Realized vs Implied Volatility

You already speak options. You know a call from a put, you can recite the Greeks in your sleep, you delta-hedge without flinching, and you accept that returns are basically a random walk with attitude. So here’s the uncomfortable question that separates option users from volatility traders: when someone quotes you “the vol is 20,” which vol do they mean? Because the single word “volatility” is hiding two completely different numbers — one that already happened and one that hasn’t happened yet — and the entire business of volatility trading lives in the gap between them. This lesson nails down both, shows you the arithmetic, and tees up the trade.

Two volatilities, one number that isn’t enough

Analogy. Picture driving a long road trip. Realized volatility is the speedometer reading you wrote down every minute and averaged afterward — it’s a record of how erratically you actually drove. Implied volatility is the speed limit the highway sign is quoting for the stretch ahead — a forward statement about how fast things are expected to get. They’re both “speed,” both in the same units, and yet they answer opposite-facing questions: one looks in the rearview mirror, one looks through the windshield. Confusing them is how you crash.

The definitions.

• Realized volatility (RV), also called historical or actual volatility, is computed from the asset’s own past returns. It is purely backward-looking and entirely model-free: you take what happened and measure its dispersion. No option prices needed.
• Implied volatility (IV) is extracted from option prices. It is the volatility number that, fed into an option-pricing model, reproduces the price the market is actually charging. It is forward-looking and is, in effect, the market’s forecast of realized volatility over the option’s remaining life.

Both are conventionally quoted as an annualized standard deviation of returns, in percent — so a “20 vol” stock and a “20” implied both mean roughly a one-standard-deviation move of 20% over a year. Because they share units, you can put them side by side and subtract, which is exactly what a vol trader does all day.

Volatility — realized or implied — is fundamentally about the spread of outcomes, not the direction. A stock that drifts steadily up 1% a day is barely volatile; one that lurches ±5% in random directions is extremely volatile even if it ends flat. The animation below makes that visceral: identical drift, different vols, wildly different fans of where the price could land.

Measuring realized volatility

The recipe. Realized volatility is just a standard deviation, dressed up with two conventions: we measure it on log returns, and we annualize it. Step by step:

1. Take a series of prices $P_0, P_1, \dots, P_n$ (say, daily closes).
2. Compute each period’s log return $r_t = \ln(P_t / P_{t-1})$. Logs are used because they’re additive across time and symmetric (a $+10\%$ then $-10\%$ round-trip behaves sensibly), which is exactly the well-behaved-noise assumption underneath option models.
3. Take the standard deviation of those returns — that’s the per-period (e.g. daily) volatility:

$\sigma_{\text{daily}} = \sqrt{\frac{1}{n-1}\sum_{t=1}^{n}\left(r_t - \bar{r}\right)^2}$

4. Annualize by scaling up to a one-year horizon:

$\sigma_{\text{annual}} = \sigma_{\text{daily}}\,\sqrt{252}$

Why √time and not time? Because for independent returns, variance (vol squared) is what adds up linearly over time, not volatility itself. Stack $T$ independent days and the variances sum: $\sigma_T^2 = T\,\sigma_{\text{daily}}^2$. Take the square root of both sides and you get $\sigma_T = \sigma_{\text{daily}}\sqrt{T}$. So volatility grows with the square root of the horizon. This is the single most-used identity in the whole field — the “square-root-of-time rule.”

Why 252? That’s the approximate number of trading days in a year (markets are shut on weekends and holidays). Volatility is generated by trading activity, so we count trading days, not the 365 calendar days. Some desks use 250 or 260; the difference is small. $\sqrt{252} \approx 15.87$.

Worked example. Suppose you’ve measured a stock’s daily log-return standard deviation at $1.25\%$. Annualize it:

$\sigma_{\text{annual}} = 1.25\% \times \sqrt{252} \approx 1.25\% \times 15.87 \approx 19.8\%$

So this stock has a realized volatility of about $19.8\%$ — a one-standard-deviation annual move of roughly a fifth of its price. Run it the other way to sanity-check IV quotes: a stock the options market is pricing at $32\%$ implied is being forecast to print a daily standard deviation of $32\% / 15.87 \approx 2.0\%$ per day. That little division is how traders translate an annualized quote into “how much should this thing move tomorrow.”

Notice the round trip: annualize by ×15.87, de-annualize by ÷15.87. The two columns of daily moves match the inputs exactly, which is the point — annualizing and de-annualizing are inverses.

The chart below shows exactly that time-varying, clustering behavior — realized vol is a process, not a constant, and it tends to mean-revert (spikes calm down, lulls eventually wake up). This is precisely why a single backward number is a shaky forecast, and why the market needs a forward one.

Beyond close-to-close. The simplest RV estimator uses only closing prices (close-to-close), which is easy but throws away the day’s range. Smarter estimators squeeze more information from each bar: the Parkinson estimator uses the high–low range, and the Garman–Klass estimator uses open, high, low, and close together — both are far more efficient (lower estimation error for the same sample) than close-to-close because intraday extremes carry real signal about how much the asset actually swung. The trade-off: range-based estimators can be biased by overnight gaps and microstructure noise, so close-to-close remains the honest default for a first pass.

Implied volatility: the market’s forecast, backed out of prices

The core idea. Black–Scholes (and every option model) takes a handful of observable inputs — spot price, strike, time to expiry, the risk-free rate — plus one input you can’t observe directly: the future volatility $\sigma$. Feed in a $\sigma$ and the model spits out a price. Implied volatility runs that machine backward: you take the market price of the option as given and solve for the $\sigma$ that makes the model reproduce it.

$\text{Model Price}(\sigma_{\text{implied}}) = \text{Observed Market Price}$

There’s no closed-form solution for $\sigma_{\text{implied}}$ — you find it numerically (iterate until the model price matches). But conceptually it’s clean: implied vol is the volatility the option’s price implies. It is the market’s collective, risk-neutral forecast of how volatile the underlying will be over the option’s remaining life. Buy a 30-day option and its IV is the market’s vol forecast for the next 30 days.

Vega is the bridge. How do you turn a price into a vol, or a vol view into a price? Through vega — the option’s sensitivity to volatility:

$\nu = \frac{\partial(\text{Option Price})}{\partial \sigma}$

Vega tells you how many dollars the option’s price moves per one-percentage-point change in implied vol. It’s strictly positive for both calls and puts (more vol = more chance of a big favorable move = more valuable optionality), largest for at-the-money options with plenty of time left, and it shrinks toward expiry. Vega is the reason IV is recoverable: because price rises monotonically with σ, there’s exactly one σ that fits any given price, and vega is the slope of that price-vs-σ curve you climb to find it.

Worked intuition. Take an at-the-money option with a vega of $0.18 (i.e. the price changes about $0.18 per 1 vol point) currently priced at $3.00, consistent with an implied vol of $20\%$. Now suppose demand surges and the option’s market price jumps to $3.72. The price rose by $0.72; at $0.18 per vol point, that’s about $0.72 / 0.18 = 4$ extra vol points. So the implied vol just repriced from $20\%$ up to roughly $24\%$ — without the stock itself moving at all. Nothing happened to realized vol; the market simply raised its forecast and paid up for the options.

That linear-via-vega approximation only holds for small moves (vega itself changes as σ moves), but it captures the mechanism: price and implied vol move in lockstep, and vega is the exchange rate between them. And crucially, IV comes out annualized in percent — the same units as the realized vol you computed above — which is what lets you finally compare the two.

The volatility smile and surface

Black–Scholes pretends a single $\sigma$ describes the underlying. The market disagrees, and it tells you so through the prices it sets. When you back out implied vol from every option on the same underlying, you don’t get one number — you get a whole landscape.

• Across strikes (the smile / skew). Plot implied vol against strike for a fixed expiry and it isn’t flat. In FX it often curves up at both wings — the classic smile. In equity indices it slopes down — out-of-the-money puts carry markedly higher IV than out-of-the-money calls. This is the skew (or “smirk”), and it exists because the market fears a sharp crash more than a sharp rally, so it pays up for downside puts. That extra demand bids their prices — and therefore their implied vols — higher.
• Across maturities (the term structure). Plot ATM implied vol against time to expiry and you get the term structure of volatility — usually upward-sloping in calm markets (more uncertainty further out) and inverted (short-dated IV above long-dated) during a panic, when near-term fear spikes.
• Both at once (the surface). Stack the smile across every expiry and you get the implied volatility surface — IV as a function of both strike and maturity, a 3-D sheet that the whole options market is constantly repricing.

The chart below shows the strike dimension — the smile/skew — for a single expiry. We’re just previewing it here; later lessons in this course dissect the skew, the term structure, and how to trade the surface’s shape directly.

The gap is the trade

Here’s the payoff for all that setup. Realized and implied are the same units measuring the same thing (volatility), but one is a fact and the other is a forecast. Subtract them and you get the number vol traders actually bet on:

$\text{The vol-trader's edge} = \sigma_{\text{implied}} - \sigma_{\text{realized}}$

• If you think realized will come in below implied — the stock will be calmer than the options are pricing — then options are expensive relative to what they’ll deliver. You SELL volatility (short a straddle, sell variance) and pocket the difference.
• If you think realized will exceed implied — the stock will be wilder than priced — options are cheap. You BUY volatility (long a straddle, buy variance).

But raw options carry direction (delta) you don’t want — you have a view on how much it moves, not which way. This is where your delta-hedging muscle pays off: delta-hedging continuously strips out the directional exposure, leaving a position whose P&L depends almost entirely on the difference between realized and implied vol. A delta-hedged short option bleeds if the stock thrashes around (realized high) and earns its time-decay premium if the stock sits still (realized low). That’s the cleanest expression of the bet there is — an option, hedged, is a pure wager on $\sigma_{\text{implied}} - \sigma_{\text{realized}}$.

Putting it together

“Volatility” is two numbers wearing one name. Realized volatility is the annualized standard deviation of an asset’s past log returns — multiply the daily figure by $\sqrt{252}$ because variance, not vol, scales with time — and it’s a moving, clustering, mean-reverting process, not a constant. Implied volatility is the $\sigma$ you back out of an option’s market price; it’s the market’s forward, risk-neutral forecast of vol over the option’s life, recovered through vega, the price–vol exchange rate. Neither is a single number: across strikes you get the smile/skew, across maturities the term structure, and together the implied-vol surface. And the punchline — the thing that makes this a trade — is the gap: sell vol when you expect realized to undershoot implied, buy it when you expect the reverse, and delta-hedge to turn a messy option into a clean wager on $\sigma_{\text{implied}} - \sigma_{\text{realized}}$. Implied usually wins that bet by a margin (the volatility risk premium), which is the engine the rest of this course takes apart.

Next — we put a real instrument on this gap: the variance swap, which lets you trade realized variance against a strike with no delta-hedging messiness at all, and the VIX, the market’s headline number for 30-day implied vol.