Interpreting Daily Volatility Estimators and Log Returns

Summary

The note clarifies why a daily volatility formula may use closing prices without dividing by the sample degrees of freedom. In the geometric setting, the relevant price variable is the logarithm of the original price, so the daily change corresponds to a log return. The formula under discussion estimates variance for one day from daily return observations; it is not the sample variance computed across many returns.

This distinction explains why the denominator and inputs can differ from a familiar historical sample volatility calculation. The note points readers to a fuller explanation but provides no derivation, comparative evidence, or details about the six alternative measures named in the question. Its scope is therefore limited to distinguishing a one-day estimator from a multi-observation sample estimate.

Key ideas

• In a geometric price model, volatility concerns changes in log prices.
• A daily change in log price is a log return.
• A one-day variance estimator differs from sample variance estimated across many returns.
• The note does not explain or compare all six alternative volatility measures.

Tags

# Can someone explain the 6 alternative volatility measures?


# Can someone explain the 6 alternative volatility measures?












I'm reading this: https://www.cmegroup.com/trading/fx/files/a_estimation_of_security_price.pdf

and am a bit confused as to why the "classical equation" on page 3 does not divide by n-1 nor use the log prices. I'm assuming C is just the closing price itself.

Thank!

## Answer by Igor Pozdeev (score 2, accepted)

https://quant.stackexchange.com/a/39933

Read the document carefully:

> Thus in the geometric case, "price" would mean "logarithm of original price", and "volatility" would mean "variance of the logarithm of original prices"

so the difference is really the log-return.

Now, the formula you are referring to is the estimator of the variance on one particular day given that only daily returns are observed. What you have in mind is the sample volatility estimated from many returns.

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