Volatility — the standard deviation of returns

What volatility is

In finance, volatility is the standard deviation of returns. Almost always annualized. Some flavors you'll encounter:

• Historical (realized) volatility — computed from past returns. The most common form.
• Implied volatility — what's priced into options. Forward-looking.
• Conditional volatility — current vol estimated from a model like GARCH.

For Phase 1 we mostly mean historical.

The sample standard deviation of returns is:

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

The $N-1$ (Bessel's correction) instead of $N$ accounts for the fact that we used the sample mean to compute deviations. It makes the sample variance unbiased (the standard deviation itself remains slightly biased, but the correction helps).

Annualization

If daily returns are approximately independent with daily standard deviation $\sigma_d$, then:

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

where $T$ is 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: 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, while the volatility estimate is measuring the noise term itself.

Concretely, for (approximately) normal returns:

• Standard error of the mean: $\sigma / \sqrt{N}$
• Standard error of the standard deviation: roughly $\sigma / \sqrt{2N}$

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 make variance estimates noisier than the normal-theory formula suggests. The asymmetry survives, but it's the σ ≫ μ scale gap doing the work, 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 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 — GARCH, EWMA weighting recent observations more heavily — capture this. 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: $E[r_T] = T \cdot E[r_1]$.

Volatility scales with the square root: $\sigma_T = \sqrt{T} \cdot \sigma_1$.

So as $T$ grows, the signal-to-noise ratio improves as $\sqrt{T}$:

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

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 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?