When Volatility Scales with the Square Root of Time
Summary
The document explains the assumptions behind square-root-of-time volatility scaling. It represents a multi-period return as the sum of single-period returns, then uses the variance of a sum. When returns are uncorrelated, covariance terms vanish; when each period also has the same variance, total variance grows in proportion to the number of periods. Taking the square root yields the familiar scaling rule.
The explanation applies most directly to log returns, for which returns add exactly over time; the source notes that simple returns only add approximately. It stresses that exponential Brownian motion is not the only route to the result: uncorrelated returns and stationary variance are the key assumptions described. If returns are correlated, covariance terms alter the calculation, and changing variance also breaks the stated derivation. The question about variance of price levels is not answered, so the result should not be transferred from returns to prices without a separate analysis.
Key ideas
- Square-root-of-time scaling follows when return variance accumulates linearly across periods.
- Uncorrelated returns make the covariance terms in the variance of a sum equal to zero.
- Equal period-by-period variance is also required for the stated scaling result.
- Log returns add exactly across time, while simple returns only do so approximately.
- The derivation concerns returns and does not establish a matching rule for price-level variance.
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# Basics about the scaling property of volatility
# Basics about the scaling property of volatility
It is a usual practice to calculate realized volatility $\sigma$ using the square root of the usual variance estimator $\hat{{\sigma}²}$. This is done using the stock log returns (practitioners sometimes BS variance). It is well known that the volatility scales as square root of time $\sigma_T = \sqrt{T} \cdot \sigma_1$. This is a trivial result when you model the stock dynamics as exponential brownian motion.
My questions are now the following, would any scaling property hold if you calculate the volatility as square root of variance of stock prices, as after all one can calculate the variance of a exponential brownian motion.
## Answer by Richi Wa (score 2)
https://quant.stackexchange.com/a/9573
I don't know what you mean by "any scaling" rule. For the square-root of time I can say that it only needs uncorrelated returns.
Assume that the return from time point $1$ to $T$ is called $r_{1,T}$ and that it is given as $r_{1,T} = r_1 + r_2 + \cdots + r_T = \sum_{t=1}^T r_t$ where $r_t, t=1,\ldots,T$ are the one-period (e.g. one day) returns. The condition (the sum) holds excactly true for log-returns and approximately for simple returns.
Then we can calculate the variance $$ VAR(r_{1,T}) = VAR(\sum_{t=1}^T r_t) = \sum_{t=1}^T VAR(r_t), $$ where the covariance terms vanish as we assume that the returns are uncorrelated. Otherwise we would have more covariance terms $COV(r_i,r_j)$ for $i \neq j$.
If we furthermore assume that $VAR(r_t) = \sigma^2$ for $t=1,\ldots,T$ (this is stationary variance) then we get $$ VAR(r_{1,T}) = \sum_{t=1}^T \sigma^2 = T \sigma^2. $$ and by taking the square-root we get the square-root of time scaling for volatility. Note that we have used log-returns and that we have assume uncorrelated returns and stationary variance - nothing more.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.