Part V · Options · Chapter 5E

Volatility

From daily moves to implied volatility

Where we are

Every Greek in Chapters 5C and 5D took volatility as an input. This chapter defines volatility precisely and shows how it is measured from past prices and priced forward by the market. It also introduces the gap between the two, which option sellers are paid to bear.

What you need first

Log returns, the square-root-of-time rule and annualizing (Chapter 1A).
The standard deviation, the normal curve and fat tails (Chapter 1B).
The Black-Scholes price and its volatility input $\sigma$ (Chapter 5C).

1 · What volatility is

Volatility is the most important driver of option premium and the most misunderstood single concept in retail options trading, so it needs a precise definition. In finance, volatility is the annualized standard deviation of logarithmic returns. It is directionless: it measures the magnitude of price swings, not their direction.

Write $r_t = \ln(S_t/S_{t-1})$ for the log return of day $t$ , where $S_t$ is the day's closing price. If $\sigma_{\text{daily}}$ is the standard deviation of those daily log returns, the volatility is

\sigma = \sigma_{\text{daily}} \times \sqrt{252}

(5E.1)

In plain terms

A volatility of 20% says that a typical day's move is about 1.26% and a typical year's move about 20%, up or down. It does not say which way: two stocks with the same volatility can have opposite trends, one drifting up and one drifting down, because volatility measures only how widely returns spread around their average.

2 · Why log returns

Professionals compute and annualize volatility with log returns, not simple percentage returns. The critical property that makes log returns indispensable here is additivity, taught in Chapter 1A: a multi-period log return is the sum of the single-period log returns. Under independence, the variance of a sum is the sum of the variances, so the variance over $n$ days is $n$ times the daily variance:

\begin{aligned} \operatorname{Var}(r_1 + \cdots + r_n) &= n\,\sigma_{\text{daily}}^2 \\ \sigma_{n\text{-day}} &= \sigma_{\text{daily}}\sqrt{n} \end{aligned}

(5E.2)

This is the square-root-of-time rule that pervades the entire volatility framework. Nine days carry three times the standard deviation of one day, not nine times.

3 · Annualization conventions

The standard annualization for daily volatility in equity markets uses 252 trading days a year, with weekly and monthly figures scaled the same way:

data	multiply by	which is about
daily	$\sqrt{252}$	15.87
weekly	$\sqrt{52}$	7.21
monthly	$\sqrt{12}$	3.46

A standard deviation of daily log returns of 1.26% gives $1.26\% \times 15.87 = 20.0\%$ a year. A weekly standard deviation of 2.77% gives $2.77\% \times 7.21 = 20.0\%$ as well, so both describe the same volatility.

4 · The rule of 16

Since $\sqrt{252} \approx 15.87$ , traders use divide by 16, the rule of 16, as a mental shortcut to convert annual implied volatility into an expected daily move:

\text{daily move (1 SD)} \approx \frac{\text{annual IV}}{16}

(5E.3)

A stock with IV of 32% is pricing in roughly 2% daily one-sigma moves, since $32/16 = 2$ . A VIX of 80 is bracing for 5% daily moves ( $80/16 = 5$ ). A VIX of 16 is at parity with a 1% daily expected move, the namesake equality that makes the rule memorable. The VIX is itself an annualized implied volatility for the S&P 500 over the next 30 days, so the rule applies to it directly.

"One sigma" has a precise meaning: if daily log returns were normal, about two days in three would move less than one standard deviation, while about one in three would move more (Chapter 1B). Real returns have fat tails, so the biggest days come more often than the normal curve allows.

This single conversion lets you instantly judge whether option prices feel cheap or expensive relative to your forecast of the underlying's daily range. Suppose you think SPX is going to move 1.5% a day for the next month, but VIX is at 12. IV is then pricing in $12/16 = 0.75\%$ a day, less movement than you expect, which makes a long-vol setup. If VIX is at 24, pricing $1.5\%$ a day, while you think SPX is going to be quiet, that is a short-vol setup. The rule of 16 makes IV legible to discretionary traders without requiring a vol surface.

The figure turns an annual implied vol into the moves it prices. Its one slider sets the annual IV (5 to 100, starting at 20). Three bars give the one-sigma move over a day, a week and a month. Set it to 16, then to 32 and to 80, comparing the day bar with the examples above.

annual IV (e.g. VIX) 20%

1-day move (1σ) = 1.25%

1-week move (1σ) = 2.77%

1-month move (1σ) = 5.77%

Above rule-of-16 baseline. With σ_annual = 20%, the rule of 16 estimates the daily move at 20 / 16 = 1.25%.

Annual implied vol, read as the moves it prices. Set an annualized implied volatility such as the VIX; the bars are the one-sigma moves it implies over one day (IV / 16, the rule of 16), one week (IV / √52) and one month (IV / √12). The moves grow with the square root of time: a month of 21 trading days moves about 4.6 times as far as a day, not 21 times. At 16 the daily move is exactly 1%.

At the starting 20, the bars read 1.25% for a day, 2.77% for a week and 5.77% for a month, which the readout calls above the rule-of-16 baseline. At 16 the day bar reads exactly 1%, which the readout calls exact parity at VIX 16. The month bar is always about 4.6 times the day bar, because a month of 21 trading days scales by $\sqrt{21} \approx 4.6$ , not by 21. The figure divides by 16 for the day and by $\sqrt{52}$ and $\sqrt{12}$ for the week and the month.

5 · Measuring volatility: the estimators

Realized volatility is always an estimate, because you see only discrete prices and never the process that generated them. No one ever knows the true volatility. Different estimators use different subsets of the day's OHLC data (its open, high, low and close), and they have different statistical efficiency.

An estimator's efficiency compares the variance of its estimates with that of a baseline. When one estimator is five times as efficient as another, it reaches the same precision with a fifth of the data. Write $O_i$ , $H_i$ , $L_i$ and $C_i$ for day $i$ 's open, high, low and close, with $N$ for the number of days.

The close-to-close estimator (CC) uses only closing prices:

\sigma^2_{\text{CC}} = \frac{1}{N}\sum_{i=1}^{N} \Big[\ln \frac{C_i}{C_{i-1}}\Big]^2

(5E.4)

Simple as it is, it ignores the intraday range and the overnight gap as separate pieces of information. Being statistically inefficient, it needs many days of data to converge to a reliable estimate. The formula squares each return without subtracting the mean, because over short windows the daily mean is too small to matter.

The Parkinson estimator (1980) uses the daily high-low range:

\sigma^2_{\text{P}} = \frac{1}{4N\ln 2}\sum_{i=1}^{N} \Big[\ln \frac{H_i}{L_i}\Big]^2

(5E.5)

It is approximately five times as efficient as close-to-close, in the sense that it achieves the same precision with a fifth of the data. It ignores opens and closes, which leaves it biased downward by overnight gaps: the high-low (H-L) range does not capture jumps that happen between sessions. The factor $4\ln 2 = 2.77$ converts the average squared range into a variance.

The Garman-Klass estimator (1980) uses the full OHLC:

\begin{aligned} \sigma^2_{\text{GK}} = \frac{1}{N}\sum_{i=1}^{N} \Big[ &\tfrac12 \big(\ln \tfrac{H_i}{L_i}\big)^2 \\ &- (2\ln 2 - 1)\big(\ln \tfrac{C_i}{O_i}\big)^2 \Big] \end{aligned}

(5E.6)

It is approximately seven to eight times as efficient as close-to-close, although it still ignores overnight gaps. It is the industry standard for liquid intraday names where the high-low range carries most of the information. Its constant, $2\ln 2 - 1 = 0.386$ , sets how much the open-to-close move counts against the range.

The Yang-Zhang estimator (2000), which handles overnight gaps explicitly, has three components: an overnight component (from each close to the next open), an open-to-close component and a Rogers-Satchell component adjusted for drift. The most efficient and robust of the four under standard assumptions, it is the default for serious vol research. The Rogers-Satchell part, on its own, averages $\ln(H_i/C_i)\ln(H_i/O_i) + \ln(L_i/C_i)\ln(L_i/O_i)$ over the days, and Yang-Zhang weights the three parts to make the combined estimate as precise as possible.

Worked example: Three days, three estimators

Take a stock that closed at 100 and then traded for three days as follows.

day	open	high	low	close
1	100.5	102.0	99.5	101.0
2	101.0	101.5	99.0	99.5
3	99.0	100.5	98.5	100.0

Step 1: close to close. The three log returns, 0.0100, −0.0150 and 0.0050, have squares that average 0.000116, a daily standard deviation of 1.08%, or $1.08\% \times 15.87 = 17.1\%$ a year.

Step 2: Parkinson. The three log ranges, $\ln(102/99.5) = 0.0248$ , 0.0249 and 0.0201, have squares that sum to 0.001642, which divided by $4 \times 3 \times \ln 2 = 8.32$ gives 0.000197. That is a daily standard deviation of 1.41%, or 22.3% a year.

Step 3: Garman-Klass. With open-to-close log returns of 0.0050, −0.0150 and 0.0101, take half the squared range for each day and subtract 0.386 times the squared open-to-close return, then average: the result is 0.000229, a daily 1.51%, or 24.0% a year.

Step 4: read the gap. The closes moved little, while the days swung widely inside their ranges, so the close-to-close estimator saw 17% where the range estimators saw 22% to 24%. Three days are far too few for any estimate to be reliable. Even so, the range estimators use more of what each day showed. None of these three sees the half-point gap between day 2's close and day 3's open; Yang-Zhang would.

Familiar ground

Throwing away the intraday range is like dichotomizing a continuous endpoint. A trial that records only whether a tumor shrank by 30% needs more patients than one that analyzes the measured shrinkage, because the cut-off discards information each patient carried. The close-to-close estimator keeps one number from each day, while the range estimators keep more of it. That is why they need fewer days for the same precision.

6 · Implied volatility and realized volatility

Implied volatility (IV) is the market's forecast of future realized volatility, extracted by inverting an option pricing model on the observed premium. If the market is quoting a 30-day ATM SPX call at $35, the IV is the volatility input that, plugged into Black-Scholes, produces a price of $35. Where the model turns a volatility into a price, IV runs it backward, from the price to the volatility.

For an at-the-money (ATM) option with no interest rate, the inversion has a handy shortcut:

C \approx 0.4\, S\,\sigma\sqrt{T}

(5E.7)

Here $C$ is the call's price, $S$ is spot and $T$ is the time to expiry in years. The 0.4 is the height of the bell curve at its center, $\varphi(0) = 0.3989$ . With no interest rate the ATM put has the same price, so the shortcut holds for it too.

Where the ATM shortcut comes from

Step 1: the ATM call. With $K = S$ and $r = 0$ , the Black-Scholes quantities of Chapter 5C are $d_1 = \tfrac12\sigma\sqrt{T}$ and $d_2 = -\tfrac12\sigma\sqrt{T}$ , so the price is

C = S\,\big[N(d_1) - N(-d_1)\big]

(5E.8)

Step 2: read it as an area. $N(d_1) - N(-d_1)$ is the area under the bell curve between $-d_1$ and $+d_1$ , a thin strip of width $\sigma\sqrt{T}$ around the center.

Step 3: height times width. A thin strip's area is close to its height times its width. The height at the center is $\varphi(0) = 0.3989$ , so the area is about $0.3989\,\sigma\sqrt{T}$ , and $C \approx 0.3989\,S\sigma\sqrt{T}$ . For a 30-day option at 20% the strip is only 0.057 wide, narrow enough for the shortcut to be right to the fourth decimal place.

Worked example: Backing out an implied volatility

A 3-month ATM call on an underlying at $100, with no interest rate, trades at $4.50.

Step 1: try two volatilities. Black-Scholes prices the call at $3.99 at 20%, too low, and at $4.98 at 25%, too high. The IV lies between the two, where repeated halving of the interval pins it at 22.6%.

Step 2: use the shortcut. Rearranging (5E.7) gives $\sigma \approx C/(0.4\,S\sqrt{T}) = 4.50/(0.3989 \times 100 \times 0.5) = 0.226$ , the same 22.6% in one line.

Because different option prices imply different volatilities, IV is a function of strike and expiration: the implied volatility surface, the subject of the last part of this chapter and of Chapter 5G.

Realized volatility (RV) is what actually happened, computed after the fact with the estimators above. Implied volatility looks forward and is a price, while realized volatility looks back and is a measurement.

7 · The variance risk premium

The persistent empirical finding across decades and markets is that IV is, on average, higher than subsequent RV. This is the variance risk premium (VRP) first met in Chapter 4C. The premium exists for three reasons, two about buyers and one about sellers:

Institutional hedgers, such as pension funds, endowments and structured-product issuers, systematically overpay for downside protection, because their mandates require them to remain hedged regardless of price.
Retail traders systematically overpay for upside speculation: lottery-ticket calls on meme stocks, and OTM SPY calls on event days.
Option sellers must be compensated for the convex tail risk they are warehousing.

Bondarenko's analysis of the CBOE PUT Index quantifies the VRP. Over the period 1990 to 2018, the average difference between VIX (implied) and the subsequent 1-month realized vol on SPX was approximately +4.2 vol points. That is the size of the VRP on average: implied has been about 4.2 percentage points higher than what subsequently realized.

The VRP is the most important source of edge available to retail options traders. It is the structural reason that systematic short-premium strategies, such as the PUT Index, ATM put writing and iron condor writing, have historically produced positive returns over multi-decade horizons. Part XIII returns to this, and Chapter 10C treats the VRP as an edge in its own right. Chapter 5C showed the mechanism: a short option position earns only if realized volatility comes in below the implied volatility at which it was sold.

The two traditions this book draws on disagree about one item in that list, and both positions stand. The dealer-mechanics tradition of this part counts iron condor writing among the short-premium strategies with positive long-run returns. The edge-by-edge case of Part X rejects systematic iron condors for retail accounts. There, an index of mechanical condors has lagged the S&P 500 by a wide margin, because a few wing breaks a year erase the premium. Chapter 10I gives that evidence, and Chapter 12C draws the condor index against SPX.

8 · The volatility surface

IV is not one number: it varies across strikes, giving the smile or skew, and across expirations, giving the term structure.

For equities (SPX, SPY, individual stocks), the IV curve across strikes typically slopes downward from low strikes, with high IV, to high strikes, with lower IV. This is the "equity skew" or "smirk": OTM puts trade at higher IV than OTM calls. The economic intuition is that crash insurance is more in demand than upside speculation in equity markets, so the put wing is structurally bid. For commodities and FX, the skew can have different shapes, depending on the structural demand profile of hedgers in each market. It can be a smile, with both wings up, or a reverse skew, with calls higher than puts.

The IV curve across expirations is usually in contango for equity markets: long-dated IV is above short-dated IV, reflecting more uncertainty about events further out. In stress regimes it can invert into backwardation, with short-dated IV above long-dated IV, because panic concentrates demand into near-dated puts. The VIX/VIX3M ratio is the canonical proxy for the shape of the SPX term structure, with values above 1.00 indicating backwardation and values below 1.00 indicating contango.

Worked example: A small surface, calm and stressed

The table gives hypothetical implied vols for an index at 100.

expiry	strike 90	strike 100	strike 110
30 days, calm	24%	18%	15%
90 days, calm	25%	20%	17%
30 days, stressed	48%	40%	34%
90 days, stressed	38%	32%	28%

Step 1: the skew. In every row the 90-strike put wing carries more IV than the 110-strike call wing, 24% against 15% in the calm 30-day row, which is the equity smirk.

Step 2: the term structure. At the 100 strike, the calm ratio of 30-day to 90-day IV is $18/20 = 0.90$ , below 1.00, which is contango. Under stress the ratio is $40/32 = 1.25$ , above 1.00, which is backwardation: the front has repriced more than the back.

The vol surface is what professionals actually trade. A pure "long vol" position is rare, while most options trades implicitly take a view on the surface. Examples are long upside skew (long calls relative to puts), long downside skew (long puts relative to calls) and long forward vol, which holds longer-dated options against shorter-dated ones. Chapter 5G makes the surface operational, and Part IX returns to it in detail with the VIX and 0DTE options.

9 · Skew as a leading indicator

A subtle and important pattern is that the SPX put-wing skew, how much higher OTM put IV is than ATM IV, often leads VIX itself. Periods of skew steepening, even with VIX flat, suggest that institutional positioning is becoming more defensive. Periods of skew flattening, often during low-volatility regimes when no one is buying puts because nothing is happening, precede subsequent vol spikes more often than chance would suggest.

The CBOE SKEW index, published since 1990, formalizes this into a single number based on the 30-day SPX skew curve. SKEW values above 130 indicate steep skew, meaning heavy demand for crash protection. SKEW above 145 has historically preceded mean-reverting vol spikes, and SKEW below 115 suggests complacency. The SKEW index is a useful regime indicator that, combined with the VIX level, captures more about market state than either does alone. Chapter 5G turns skew into a trading check.

In practice

Before any options trade, put the IV into daily terms with the rule of 16. Then set it beside a realized-vol estimate of your own over a comparable window. A range-based estimator gives a steadier reading than closes alone from the same number of days. If the gap between the two is small, the trade has no volatility edge, whatever its direction.

When you sell premium you are selling IV, which pays only if realized volatility comes in below it (Chapter 5C). The VRP says that happens on average, not every month. Size a short-premium position for the months in which it does not (Chapter 1E).

When it fails

The rule of 16 and the square-root-of-time rule assume independent days with a constant variance. When volatility clusters, as it does in stress, a quiet week says little about the next one, and $\sigma\sqrt{n}$ misstates multi-day risk. Fat tails make large days more common than a one-sigma band suggests.

The range estimators assume the price moves continuously within the day. Overnight gaps bias Parkinson and Garman-Klass downward, which matters most around earnings and weekends. The VRP is an average over decades, which means that in the months when realized volatility jumps above implied, sellers lose. The skew signals are weaker still, since "more often than chance would suggest" says nothing about how often a high reading is followed by nothing. A signal's hit rate also depends on how common vol spikes are in the first place.

Decision rule

Compare before you act: buy options when your forecast of the daily move clearly exceeds IV / 16, or sell them when it clearly falls short. Base the forecast on a realized-vol estimate, not on a feeling.
No gap, no trade: when IV / 16 and your forecast are close, the option is fairly priced for your view, which leaves no volatility reason to trade it.
Read the surface before the level: check skew and the VIX/VIX3M ratio before a trade that depends on the IV level alone, using the table of structures in Chapter 5G.

Common mistake: comparing the average move with IV / 16

IV / 16 is a standard deviation, which is larger than the average absolute daily move. For a normal curve it is about 0.8 of the standard deviation. If SPX has moved 1% a day on average in absolute terms, its daily standard deviation is nearer 1.25%, which corresponds to an IV of 20, not 16. Comparing the average move directly with IV / 16 makes options look more expensive than they are.

Common mistake: treating IV above RV as a mispricing

IV above subsequent RV is the normal state of the market, not an error to exploit. The gap is the price of insurance, paid for bearing crash risk. A short-premium trade collects it only by accepting that risk. It has an edge only if the premium more than pays for the losses in the bad months.

What to carry forward

Volatility is the annualized standard deviation of log returns: the daily figure times $\sqrt{252} = 15.87$ . It measures the size of moves, not their direction.
The rule of 16 turns annual IV into a daily one-sigma move, so a VIX of 16 prices a 1% day. Moves grow with the square root of time.
Because realized volatility can only be estimated, the estimator matters. Range-based estimators (Parkinson, Garman-Klass, Yang-Zhang) reach a given precision with far fewer days than close-to-close, and Yang-Zhang also handles overnight gaps.
IV is the volatility that reproduces an option's market price. On average it has exceeded subsequent RV, by about 4.2 vol points on SPX over 1990 to 2018: the variance risk premium.
IV varies across strikes (equity skew: puts above calls) and across expiries, in contango in calm markets and in backwardation under stress. The whole surface carries information.

10 · Check yourself

Question 1

What one-sigma moves does an implied volatility of 40% price over a day, a week and a month?

Answer

The day's move is $40/16 = 2.5\%$ , the week's is $40/\sqrt{52} = 40/7.21 = 5.55\%$ , and the month's is $40/\sqrt{12} = 40/3.46 = 11.5\%$ . The month is about 4.6 times the day, not 21 times, because moves grow with the square root of time.

Question 2

What is the annualized volatility of a stock whose daily log returns have a standard deviation of 0.9%?

Answer

By (5E.1), the annual figure is $0.9\% \times \sqrt{252} = 0.9\% \times 15.87 = 14.3\%$ .

Question 3

Why does the Parkinson estimator need fewer days than close-to-close for the same precision, and when is it biased?

Answer

Each day's high-low range carries more information about how far the price wandered than the single close-to-close change does, which makes Parkinson about five times as efficient. It is biased downward when prices jump between sessions, because the day's range does not include the overnight gap.

Question 4

Estimate the implied volatility of a 3-month ATM call on a $50 stock that costs $2.00, with no interest rate.

Answer

The ATM shortcut gives $\sigma \approx C/(0.4\,S\sqrt{T}) = 2.00/(0.3989 \times 50 \times 0.5) = 0.201$ , about 20%, which Black-Scholes confirms: at 20.06% it prices the call at $2.00 exactly.

Question 5

How would you describe the term structure when VIX is 15 and VIX3M is 17, and again a month later, when VIX is 30 and VIX3M is 25?

Answer

The first ratio is $15/17 = 0.88$ , below 1.00, so the term structure is in contango, the usual calm shape. The second is $30/25 = 1.20$ , above 1.00, which is backwardation: near-dated protection has been bid up more than longer-dated, as happens in stress.