Implied against realised, and the surface

Volatility comes in two kinds: what the underlying actually did, and what the market is charging for what it might do. An options desk is in the business of the gap between them, and almost every volatility question reduces to which of the two is being asked about.

Aspect	Realised	Implied
What it is	What the underlying did	What the option price says it might do
Where it comes from	A time series of returns	Inverting an observed premium
Looks	Backwards	Forwards
Traded through	Delta-hedging an option	Buying or selling the option itself

Table 1.1 · The two volatilities.

Equation 1.2

Measuring realised volatility

Close-to-close volatility, annualised. Variance scales with time, so volatility scales with its square root.

\sigma_{\text{ann}} = \sqrt{\frac{252}{n}\sum_{i=1}^{n} r_i^2}, \qquad r_i = \ln\frac{S_i}{S_{i-1}}

$252$: Trading days in a year — the convention, not the calendar count.
$r_i$: Log returns, which add across periods in a way simple returns do not.

Proposition 1.3

The square-root-of-time rule

To move volatility between horizons, scale by the square root of the time ratio: a $16\%$ annual volatility is about $1\%$ a day, because $\sqrt{252} \approx 16$ . That single fact converts between the two units traders quote in, and it is worth being able to do in both directions instantly.

Holds when

It assumes independent returns. Trending or mean-reverting series break it, which is what a variance ratio test measures.
Divide annual volatility by 16 for a daily move; multiply a daily move by 16 for the annual figure.

Example 1.4

A name has $32\%$ annualised implied volatility. Roughly how large is a one-standard-deviation daily move, and what does the market expect over the next week?

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Worked solution

Formula $\sigma_{\text{daily}} = \frac{\sigma_{\text{ann}}}{\sqrt{252}}, \qquad \sigma_{\text{period}} = \sigma_{\text{daily}}\sqrt{d}$
Substitute $= \frac{32\%}{16}, \qquad d = 5$
Solve $\sigma_{\text{daily}} = 2\%$
$\sigma_{\text{week}} = 2\% \times \sqrt{5} \approx 4.5\%$
Answer $\text{about } 2\% \text{ a day and } 4.5\% \text{ over a week}$

Sanity check. Five days is not five times a daily move — it is $\sqrt{5} \approx 2.24$ times it.

Proposition 1.5

What a delta-hedged option is long

Hedge the delta of a long option and your P&L is gamma earning the realised variance, minus theta paying for the implied variance you bought. So the position is long realised volatility and short implied — it makes money precisely when the underlying moves more than the option price assumed.

Holds when

This is why "buying vol" and "buying gamma" describe the same trade from two angles.
Implied has historically exceeded realised on index options, which is the variance risk premium.

Figure 1.6 · One expiry can imply a different volatility at every strike. Illustrative prices, not a live market: lower-strike puts can carry higher implied volatility than at-the-money options. A constant-volatility model would draw a flat line, so this shape is a direct view of where that assumption fails.

Implied against realised, and the surface

After this lesson you should be able to

Measuring realised volatility

The square-root-of-time rule

What a delta-hedged option is long

The rest of this lesson is in Premium