Realized and Implied Volatility

Prerequisites

• joint-distributions-and-independence — formal definition of statistical independence
• [[confidence-levels-in-risk#The $\sqrt{T}$ Scaling Rule]] — derivation of variance scaling under iid assumptions

Training-data note

This note was written from Claude’s training data. The volatility measurement formulas are standard textbook material (Hull Ch. 15). The volatility risk premium explanation is directionally correct but simplified — it omits tail risk, VRP inversion during crises, and the distinction between equity and crypto VRP. Cross-check with Gatheral’s The Volatility Surface for depth.

Why this note exists

Volatility appears throughout derivatives pricing — it’s the key input to Black-Scholes, the driver of option premiums, and the variable that distinguishes risky assets from safe ones. This note defines volatility precisely, shows how to measure it, and explains the realized/implied distinction.

Notation

• $\sigma$ — annualized volatility (the central quantity of this note)
• $\hat{\sigma }$ — estimated (sample) volatility
• $P_i$ — closing price on day $i$
• $r_i$ — log-return on day $i$: $\ln (P_i/P_{i-1})$
• $N$ — number of observations (trading days)

Volatility in One Sentence

Volatility measures how much an asset’s price moves over time. High volatility means large swings; low volatility means the price is stable.

More precisely, volatility ($\sigma$) is the annualized standard deviation of log-returns. “Standard deviation” means: typical distance of observations from the average. “Log-returns” means: the natural logarithm of price ratios. “Annualized” means: scaled to a one-year horizon.

Realized Volatility: Measuring What Already Happened

Realized volatility (also called historical volatility) is computed from actual past prices. It answers: “how much did this asset actually move over the last $N$ days?”

The calculation

Given daily closing prices $P_0,P_1,\dots ,P_N$:

Step 1 — Compute log-returns. The log-return on day $i$ is:

$$r_i=\ln \left(\frac{P_i}{P_{i-1}}\right)$$

Why logarithms? Because log-returns are additive (a 2-day return is $r_1+r_2$, not $r_1\times r_2$) and symmetric (a +10% log-return followed by a -10% log-return gets you back to the starting price, which is not true for simple percentage returns). These properties make the math cleaner and connect directly to continuous-time finance: the Black-Scholes model assumes log-returns are normally distributed.

Step 2 — Compute the sample standard deviation.

$${\hat{\sigma }}_{\text{daily}}=\sqrt{\frac{1}{N-1}\sum _{i=1}^N(r_i-\bar{r}{)}^2}$$

where $\bar{r}=\frac{1}{N}{\sum }_{i=1}^N{r}_i$ is the mean log-return. In practice, for short windows the mean is close to zero and often dropped.

Step 3 — Annualize. Under the standard simplification, daily returns are assumed to be independent and identically distributed (iid) — each day’s return is drawn from the same distribution and is unaffected by previous days’ returns. “Independent” means knowing yesterday’s return tells you nothing about today’s. “Identically distributed” means the distribution doesn’t change over time (same mean, same variance every day). This is a strong assumption — volatility clustering violates it directly — but it is the standard convention because it gives us a clean scaling rule.

Why $\sqrt{365}$ and not just $365$? The key property is that variance is additive for independent variables: if each day’s return has variance ${\sigma }_d^2$, then 365 independent days have total variance $365\cdot {\sigma }_d^2$. Volatility is the standard deviation (the square root of variance), so:

$${\sigma }_{\text{annual}}=\sqrt{365\cdot {\sigma }_d^2}={\sigma }_d\cdot \sqrt{365}$$

The $\sqrt{365}$ comes from this two-step process: variance scales linearly with time (because of independence), then you take the square root to get back to standard deviation units. For the full derivation and its connection to VaR scaling, see [[confidence-levels-in-risk#The $\sqrt{T}$ Scaling Rule]].

For traditional equities, the convention is $\sqrt{252}$ (trading days per year). Crypto uses $\sqrt{365}$ because trading is 24/7/365.

Worked example

ETH daily closing prices for 5 days: $2,000, $2,060, $1,980, $2,040, $2,100.

Day $i$	$P_i$	$r_i=\ln (P_i/P_{i-1})$
1	$2,060	$\ln (2060/2000)=0.0296$
2	$1,980	$\ln (1980/2060)=-0.0396$
3	$2,040	$\ln (2040/1980)=0.0299$
4	$2,100	$\ln (2100/2040)=0.0290$

Mean $\bar{r}=(0.0296-0.0396+0.0299+0.0290)/4=0.0122$.

Sample variance:

$$\begin{array}{rl}{s}^2& =\frac{1}{3}[(0.0296-0.0122{)}^2+(-0.0396-0.0122{)}^2\\ & +(0.0299-0.0122{)}^2+(0.0290-0.0122{)}^2]\\ & =\frac{1}{3}(0.000301+0.002684+0.000312+0.000282)\\ & =\frac{0.003579}{3}=0.001193\end{array}$$

Standard deviation ${\hat{\sigma }}_{\text{daily}}=\sqrt{0.001193}=0.0345$.

$${\hat{\sigma }}_{\text{annual}}=0.0345\times \sqrt{365}\approx 0.660=66.0\%$$

This is typical for ETH — crypto assets commonly have annualized volatility in the 60–120% range. For comparison, the S&P 500 index typically has realized vol around 15–20%.

What volatility numbers mean in practice

Asset class	Typical realized vol	Context
S&P 500	15–20%	Low by crypto standards
Individual tech stocks	30–50%	Moderate
ETH	60–100%	High; normal for major crypto
Memecoins (Pump.fun)	200–500%+	Extreme; most of the price history is a spike followed by a crash

These numbers matter for derivatives pricing: an option pricing model needs $\sigma$ as input. Using 80% when the true vol is 60% will massively overprice options. Using 20% for a memecoin will massively underprice them.

Implied Volatility: The Market’s Forecast

Implied volatility (${\sigma }_{\text{imp}}$) is not measured from past prices. It is extracted from option prices — it is the volatility value that, when plugged into an option pricing model (typically Black-Scholes — the standard closed-form model for pricing European options, published by Fischer Black and Myron Scholes in 1973), produces the option’s current market price.

Think of it this way:

1. The Black-Scholes formula takes several inputs (stock price, strike, time to expiry, interest rate, and volatility $\sigma$) and outputs an option price $V$.
2. In the real market, you can observe the option price $V_{\text{market}}$.
3. Implied volatility is the $\sigma$ that makes Black-Scholes match the market: solve $\text{BS}(S,K,T,r,{\sigma }_{\text{imp}})=V_{\text{market}}$.

It is the market’s consensus forecast of how much the underlying will move between now and expiration. If traders expect big moves (earnings announcement, macro event, Merge upgrade), they bid up option prices, which pushes up implied vol. If they expect calm, implied vol drops.

Why implied > realized (usually)

Empirically, implied volatility tends to be higher than subsequently realized volatility. This gap is called the volatility risk premium (VRP).

$$\text{VRP}={\sigma }_{\text{implied}}-{\sigma }_{\text{realized}}>0\text{(on average)}$$

Why? Option buyers pay a premium for insurance — protection against large moves. That insurance has a cost, just like car insurance costs more than your expected accident losses. Option sellers (who bear the risk) collect this premium.

This is directly relevant to option sellers: anyone who sells options (writing calls or puts) is collecting a premium that embeds the market’s volatility forecast. If realized vol comes in below implied vol, the seller profits. The VRP is why systematic option-selling strategies have historically earned positive returns.

Measuring Volatility: Practical Considerations

Rolling windows

Realized vol is computed over a rolling window — the last 7 days, 30 days, or 90 days. Short windows capture recent conditions but are noisy. Long windows are stable but may miss regime changes (a calm market turning volatile).

Volatility clustering

Volatility is autocorrelated: high-vol days tend to follow high-vol days, and calm periods tend to persist. This is called volatility clustering and is one of the most robust empirical facts in finance (first documented by Benoit Mandelbrot in the 1960s). It means:

• If the last week was calm, next week is probably calm too (good for option sellers).
• If the last week was wild, next week is probably wild too (bad for option sellers).
• A simple 30-day rolling average understates future vol during regime transitions.

Intraday vs close-to-close

The standard formula uses daily closing prices. But much of the price action happens intraday. More sophisticated measures like the Garman-Klass or Yang-Zhang estimators use open/high/low/close data and are more efficient (give better estimates from fewer data points). Garman-Klass uses high-low-open-close prices (not just close-to-close) to capture intraday price range, extracting more information from each trading day. Yang-Zhang combines overnight returns (close-to-open) and intraday returns (open-to-close) into a minimum-variance estimator — it is unbiased and handles overnight gaps that Garman-Klass ignores. For crypto, where there is no “close” (markets run 24/7), the convention is to sample at midnight UTC.

---

Companion notebook: notebook — compute realized vol from actual ETH price history; visualize rolling vol windows; compare realized vs implied vol across asset classes.

Questions to sit with:

1. If implied vol is 30% and 30-day realized vol is 15%, should you sell options? What additional information would change your answer?
2. Volatility clustering means high-vol days follow high-vol days. How does this affect the reliability of a 30-day rolling vol estimate as a forecast for the next 30 days?
3. The VRP says implied > realized on average. But averages include calm periods. How would you construct a strategy that captures the VRP while limiting tail losses?

See also

• options-basics — calls, puts, and straddles
• the-greeks — delta, gamma, and vega
• black-scholes — the standard option pricing model that takes $\sigma$ as input