Parkinson Volatility

Formula

σ²_period = (1 / (4 N ln 2)) · Σ_{t=1..N} (ln(H_t / L_t))²
σ_annual  = σ_period · sqrt(P)

H_t ≥ L_t > 0.  N is the number of ranges you pass.
4 ln 2 ≈ 2.7726, so a day whose high is about 1.7% above its low
(ln(H/L) ≈ 0.0166, squared ≈ 2.77e-4) contributes about 1e-4 of daily variance
before annualisation, i.e. 1 percent daily, before the average.

P defaults to 252 and scales the mean daily variance, which is the right
shape when each row is one session.

Why a range

• The high and the low summarise the whole path, not only the two endpoints. For a driftless Brownian motion the range estimator has a lower variance than the close-to-close estimator on the same number of days.
• It is a natural check on realized volatility when you have bars but not a full trade tape: daily high and low are enough.
• It is still a historical number. An option-implied number is implied volatility, and a recursive filter is GARCH.

Pitfalls

• Drift. A strong trend widens the range for a reason other than diffusion variance. Parkinson is derived at zero drift. Rogers-Satchell and Yang-Zhang exist because of that; this page does not implement them.
• Overnight gaps. If high and low are the cash-session extrema, the gap from yesterday's close is invisible. The estimator then misses a piece of the day's variance that a close-to-close number includes.
• Bad prints. One erroneous high dominates a squared range. The average does not trim it.
• Do not feed intraday bars and also multiply by 252 unless each row is one session. The function annualises the mean of the rows you pass.

Python reference implementation

Requires numpy. Tested before publication; the generator re-executes it on every rebuild. MIT licensed. Download parkinson-volatility.py · All snippets

import numpy as np

def parkinson_vol(high, low, periods=252) -> float:
    """Annualised Parkinson volatility from high and low arrays.

    Each pair is one period. `periods` is how many such periods make a year.
    Requires 0 < low <= high.
    """
    h = np.asarray(high, dtype=float)
    lo = np.asarray(low, dtype=float)
    if h.shape != lo.shape or h.ndim != 1 or h.size == 0:
        raise ValueError("high and low must be non-empty and aligned")
    if not np.all(np.isfinite(h)) or not np.all(np.isfinite(lo)):
        raise ValueError("high and low must be finite")
    if np.any(lo <= 0) or np.any(h < lo):
        raise ValueError("require 0 < low <= high")
    if periods <= 0:
        raise ValueError("periods must be positive")
    x = np.log(h / lo)
    return float(np.sqrt(np.mean(x * x) / (4.0 * np.log(2.0)) * periods))

FAQ

Why is the divisor 4 ln 2?

It is the expected value of the squared log range of a standard Brownian motion over one unit of time. Dividing by it converts an average squared range into a variance estimate. The derivation assumes no drift and no jumps.

Is Parkinson always better than close-to-close?

No. It is more efficient in the zero-drift diffusion it was derived for. A large overnight gap, a trading halt, or a bad high print can make it worse than a close-to-close estimator that at least uses the official closes.