QuanTAlib/lib/volatility/natr/Natr.md

NATR: Normalized Average True Range

The same volatility reads different on different price scales. NATR speaks the universal language of percentages.

Property	Value
Category	Volatility
Inputs	OHLCV bar (TBar)
Parameters	period (default 14)
Outputs	Single series (Natr)
Output range	\geq 0
Warmup	1 bar
PineScript	natr.pine

• NATR normalizes the Average True Range (ATR) as a percentage of the closing price.
• Similar: ATR, ATRN | Complementary: Cross-asset comparison | Trading note: Normalized ATR as percentage of close.
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

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.

Historical Context

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.

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.

Both names describe the same mathematical transformation: making volatility comparable across instruments with different price levels.

Architecture & Physics

NATR consists of three cascaded components:

1. True Range (TR)

Captures the actual price movement including gaps:

TR_t = \max(H_t - L_t, |H_t - C_{t-1}|, |L_t - C_{t-1}|)

Where:

• H_t: Current high
• L_t: Current low
• C_{t-1}: Previous close

First bar uses simple range: TR_0 = H_0 - L_0

2. RMA Smoothing (Wilder's Method)

ATR smooths TR using Wilder's RMA with \alpha = 1/N:

ATR_t = \alpha \cdot TR_t + (1 - \alpha) \cdot ATR_{t-1}

With warmup compensation to eliminate initialization bias:

e_t = e_{t-1} \cdot (1 - \alpha), \quad e_0 = 1

ATR_{compensated} = \frac{ATR_{raw}}{1 - e_t} \quad \text{when } e_t > \epsilon

3. Percentage Normalization

NATR_t = \frac{ATR_t}{C_t} \times 100

This transforms absolute volatility into relative volatility, enabling cross-asset comparison.

Mathematical Foundation

Complete Formula Chain

Given period N:

1. Parameters: \alpha = \frac{1}{N}, \text{decay} = 1 - \alpha

2. True Range:

TR_t = \begin{cases}
H_t - L_t & \text{if } t = 0 \\
\max(H_t - L_t, |H_t - C_{t-1}|, |L_t - C_{t-1}|) & \text{otherwise}
\end{cases}

3. RMA with FMA optimization:

ATR_{raw,t} = \text{FMA}(ATR_{raw,t-1}, \text{decay}, \alpha \cdot TR_t)

4. Warmup compensation:

ATR_t = \frac{ATR_{raw,t}}{1 - e_t}

5. Normalization:

NATR_t = \frac{ATR_t}{C_t} \times 100

Warmup Period

Convergence threshold: e < 0.05 (5% remaining bias)

\text{WarmupPeriod} = \left\lceil \frac{\ln(0.05)}{\ln(1 - \alpha)} \right\rceil

For N = 14: \text{WarmupPeriod} \approx 42 bars.

Performance Profile

Metric	Score	Notes
Throughput	10/10	O(1) calculation via RMA + single division
Allocations	0	Zero-allocation streaming; state in record struct
Complexity	O(1)	Constant time regardless of period
Accuracy	10/10	Exact mathematical computation
Timeliness	4/10	Inherits ATR's lag from RMA smoothing
Overshoot	0/10	Mathematically bounded
Smoothness	8/10	Smooth RMA decay; minor noise from close price variation

Operation Count (Streaming Mode)

Operation	Count	Notes
SUB	3	H-L, H-PrevC, L-PrevC
ABS	2	Gap calculations
MAX	2	True Range selection
FMA	1	RMA update
MUL	1	Decay for warmup
DIV	2	Warmup compensation + percentage
MUL	1	× 100
Total	~12 ops	Dominated by FMA and divisions

Validation

NATR is validated by computing ATR from external libraries and applying the same percentage formula.

Library	Status	Notes
QuanTAlib	✅	Native implementation
TA-Lib	✅	Via (ATR / Close) × 100; tolerance 0.10 for warmup divergence
Skender	✅	Via (GetAtr / Close) × 100
Tulip	✅	Via (atr / Close) × 100
Ooples	✅	Via (CalculateAverageTrueRange / Close) × 100

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.

Use Cases

Cross-Asset Volatility Comparison

Compare volatility across different price scales:

Asset	Price	ATR	NATR
Penny Stock	$2.50	0.25	10.0%
Mid-Cap	$150	4.50	3.0%
Blue Chip	$500	5.00	1.0%

ATR suggests Blue Chip is most volatile. NATR reveals Penny Stock has 10× the relative volatility.

Volatility-Adjusted Position Sizing

Position Size = (Account Risk %) / NATR

Ensures equal percentage risk per position regardless of asset price.

Regime Detection

NATR Range	Interpretation	Strategy Implication
< 1%	Low volatility	Mean reversion, tight stops
1-3%	Normal	Standard trend-following
3-5%	Elevated	Wider stops, reduced size
> 5%	High volatility	Crisis mode, capital preservation

Common Pitfalls

1. Lag Inheritance: NATR inherits ATR's smoothing lag. It measures recent volatility, not current or future volatility.

2. Close Price Spikes: A sharp close creates transient NATR spikes since it affects both TR (numerator) and the denominator simultaneously.

3. Near-Zero Prices: Assets approaching zero produce extreme NATR values. Implement minimum price thresholds.

4. Gap Sensitivity: Large overnight gaps inflate TR significantly. Consider using gap-adjusted data for equity analysis.

5. Warmup Period: The first 40+ bars (for period=14) contain warmup bias. Use IsHot to filter unreliable values.

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).

Related Indicators

• ATR: Absolute volatility measure NATR normalizes
• ATRN: ATR normalized to [0,1] based on historical min/max (different algorithm)
• CV: Coefficient of Variation—alternative percentage volatility measure
• HV: Historical Volatility—annualized standard deviation approach

References

• Wilder, J.W. (1978). New Concepts in Technical Trading Systems. Trend Research.
• TA-Lib documentation: NATR function specification
• TradingView PineScript: ta.natr() implementation