Quantitative Risk Volatility Estimation Yang-Zhang (2000)

Yang-Zhang Range-Based Volatility Lab

Estimate historical volatility with 8x to 14x greater statistical efficiency than Close-to-Close using Dennis Yang and Qiang Zhang's (2000) jump- and drift-adjusted minimum-variance OHLC model.

Asset Class Archetypes:
Yang-Zhang Volatility
34.4%
Abs Error: 0.2% | Eff: 11.2x
Close-to-Close Baseline
31.8%
Abs Error: 2.6% | Eff: 1.0x (Baseline)
Garman-Klass Volatility
28.9%
Parkinson: 27.4% | Eff: 7.4x
True Generating Target
34.4%
Cont: 28% | Jump: 20%
Model Controls
Yang & Zhang (2000)
Initial Security Price ($S_0$) $150.00
Continuous Volatility ($\sigma_{\text{cont}}$) 28%
Annualized intraday continuous diffusion volatility.
Overnight Jump Volatility ($\sigma_{\text{jump}}$) 20%
Annualized opening jump volatility between previous close and open.
Directional Drift Rate ($\mu$) 12%
Annual continuous trend rate. Non-zero drift biases classical estimators.
Sample Window Length ($n$) 30 days
Lookback window in trading days for calculating estimators.
Real-Time Canvas

Historical Volatility Estimator Comparison Matrix

Analytical & Empirical
Estimator Name Mathematical Formula Calculated Vol Efficiency Theoretical Robustness
Total Target Volatility Matrix

True Total Volatility ($\sigma_{\text{true}} = \sqrt{\sigma_{\text{cont}}^2 + \sigma_{\text{jump}}^2}$) across overnight jump and continuous intraday variance.

Yang-Zhang Efficiency Multiplier Matrix

Relative Efficiency Multiplier ($\text{Var}(\sigma_{C2C}) / \text{Var}(\sigma_{YZ})$) across Sample Window ($n$ days) and Continuous Drift ($\mu$).

Range-Based Volatility Econometrics Masterclass

1. The Inefficiency of Close-to-Close Volatility

In quantitative risk management and options market making, historical volatility is traditionally measured using the sample variance of daily logarithmic close-to-close returns:

$$\sigma_{C2C}^2 = \frac{1}{n-1}\sum_{t=1}^n \left( \ln\left(\frac{C_t}{C_{t-1}}\right) - \bar{r} \right)^2$$

While simple and unbiased, the Close-to-Close estimator discards approximately 90% of price path information generated during trading hours. An asset could rally +5% intraday and collapse -5% to finish completely flat ($C_t = C_{t-1}$), producing a measured Close-to-Close return of zero despite immense market turbulence.

2. The Evolution of Extreme-Value & OHLC Estimators

To capture intraday volatility, financial econometricians developed range-based estimators utilizing daily Open ($O_t$), High ($H_t$), Low ($L_t$), and Close ($C_t$) prices:

Parkinson (1980) & Garman-Klass (1980)

Parkinson introduced extreme-value estimation using daily high and low prices: $\sigma_P^2 = \frac{1}{4n\ln 2}\sum (\ln(H_t/L_t))^2$, offering 5.2x efficiency. Garman and Klass added open and close prices, raising efficiency to 7.4x.

Critical Flaw: Both assume continuous diffusion with zero drift ($\mu = 0$) and zero overnight gap. Opening gaps and trending runs introduce substantial upward bias.

Rogers-Satchell (1991)

Rogers and Satchell developed a drift-independent estimator:

$$\sigma_{RS}^2 = \frac{1}{n}\sum_{t=1}^n \left[ \ln\left(\frac{H_t}{C_t}\right)\ln\left(\frac{H_t}{O_t}\right) + \ln\left(\frac{L_t}{C_t}\right)\ln\left(\frac{L_t}{O_t}\right) \right]$$

Rogers-Satchell remains unbiased even under strong directional trends ($\mu \ne 0$), but still assumes zero overnight jump between $C_{t-1}$ and $O_t$.

3. The Yang-Zhang (2000) Unbiased Minimum-Variance Solution

Dennis Yang and Qiang Zhang solved the open problem in volatility estimation by creating a linear combination of overnight jump variance, open-to-close continuous variance, and Rogers-Satchell drift-independent variance that achieves minimum variance:

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

where the components and optimal weighting factor $k$ are defined as:

$$\sigma_o^2 = \frac{1}{n-1}\sum_{t=1}^n (o_t - \bar{o})^2, \quad \sigma_c^2 = \frac{1}{n-1}\sum_{t=1}^n (c_t - \bar{c})^2, \quad k = \frac{0.34}{1.34 + \frac{n + 1}{n - 1}}$$

Because $k$ minimizes the total variance of the linear combination, Yang-Zhang is proven to be the unique minimum-variance unbiased estimator among all quadratic combinations of OHLC prices, achieving up to 14x efficiency over Close-to-Close.

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Frequently Asked Questions

The Yang-Zhang (2000) estimator, published by Dennis Yang and Qiang Zhang in the Journal of Business, is a multi-price historical volatility estimator that utilizes Open, High, Low, and Close (OHLC) prices. It is the first and only unbiased minimum-variance estimator that simultaneously handles both non-zero continuous trend/drift and overnight opening price jumps, achieving 8x to 14x greater statistical efficiency than classical Close-to-Close standard deviation.

The classic Close-to-Close (C2C) sample standard deviation discards all price movement between closing bells. By ignoring intraday High, Low, and Open prices, it requires large sample sizes (e.g., 60 to 252 trading days) to achieve acceptable statistical precision. Furthermore, during market regime changes or volatility spikes, Close-to-Close reacts sluggishly, whereas range-based estimators detect volatility shifts immediately.

Parkinson (1980) uses extreme values (daily High and Low prices) and achieves approximately 5.2 times the statistical efficiency of Close-to-Close. Garman-Klass (1980) extends Parkinson by incorporating Open and Close prices as well, raising efficiency to 7.4 times Close-to-Close. However, both Parkinson and Garman-Klass assume continuous geometric Brownian motion with zero drift and zero opening jumps; if opening gaps or strong trends occur, both estimators exhibit upward bias.

Rogers and Satchell (1991) formulated a closed-form estimator using Open, High, Low, and Close prices that is analytically invariant to non-zero continuous drift (mu != 0). While standard estimators inflate volatility estimates when a stock experiences strong directional upward or downward trending runs, Rogers-Satchell isolates pure diffusion volatility without trend distortion. However, it still assumes continuous trading without overnight opening gaps.

Statistical efficiency is defined as the ratio of estimator sample variances: Eff = Var(sigma_C2C) / Var(sigma_YZ). An efficiency multiplier of 10x means that 20 days of OHLC data using the Yang-Zhang estimator provides the same statistical precision and confidence interval width as 200 days of Close-to-Close returns. This allows quantitative risk managers, VaR models, and algorithmic execution algorithms to calibrate to market volatility using short, responsive lookback windows without suffering from noisy estimation error.