QuanTAlib/lib/volatility/yzv/Yzv.md

YZV: Yang-Zhang Volatility

Property	Value
Category	Volatility
Inputs	OHLCV bar (TBar)
Parameters	period (default 20)
Outputs	Single series (Yzv)
Output range	\geq 0
Warmup	period bars

TL;DR

• Yang-Zhang Volatility is a sophisticated volatility estimator that combines overnight (close-to-open) returns with Rogers-Satchell intraday volatil...
• Parameterized by period (default 20).
• Output range: \geq 0.
• Requires period bars of warmup before first valid output (IsHot = true).
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

"The best volatility estimator uses all the information the market gives you—overnight gaps, intraday swings, and everything in between."

Yang-Zhang Volatility is a sophisticated volatility estimator that combines overnight (close-to-open) returns with Rogers-Satchell intraday volatility to capture the full spectrum of price dynamics. Unlike simple close-to-close volatility that misses overnight gaps, or purely intraday measures that ignore opening moves, Yang-Zhang provides a theoretically unbiased estimate that remains consistent whether markets gap or drift.

Historical Context

Introduced by Dennis Yang and Qiang Zhang in their 2000 paper "Drift-Independent Volatility Estimation Based on High, Low, Open, and Close Prices," this estimator addressed a fundamental gap in volatility measurement. Traditional close-to-close volatility understates true volatility when significant price movements occur outside trading hours. The Parkinson (1980) and Garman-Klass (1980) estimators used high-low information but assumed continuous trading with no overnight gaps.

Yang and Zhang combined three components:

1. Overnight volatility (\sigma_o^2): Captures close-to-open gaps
2. Open-to-close volatility (\sigma_c^2): Captures standard intraday drift
3. Rogers-Satchell volatility (\sigma_{RS}^2): Captures intraday high-low range accounting for drift

The key innovation was deriving optimal weights that minimize variance while remaining independent of price drift. The resulting estimator is approximately 8× more efficient than close-to-close for capturing true volatility.

Architecture & Physics

1. Log Return Components

For each bar, compute four log returns relative to the previous close and current open:

r_o = \ln\left(\frac{O_t}{C_{t-1}}\right) \quad \text{(overnight return)}

r_c = \ln\left(\frac{C_t}{O_t}\right) \quad \text{(open-to-close return)}

r_h = \ln\left(\frac{H_t}{O_t}\right) \quad \text{(high relative to open)}

r_l = \ln\left(\frac{L_t}{O_t}\right) \quad \text{(low relative to open)}

2. Yang-Zhang Weighting Factor

The optimal weight k that minimizes estimator variance:

k = \frac{0.34}{1.34 + \frac{n+1}{n-1}}

where n is the smoothing period. For typical values:

• n = 10: k \approx 0.196
• n = 20: k \approx 0.215
• n = 30: k \approx 0.222

3. Daily Variance Components

Overnight variance:

\sigma_o^2 = r_o^2

Open-to-close variance:

\sigma_c^2 = r_c^2

Rogers-Satchell variance (drift-independent intraday measure):

\sigma_{RS}^2 = r_h \cdot (r_h - r_c) + r_l \cdot (r_l - r_c)

4. Combined Daily Variance

\sigma_{daily}^2 = \sigma_o^2 + k \cdot \sigma_c^2 + (1 - k) \cdot \sigma_{RS}^2

5. Smoothed Volatility Output

Apply exponential smoothing (RMA) to daily variance with bias correction, then take square root:

\text{YZV}_t = \sqrt{\text{RMA}(\sigma_{daily}^2, n)}

Mathematical Foundation

Bias-Corrected RMA

The implementation uses RMA (Relative Moving Average, equivalent to EMA with \alpha = 1/n) with bias correction to handle the startup period:

\text{RMA}_t = \alpha \cdot x_t + (1 - \alpha) \cdot \text{RMA}_{t-1}

where \alpha = 1/n.

Bias compensator:

e_t = (1 - \alpha)^t

Corrected output:

\text{RMA}_{corrected} = \frac{\text{RMA}_{raw}}{1 - e_t}

This ensures the first few bars don't suffer from initialization bias.

Rogers-Satchell Properties

The Rogers-Satchell component has elegant properties:

• Drift-independent: Provides consistent estimates regardless of price trend
• Efficiency: Uses high and low prices for information gain
• Non-negativity: Always ≥ 0 when calculated correctly

The formula r_h(r_h - r_c) + r_l(r_l - r_c) can be rewritten as:

\sigma_{RS}^2 = r_h \cdot r_l - r_l \cdot r_c - r_h \cdot r_c + r_h^2 + r_l^2 - r_l^2

Example Calculation

Period = 2, Bars: [(O=100, H=105, L=98, C=103), (O=102, H=108, L=101, C=106)]

Bar 1 (assuming previous close = 99):

• r_o = \ln(100/99) = 0.01005
• r_c = \ln(103/100) = 0.02956
• r_h = \ln(105/100) = 0.04879
• r_l = \ln(98/100) = -0.02020
• \sigma_o^2 = 0.0001010
• \sigma_c^2 = 0.0008738
• \sigma_{RS}^2 = 0.04879(0.04879-0.02956) + (-0.02020)((-0.02020)-0.02956) = 0.001935
• k = 0.34/(1.34 + 3/1) = 0.0783
• \sigma_{daily}^2 = 0.0001010 + 0.0783(0.0008738) + 0.9217(0.001935) = 0.001953

Bar 2 (previous close = 103):

• Similar calculation...
• Apply RMA to variance sequence
• Output = sqrt(smoothed variance)

Performance Profile

Operation Count (Streaming Mode, Scalar)

Per-bar operations:

Operation	Count	Cost (cycles)	Subtotal
LN (natural log)	4	50	200
MUL	12	3	36
ADD/SUB	8	1	8
DIV	3	15	45
SQRT	1	15	15
FMA candidates	3	5	15
Total	—	—	~319 cycles

The logarithm operations dominate the cost.

Batch Mode (512 values, SIMD/FMA)

Operation	Scalar Ops	SIMD Ops (AVX2)	Speedup
LN	2048	256	8×
Arithmetic	6144	768	8×
SQRT	512	64	8×

Per-bar savings with SIMD/FMA:

Optimization	Cycles Saved	New Total
SIMD LN	~175	~144
FMA for compound ops	~10	~134
Total SIMD/FMA	~185 cycles	~134 cycles

Memory Profile

• Per instance: ~120 bytes (state record + backup)
• 100 instances: ~12 KB
• Minimal footprint: No ring buffers required (RMA is recursive)

Quality Metrics

Metric	Score	Notes
Accuracy	10/10	Theoretically optimal, unbiased estimator
Timeliness	8/10	Responds within period bars
Efficiency	9/10	~8× more efficient than close-to-close
Gap Handling	10/10	Explicitly models overnight returns
Drift Independence	10/10	Rogers-Satchell component is drift-free

Validation

Library	Status	Notes
TA-Lib	N/A	Not implemented
Skender	N/A	Not implemented
Tulip	N/A	Not implemented
OoplesFinance	N/A	Not implemented
PineScript	✅	Matches yzv.pine reference
Self-consistency	✅	Streaming = Batch modes match

Common Pitfalls

1. First bar handling: On the very first bar, there's no previous close. The implementation uses the current open as the "previous close" for this bar only, meaning r_o = 0 for bar 0.

2. Warmup period: YZV needs approximately Period bars before producing stable estimates. The bias-corrected RMA helps, but early values during warmup may still be less reliable.

3. Negative variance guard: Due to floating-point precision, the Rogers-Satchell component can theoretically go slightly negative in edge cases. The implementation guards against this by clamping variance to zero before taking the square root.

4. Scale interpretation: YZV output is in the same units as the log-return standard deviation (essentially a percentage in decimal form). A value of 0.02 means ~2% daily volatility.

5. Parameter sensitivity: The optimal k weight depends on period. Don't reuse k values calculated for different periods—the formula must be recomputed.

6. Gap vs no-gap markets: For instruments that trade 24/7 (crypto, forex), the overnight component may be less meaningful. Consider using only the Rogers-Satchell component for such markets.

Trading Applications

Volatility Forecasting

Yang-Zhang provides more accurate current volatility estimates, improving forecasts:

Forecast accuracy: YZV > Close-to-close > Parkinson
Use for: Option pricing, VaR calculations, position sizing

Regime Detection

Monitor YZV for volatility regime changes:

Rising YZV: Increasing market uncertainty
Falling YZV: Settling market conditions
YZV > 2 × historical average: High-volatility regime

Options Trading

Better IV estimation for pricing and hedging:

If Realized_YZV > Implied_Vol: Options may be underpriced
If Realized_YZV < Implied_Vol: Options may be overpriced

Position Sizing

Scale positions inversely with volatility:

Position Size = Target $ Risk / (Entry Price × YZV × Multiplier)

Gap Risk Assessment

Compare overnight vs intraday components:

If overnight_component > intraday_component: Gap risk elevated
Consider reducing overnight positions or hedging

Relationship to Other Volatility Measures

Measure	Compared to YZV
Close-to-Close	YZV ~8× more efficient; C2C ignores gaps
Parkinson	Parkinson ignores gaps; YZV handles them
Garman-Klass	GK handles overnight but not as optimally weighted
Rogers-Satchell	RS is a component of YZV; doesn't handle gaps
ATR	ATR is absolute price-based; YZV is log-return based
Historical Volatility	YZV is a better HV estimator

Implementation Notes

State Management

The indicator maintains a compact state record:

• RawRma: Running RMA value (before bias correction)
• ECompensator: Bias compensator (1-\alpha)^n
• PrevClose: Previous bar's close for overnight return
• LastValidYzv: Last valid output for NaN handling
• Count: Bar count for warmup tracking
• HasPrevClose: Flag for first-bar handling

NaN/Infinity Handling

Invalid OHLC inputs are detected and the last valid YZV is substituted. This prevents NaN propagation through the RMA chain.

Numerical Stability

The implementation uses:

• Epsilon guard (1e-10) for division safety in bias correction
• Clamping of variance to ≥ 0 before sqrt
• Last-valid substitution for non-finite results

References

• Yang, D., & Zhang, Q. (2000). "Drift-Independent Volatility Estimation Based on High, Low, Open, and Close Prices." Journal of Business, 73(3), 477-491.
• Rogers, L. C. G., & Satchell, S. E. (1991). "Estimating Variance from High, Low and Closing Prices." Annals of Applied Probability, 1(4), 504-512.
• Parkinson, M. (1980). "The Extreme Value Method for Estimating the Variance of the Rate of Return." Journal of Business, 53(1), 61-65.
• Garman, M. B., & Klass, M. J. (1980). "On the Estimation of Security Price Volatilities from Historical Data." Journal of Business, 53(1), 67-78.