Why Historical Volatility Uses the Standard Deviation of Log Returns

Summary

The document asks why historical volatility is commonly measured as the standard deviation of log returns. The author points out that a sequence of identical daily percentage changes would have constant log returns and thus zero standard deviation, even though repeated large moves might feel volatile in ordinary language. This raises a distinction between variability in returns and the size or persistence of returns.

The text asks whether the standard measure is chosen because it aligns with Black-Scholes assumptions, or whether other reasons explain its widespread use. It provides no answer, formula comparison, or empirical evidence; it is a conceptual question rather than a full treatment. The example helps expose that standard deviation measures dispersion around the average return, not whether returns are consistently large. The document does not specify alternative volatility measures, sampling choices, or how different measures affect risk estimates, so it leaves those practical issues open.

Key ideas

• Historical volatility is commonly defined as the standard deviation of log returns.
• A constant sequence of log returns has zero standard deviation even when each return is large.
• Standard deviation measures return variability around the average, rather than return magnitude alone.
• The document asks about the link to Black-Scholes but does not answer it or compare alternatives.

Tags

# Rationale for Historical Volatility definition


# Rationale for Historical Volatility definition












I think I understand the most common definition for historical volatility (standard deviation of log returns), but it has me puzzled because it conflicts with my intuitive idea of what volatility is. If we use standard deviation of log returns, a stock that doesn't move has 0 volatility which makes sense. But a stock that jumps up 10% daily (or drops 10% daily) everyday consistently would have constant log return which would also produce a standard deviation of 0.

This conflicts with my intuitive sense of volatility, as these stocks seem quite volatile to me. Is this definition in place specifically so that it integrates with the BS model? Why else is this one of the more standard measurements of volatility?

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