Volatility of Log Returns Versus Percentage Price Changes
Summary
The document asks how several definitions of volatility relate: the standard deviation of log returns, the standard deviation of continuously compounded rates, and the standard deviation of percentage price changes. It notes that log returns and continuously compounded rates aggregate across periods by addition. Under independent, identically distributed returns, their variance adds across periods, giving the familiar square-root-of-time scaling for volatility.
The question is whether simple percentage changes aggregate in the same way. It proposes that small percentage changes approximate log returns, but does not provide a resolution. The useful distinction is that simple returns compound multiplicatively, whereas log returns add exactly; the approximation between them is strongest for small changes. The discussion is conceptual and supplies no empirical comparison or assumptions beyond the stated normality and independence example, so it should not be read as establishing a universal scaling rule for observed returns.
Key ideas
- Log returns add across time periods, while simple percentage returns compound multiplicatively.
- For independent returns with equal variance, log-return variance adds across periods and volatility scales with the square root of time.
- Simple percentage changes approximate log returns when changes are small.
- The document poses the aggregation question but does not include an answer or empirical evidence.
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Full text
# Different definitions of volatility (simple question)
# Different definitions of volatility (simple question)
I have a basic question on volatility that I wanted some clarification on.
In finance books (such as Hull), there's a few different ways volatility is defined. One of them is the standard deviation of yearly log returns, $R_i = \ln \frac{S_i}{S_{i-1}}$. Another is the standard deviation of the annual continuously compounded return rate, so the $x$ in $\exp(xT)$.
Both of these definitions make sense to me in terms of how they are derived and since the log returns and continuously compounded rates can be added up to give log returns and rates over longer time periods. For example: If we assume log returns (or rates) are normally distributed and independent, then if the random log return for year 1 is $R_1$, the random log return for year 2 is $R_2$ (each with volatility $\sigma$), and the random log return for years 1 and 2 combined is $R^* = R_1 + R_2$ , then we can calculate the volatility over two years using $\text{Variance}(R_1 + R_2) = 2\sigma^2$, and so the volatility of $R^*$is $\sqrt{2}\sigma$. So we can calculate volatility for 1 year, 2 years, 1 day, and so on, by the usual square root of time scale factor.
Hull's book also mentions that volatility can be considered the standard deviation of the percentage change in price, $\frac{dS}{S}$. I know that this makes sense since the SDE for $\frac{dS}{S}$ is normally distributed with variance $\sigma{\Delta t}$, but I was wondering why the values $\frac{S_i - S_{i-1}}{S_{i-1}}$ when added up don't have to lead to percentage changes in price of the longer time periods like log returns and continously compounded rates do. I was thinking it could be since for small percentage changes the log returns are approximately equal to the percentage changes. I'm not sure if this correct though.
I might be missing something obvious here, so I'd appreciate any help. Thanks!Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.