Volatility — the standard deviation of returns

What volatility is

In finance, volatility is the standard deviation of returns — the spread number $\sigma$ from A1-04, applied to a return series. It's almost always quoted annualized (converted to a per-year figure). Some flavors you'll encounter:

• Historical (realized) volatility — computed from past returns. The most common form.
• Implied volatility — the volatility that option prices imply the market expects. Forward-looking.
• Conditional volatility — today's vol estimated from a model like GARCH (a model that updates its volatility estimate day by day; it arrives in the time series track).

For Phase 1 we mostly mean historical.

The sample standard deviation of returns is:

$\hat{\sigma} = \sqrt{\frac{1}{N-1} \sum_{t=1}^{N} (r_t - \bar{r})^2}$

In words: "sigma-hat" (our estimate of $\sigma$) is the square root of the average squared deviation. Recipe: take each return $r_t$, subtract "r-bar" — the sample average from A2-01 — to get that day's deviation. Square it. Sum the squares over all $N$ days. Divide by $N - 1$. Take the square root.

Concrete: for returns +1% and −1%, the average is 0, the squared deviations are 0.0001 each, their sum is 0.0002, dividing by $N - 1 = 1$ gives 0.0002, and the square root is about 1.41%.

Why divide by $N-1$ instead of $N$? This is Bessel's correction. We measured deviations from the sample mean — a number computed from the same data — which makes the raw deviations slightly too small on average. Dividing by the smaller $N-1$ compensates. It makes the sample variance unbiased (the standard deviation itself remains slightly biased, but the correction helps).

Annualization

If daily returns are approximately independent — no day influencing another — with daily standard deviation $\sigma_d$ (read "sigma sub d," the daily sigma), then:

$\sigma_{\text{annual}} = \sigma_d \cdot \sqrt{T}$

In words: annual volatility equals daily volatility times the square root of $T$, where $T$ is the number of trading days per year (252 for US equities, sometimes 252 or 250 for other markets).

For SPY: daily $\sigma$ around 1% → annual $\sigma$ around 16%. Memorize the multiplier $\sqrt{252} \approx 15.87$.

Why volatility is easier to estimate than the mean

In A2-01 we said estimating the mean from a small sample is essentially hopeless. Volatility is the opposite — it's estimable from short samples.

Why? Not because variance is intrinsically easier. The reason is a scale mismatch — the two quantities being estimated are wildly different sizes. At daily frequency, volatility (~1%) is roughly 30× larger than the mean (~0.03%). Both estimators shrink their error as $\sqrt{N}$. But the mean estimate has to resolve a tiny signal buried under a much larger noise term. The volatility estimate is measuring the noise term itself.

Concretely, for (approximately) normal returns:

• Standard error of the mean: $\sigma / \sqrt{N}$ — sigma divided by root-N, as in A2-01. (Standard error, from A2-01: the typical wobble of the estimate itself.)
• Standard error of the standard deviation: roughly $\sigma / \sqrt{2N}$ — same shape, but with $2N$ under the square root.

So for a 1-year sample of daily returns ($N = 252$, daily $\sigma = 1\%$, daily mean ≈ 0.03%):

• Estimate of mean: SE $= 1\%/\sqrt{252} \approx 0.06\%$ — twice the size of the mean itself. Useless for confirming the mean is positive.
• Estimate of volatility: SE $= 1\%/\sqrt{504} \approx 0.045\%$ — only ~4.5% relative error on a 1% quantity. Tight enough to size positions.

(One caveat cuts the other way: fat tails — extreme days being more common than the bell curve predicts — make variance estimates noisier than the normal-theory formula suggests. The asymmetry survives. But it's the σ ≫ μ scale gap doing the work — the symbol $\gg$ reads "much greater than" — not any magic of squared deviations.)

This asymmetry is enormous for trading. It means:

• We cannot reliably know whether a strategy's positive return is real or noise.
• We can reliably know how volatile a strategy is.

This is why disciplined quants focus relentlessly on risk: it's the part of strategy performance we can measure honestly.

The dark side: volatility isn't constant

Empirically, daily stock-return volatility:

• Clusters in time — high-vol days follow high-vol days. The 2008 crisis had months of 3%+ daily volatility; calm periods can have months of 0.5% daily volatility.
• Spikes around news — earnings, Fed announcements, geopolitical shocks.
• Mean-reverts — drifts back toward its long-run average — but slowly. Vol regime changes can persist for months.

A constant-vol assumption (which the basic formulas above implicitly make) underestimates risk during stress periods. More sophisticated models capture the clustering — GARCH, and EWMA (exponentially weighted moving average, which counts recent observations more heavily than old ones). We'll meet them in the time series track.

For now: when you compute "the volatility" of an asset over a single window, you're computing an average across regimes. The vol you'll face going forward could be much higher or lower than the historical average.

A useful intuition: vol scales with $\sqrt{T}$, returns scale with $T$

Expected return scales linearly with time: $E[r_T] = T \cdot E[r_1]$. In words: the expected return over $T$ days is just $T$ times the expected one-day return — it stacks up in a straight line. Concrete: 0.03% per day × 252 days ≈ 7.6% per year.

Volatility scales with the square root: $\sigma_T = \sqrt{T} \cdot \sigma_1$ — the $T$-day sigma is the one-day sigma times root-$T$, as in the annualization rule above.

So as $T$ grows, the signal-to-noise ratio — expected return divided by volatility, i.e. how loud the signal is relative to the random wiggle — improves as $\sqrt{T}$:

$\frac{E[r_T]}{\sigma_T} = \sqrt{T} \cdot \frac{E[r_1]}{\sigma_1}$

In words: the $T$-day ratio of expected return to volatility equals the one-day ratio times the square root of $T$. The linear-growth numerator outruns the square-root-growth denominator.

This is why time itself is one of the biggest sources of edge in investing. Over a single day, returns are dominated by noise. Over 30 years, returns are dominated by expected return. The whole institutional-investing playbook (buy-and-hold equity, harvest the equity risk premium — the extra return stocks pay over safe assets — over decades) is just exploiting this fact.

It's also why intraday strategies are so much harder — your signal-to-noise is in the worst part of the curve.

Try it

Write the annualization yourself — this two-liner is one you'll type for the rest of the curriculum:

Explain it back

Without scrolling up, write 2 sentences: why is volatility easier to estimate than the mean, and why does this matter for trading?