L_t - C_{t-1}|) & \text{otherwise}
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\end{cases}
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$$
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3. **RMA with FMA optimization**:
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$$
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ATR_{raw,t} = \text{FMA}(ATR_{raw,t-1}, \text{decay}, \alpha \cdot TR_t)
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$$
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4. **Warmup compensation**:
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$$
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ATR_t = \frac{ATR_{raw,t}}{1 - e_t}
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$$
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5. **Normalization**:
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$$
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NATR_t = \frac{ATR_t}{C_t} \times 100
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$$
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### Warmup Period
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Convergence threshold: $e < 0.05$ (5% remaining bias)
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$$
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\text{WarmupPeriod} = \left\lceil \frac{\ln(0.05)}{\ln(1 - \alpha)} \right\rceil
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$$
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For $N = 14$: $\text{WarmupPeriod} \approx 42$ bars.
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## Performance Profile
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| Metric | Score | Notes |
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| :--- | :---: | :--- |
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| **Throughput** | 10/10 | O(1) calculation via RMA + single division |
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| **Allocations** | 0 | Zero-allocation streaming; state in record struct |
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| **Complexity** | O(1) | Constant time regardless of period |
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| **Accuracy** | 10/10 | Exact mathematical computation |
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| **Timeliness** | 4/10 | Inherits ATR's lag from RMA smoothing |
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| **Overshoot** | 0/10 | Mathematically bounded |
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| **Smoothness** | 8/10 | Smooth RMA decay; minor noise from close price variation |
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### Operation Count (Streaming Mode)
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| Operation | Count | Notes |
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| :--- | :---: | :--- |
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| SUB | 3 | H-L, H-PrevC, L-PrevC |
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| ABS | 2 | Gap calculations |
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| MAX | 2 | True Range selection |
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| FMA | 1 | RMA update |
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| MUL | 1 | Decay for warmup |
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| DIV | 2 | Warmup compensation + percentage |
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| MUL | 1 | × 100 |
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| **Total** | ~12 ops | Dominated by FMA and divisions |
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## Validation
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NATR is validated by computing ATR from external libraries and applying the same percentage formula.
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| Library | Status | Notes |
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| :--- | :---: | :--- |
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| **QuanTAlib** | ✅ | Native implementation |
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| **TA-Lib** | ✅ | Via `(ATR / Close) × 100`; tolerance 0.10 for warmup divergence |
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| **Skender** | ✅ | Via `(GetAtr / Close) × 100` |
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| **Tulip** | ✅ | Via `(atr / Close) × 100` |
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| **Ooples** | ✅ | Via `(CalculateAverageTrueRange / Close) × 100` |
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Note: QuanTAlib's warmup-compensated RMA may diverge 4-7% from classic Wilder implementations over long histories. Both approaches are mathematically valid; QuanTAlib prioritizes accurate early-series values.
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## Use Cases
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### Cross-Asset Volatility Comparison
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Compare volatility across different price scales:
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| Asset | Price | ATR | NATR |
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| :--- | :---: | :---: | :---: |
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| Penny Stock | $2.50 | 0.25 | 10.0% |
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| Mid-Cap | $150 | 4.50 | 3.0% |
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| Blue Chip | $500 | 5.00 | 1.0% |
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ATR suggests Blue Chip is most volatile. NATR reveals Penny Stock has 10× the relative volatility.
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### Volatility-Adjusted Position Sizing
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```
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Position Size = (Account Risk %) / NATR
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```
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Ensures equal percentage risk per position regardless of asset price.
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### Regime Detection
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| NATR Range | Interpretation | Strategy Implication |
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| :--- | :--- | :--- |
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| < 1% | Low volatility | Mean reversion, tight stops |
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| 1-3% | Normal | Standard trend-following |
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| 3-5% | Elevated | Wider stops, reduced size |
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| > 5% | High volatility | Crisis mode, capital preservation |
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## Common Pitfalls
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1. **Lag Inheritance**: NATR inherits ATR's smoothing lag. It measures recent volatility, not current or future volatility.
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2. **Close Price Spikes**: A sharp close creates transient NATR spikes since it affects both TR (numerator) and the denominator simultaneously.
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3. **Near-Zero Prices**: Assets approaching zero produce extreme NATR values. Implement minimum price thresholds.
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4. **Gap Sensitivity**: Large overnight gaps inflate TR significantly. Consider using gap-adjusted data for equity analysis.
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5. **Warmup Period**: The first 40+ bars (for period=14) contain warmup bias. Use `IsHot` to filter unreliable values.
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6. **OHLC Requirement**: NATR requires bar data (Open, High, Low, Close). It cannot be computed from close prices alone. Use `Update(TBar)` not `Update(TValue)`.
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## Related Indicators
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- **ATR**: Absolute volatility measure NATR normalizes
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- **ATRP**: Mathematically identical; different naming convention
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- **ATRN**: ATR normalized to [0,1] based on historical min/max
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- **CV**: Coefficient of Variation—alternative percentage volatility measure
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- **HV**: Historical Volatility—annualized standard deviation approach
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## References
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- Wilder, J.W. (1978). *New Concepts in Technical Trading Systems*. Trend Research.
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- TA-Lib documentation: NATR function specification
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- TradingView PineScript: `ta.natr()` implementation
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