- Historical Volatility (HV), also known as close-to-close volatility or realized volatility, is the classical measure of price volatility using the ...
Historical Volatility (HV), also known as close-to-close volatility or realized volatility, is the classical measure of price volatility using the standard deviation of logarithmic returns. First formalized in the early 20th century and central to the Black-Scholes option pricing model, HV remains the benchmark against which all other volatility estimators are compared. This implementation uses population standard deviation with a rolling window and optional annualization.
## Historical Context
The close-to-close volatility estimator predates most range-based alternatives, with its mathematical foundations established alongside the development of stochastic calculus and diffusion processes. The estimator became central to quantitative finance with the publication of the Black-Scholes model in 1973, which explicitly required an estimate of stock price volatility.
Louis Bachelier's 1900 thesis "Théorie de la spéculation" laid the groundwork, modeling price changes as Brownian motion. Fischer Black, Myron Scholes, and Robert Merton formalized the use of historical standard deviation of log returns as the volatility parameter in option pricing.
Despite the development of more efficient estimators (Parkinson 1980, Garman-Klass 1980, Yang-Zhang 2000), close-to-close volatility remains the most widely used and understood measure because:
1. It requires only closing prices, universally available
2. It directly measures what options traders care about—settlement-to-settlement variation
3. It serves as the baseline efficiency benchmark (efficiency = 1.0)
## Architecture & Physics
### 1. Log Return Calculation
Each period's return is computed as the natural logarithm of price ratios:
This implementation uses **population variance** (dividing by $n$) rather than sample variance (dividing by $n-1$). For typical periods (14-30 returns), the difference is small:
| Period | Sample/Pop Ratio |
| :---: | :---: |
| 10 | 1.111 |
| 14 | 1.077 |
| 20 | 1.053 |
| 30 | 1.034 |
Population variance provides a consistent estimator for the rolling window and matches the implementation in most trading platforms.
### 4. Volatility Calculation
Convert variance to volatility (standard deviation):
$$
\sigma_t = \sqrt{variance}
$$
### 5. Optional Annualization
If annualization is enabled (default):
$$
\sigma_{annual,t} = \sigma_t \times \sqrt{N}
$$
where $N$ = annual periods (default 252 trading days).
## Mathematical Foundation
### Log Return Properties
For a geometric Brownian motion $dS = \mu S dt + \sigma S dW$:
The log return over interval $\Delta t$ is:
$$
r = \ln\left(\frac{S_t}{S_{t-1}}\right) = \left(\mu - \frac{\sigma^2}{2}\right)\Delta t + \sigma \sqrt{\Delta t} \cdot Z
| Estimator | Relative Efficiency | Data Required |
| :--- | :---: | :--- |
| **Close-to-Close (HV)** | **1.0** | **C** |
| Parkinson (HLV) | 5.2 | H, L |
| Garman-Klass (GKV) | 7.4 | O, H, L, C |
| Rogers-Satchell | 8.4 | O, H, L, C |
| Yang-Zhang | 14.0 | O, H, L, C |
HV (close-to-close) is the efficiency baseline. A Parkinson estimator with efficiency 5.2 means you need 5.2× fewer observations to achieve the same precision—or equivalently, 5.2× better precision with the same observations.
### Annualization Factor
For daily data with 252 trading days:
$$
\sqrt{252} \approx 15.875
$$
Common annualization factors:
| Data Frequency | Periods/Year | Factor |
| :--- | :---: | :---: |
| Daily | 252 | 15.875 |
| Weekly | 52 | 7.211 |
| Monthly | 12 | 3.464 |
| Hourly (6.5h/day) | 1638 | 40.472 |
### Warmup Period
HV requires `period + 1` prices to produce a valid result:
- First price establishes the baseline
- Next `period` prices generate `period` returns
- Standard deviation is calculated on these `period` returns
The `IsHot` property indicates when warmup is complete.
## Performance Profile
### Operation Count (Streaming Mode, Scalar)
Per-bar operations after warmup:
| Operation | Count | Cost (cycles) | Subtotal |
| :--- | :---: | :---: | :---: |
| LOG | 1 | 25 | 25 |
| DIV | 1 | 15 | 15 |
| MUL | 2 | 3 | 6 |
| ADD/SUB | 4 | 1 | 4 |
| DIV (variance) | 2 | 15 | 30 |
| SQRT | 1 | 15 | 15 |
| MUL (annual) | 1 | 3 | 3 |
| **Total** | — | — | **~98 cycles** |
The dominant costs are LOG (26%) and SQRT (15%). Computational formula avoids iteration over the window.
Note: Most libraries provide building blocks (STDDEV, LOG) rather than a dedicated HV function. The implementation is validated against the mathematical formula and PineScript reference.
## Common Pitfalls
1.**Warmup period**: HV requires `period + 1` prices before producing valid results. With default period=20, you need 21 prices to generate 20 returns. The `IsHot` property indicates when warmup is complete.
2.**Zero or negative prices**: Log transformation requires positive prices. Zero or negative values trigger last-valid-value substitution to prevent NaN propagation.
3.**Constant prices**: When all prices in the window are identical, returns are zero, producing zero volatility. This is mathematically correct but may indicate data issues.
4.**Annualization assumptions**: Default annualization assumes 252 trading days/year. For intraday data, cryptocurrency (365 days), or weekly data, adjust `annualPeriods` accordingly.
5.**Mean return assumption**: The standard formula implicitly subtracts the mean return. During strong trends, this captures both directional movement and noise, potentially overstating "noise" volatility.
6.**Population vs sample variance**: This implementation uses population variance (n divisor). If comparing with implementations using sample variance (n-1 divisor), expect slight differences: sample/pop ratio ≈ n/(n-1).
7.**Overnight gaps**: Unlike range-based estimators, HV fully captures overnight gaps (close-to-close movements). This can be an advantage (complete picture) or disadvantage (includes information not tradeable intraday).
8.**Comparison with range-based**: HV is 5.2× less efficient than Parkinson (HLV) and 7.4× less efficient than Garman-Klass (GKV). Use HV when:
- Only close prices are available
- You specifically want close-to-close volatility (e.g., settlement-based risk)
- Comparing with implied volatility (which prices close-to-close variation)
## Trading Applications
### Options Volatility Comparison
Compare realized HV with implied volatility (IV):
```
Volatility Risk Premium = IV - HV
If IV > HV consistently: Options are "expensive," consider selling
If IV < HV consistently: Options are "cheap," consider buying
```
This comparison is most valid with HV because IV prices close-to-close variation.
- Merton, R. C. (1980). "On Estimating the Expected Return on the Market: An Exploratory Investigation." *Journal of Financial Economics*, 8(4), 323-361.
