Dimensional Units of Returns and Volatility
Summary
The document examines an apparent conflict in assigning time dimensions to returns, variance, and volatility. Under a definition of return as log price change per unit time, returns have inverse-time units and their variance has inverse-time-squared units. Annualizing variance by multiplying monthly variance by the number of periods may seem inconsistent with those dimensions.
The discussion explains that the square-root-of-periods scaling used for volatility treats the count of observations as dimensionless, relying on aggregation of log returns under assumptions such as independent, identically distributed observations. Another answer instead treats returns as dimensionless ratios quoted at a chosen measurement frequency, or uses inverse-time as a convention tied to annual quoting; this makes volatility scale with the square root of time. The answers offer different intuitions rather than a fully unified resolution, and one explicitly notes a remaining tension with the variance formula and Black–Scholes dimensional consistency.
Key ideas
- Defining return as log price change per unit time gives it inverse-time units.
- Under that definition, return variance has inverse-time-squared units.
- Annualized volatility scales by the square root of the number of observations, treated as dimensionless.
- The answers also frame returns as dimensionless ratios quoted at a selected frequency.
- The discussion acknowledges unresolved dimensional tension across conventions and models.
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Full text
# What are the units of the variance of returns?
# What are the units of the variance of returns?
I am a little confused about the units of the variance of returns. One way to compute that would be to look at the units of returns-
$$r=\frac{1}{\Delta t}\ln\frac{P(t+\Delta t)}{P(t)}=\text{Dimension }(\text{time})^{-1}$$
$$\text{Cov}(r_i,r_j)=E[(r_i-\bar{r}_i)(r_j-\bar{r}_j)]=\text{Dimension }(\text{time})^{-2}$$
But the above seems incorrect. For starters variance scales with time, i.e. annual variance is 12 times the monthly variance (assuming iid returns). Also stochastic calculus tells us that $\sigma$ or standard deviation scales with $(\text{time})^{-1/2}$. How do I reconcile that with the above?
## Answer by skoestlmeier (score 3, accepted)
https://quant.stackexchange.com/a/41480
There is nothing incorrect with your formulas, so let's look at the units when you annualize the volatility.
As an example, assume you have 252 daily return data. Their dimension is $(time)^{-1}$ and their variance is given in $(time)^{-2}$ (as you already stated). You may look at Chris Taylor's answer here on the underlying assumptions, why one can annualize volatility as $$\sigma_{\rm annual} = \sigma_{\rm daily} \times \sqrt{252}.$$
In fact, the scaling factor $\sqrt{252}$ is expressed with no unit. This arrises from the central limit theorem:
When using log-returns, the annualized return is calculated as $R_{year} = r_1 + r_2 + \cdots + r_n$ with $n=252$ daily log-return data. This sum of daily return data converges towards a normal distribution with parameters $N\left( n \mu ,n\sigma_{daily} \right)$, where $\mu$ is the mean of daily returns. This scaling with the factor $n$ is done without any unit. So on both sides of the above formula you have the unit $(time)^{-1}$ for the standard deviation of returns and the daily one scaled dimensionless by $\sqrt{252}$.
In a statistical meaning, the factor $n$ is just the number of data points you observe, although they represent single points in time in the context of financial calculus.
## Answer by zw_ (score 0)
https://quant.stackexchange.com/a/65962
I agree with @skoestlmeier's answer that the $\sqrt{252}$ should be considered dimensionless. However, using the result from your dimensional analysis (i.e. returns and volatility having dimension $time^{-1}$), you would conclude that the black scholes (BS) formula is dimensionally inconsistent (see this question).
I find it more helpful to think of returns as dimensionless (i.e. ratios) and therefore volatility, given a measurement frequency. A naive analogy would be, if you have a car the travels at constant velocity $10\,m/s$, then it will have travelled $10\,m$ after $1\,s$. The $10\,m$ distance is the way of thinking about returns, given you always use some frequency (1s, 1m, 1y etc..).
Now if you must give them dimensions, then I suggest thinking the notions of returns and variance having a "unit" $time^{-1}$ (much like the velocity in my naive example) since they are usually quoted in annual terms (which is a choice of frequency). As a consequence volatility must have dimension $time^{-1/2}$ which is in agreement with your stochastic calculus source and you have a dimensionally correct BS formula.
(P.S. I do recognise the inconsistency there would be when plugging in the variance formula, so I welcome any better intuitions.)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.