Volatility

OpeningOne 32%, three conclusions

Stock X trades at $100, and the implied volatility (IV) of its 30-day ATM options is 32%. The ATM straddle, buying the 100 call and the 100 put together, costs $7.31. Three students look at the same quote and reach three conclusions.

• A: over the past 20 trading days, X’s realized volatility was only 22%. Priced at 22%, the same straddle is worth only $5.03. The options are 45% too expensive, so sell.
• B: over the past year, X’s IV ranged between 25% and 60%. At 32% it sits only 20% of the way up that range, so the options are cheap. Buy.
• C: X reports earnings in 12 days. Without earnings, 30-day IV would be about 26%. The extra amounts to the market pricing a one-standard-deviation move of about ±5.3% on earnings day. The real question is: how big do you think this earnings move will be?

All three calculations are correct, yet the conclusions contradict each other. That is because an IV number by itself doesn’t tell you cheap or expensive; it only means something once you place it on a set of axes. This chapter builds those axes in turn: against realized volatility (6.1–6.3), against its own history (6.4), against other terms (6.5, 6.8) and against other strikes (6.6). Finally, it uses VIX (6.7) to apply these readings to the whole market.

6.1How to calculate realized volatility

Realized volatility (also called actual or historical volatility, written RV) is the standard deviation of log returns, scaled up to a year by the square root of time.

Working it by hand

Six consecutive closing prices: 100, 101.5, 100.8, 102.2, 101.0, 101.9. The five log returns are +1.489%, −0.692%, +1.379%, −1.181% and +0.887%, averaging +0.376%. The sample standard deviation is 1.232%; multiply by √252 ≈ 15.87 to get an annualized realized volatility of 19.56%.

Many platforms and traders assume a mean of zero and use √(Σr²/n) directly. For this data that gives a daily volatility of 1.164%, or 18.49% annualized. Over a short window the mean is mostly noise, so assuming zero is actually more stable; the two methods diverge noticeably only when there is a strong trend. When comparing RV from different sources, first check which method each one uses.

Why multiply by the square root

If daily returns are independent, the variance of the cumulative return over n days is n times the one-day variance, and the standard deviation is √n times. So to annualize you multiply by √252, for a week by √5, and for a month (about 21 trading days) by √21. The assumption breaks down in two cases: when prices show a clear trend or mean reversion (returns are correlated from one day to the next), and when there are gaps (one day’s variance far exceeds the others’). When it breaks down, square-root scaling understates or overstates volatility over longer periods.

The “rule of 16” traders talk about comes from the second row: 16% annualized is about 1% a day. Conversely, a stock that typically moves about 2% a day has an annualized volatility of around 32%.

Window length: speed vs. noise

RV computed over 10 days reacts quickly but has few data points and a lot of noise; RV over 60 days is stable but always a step behind. You can estimate how big the noise is: for normally distributed data, the error of the sample standard deviation is about σ / √(2n).

In other words, for a stock whose true volatility is 20%, a 20-day RV of 17% or 23% is perfectly normal. Seeing IV 3 points above 20-day RV is not enough to say the options are expensive.

In practice people often watch both a 20-day and a 60-day window: the 20-day tells you what has been happening lately, the 60-day what is normal. IV looks ahead over the next 30 calendar days, about 21 trading days, so 20-day RV is the most natural comparison. But remember: RV is the past and IV is the future. Comparing the past with the future carries error in itself.

Besides closing prices, there are estimators that use each day’s high and low (such as Parkinson and Garman-Klass), with smaller error for the same sample length. Their drawback is sensitivity to data quality: a single bad intraday print can distort the result. When you are starting out, the closing-price method is enough.

6.2Implied vs. realized: expectation and outcome

Implied volatility is the price you lock in when you buy; realized volatility is the outcome revealed while you hold. For a Delta-hedged option, P&L depends almost entirely on the difference between the two.

Take the conclusion from Chapter 4 one step further. Buy an option and hedge its Delta every day, and one day’s P&L is the Gamma gain minus Theta:

Add up every day of the holding period, and the sum of (ΔS/S)² is σ_R²T. Then, using a Black-Scholes identity, Vega = ΓS²σT, you can swap Gamma for Vega:

In one sentence: a hedged option position is a contract struck at IV and settled at RV, with each vol point worth one Vega.

The straddle costs 6.854, with Vega of 0.228 per point (both legs combined). The simulation used 20,000 price paths, hedging once a day at the close. The averages almost match the formula, but individual paths are widely spread: with the same 20% realized volatility, an unlucky path loses more than $4 per share and a lucky one only $0.8. The reason is that Gamma is concentrated near the money: if the big moves happen while the stock is far from the strike, you earn less from them.

6.3The volatility risk premium

Over the long run, the implied volatility of index options is higher than the realized volatility that follows. This gap is called the volatility risk premium (VRP). It is the seller’s source of income and the seller’s reward for bearing tail risk.

Take the S&P 500: on long-run average, VIX runs a few percentage points above the realized volatility of the following 30 days. That is only an average: in the month a crisis breaks out, realized volatility can far exceed the implied volatility beforehand.

Where the premium comes from

• Demand for insurance: pension funds, mutual funds and other institutions hold large amounts of stock and keep buying index puts to protect their portfolios. This demand isn’t very sensitive to price.
• Crash aversion: people will pay more than the expected loss to avoid a big loss, just as they do with home insurance.
• Sellers need capital: whoever sells options ties up margin and bears occasional huge losses. Without enough compensation, nobody would take that side.

Sellers are like insurance companies

An insurer collects premiums above its average claims, so it makes money over the long run. But claims come in clusters: one hurricane can wipe out several years of profit. Selling options is the same. Here is a calculation with a hypothetical model: each month you sell a 30-day ATM straddle on an index at IV 18%, and each month’s realized volatility falls into one of four cases.

Method: if realized volatility over the holding period is σ_R, the average of |S_T − K| at expiration is about the price of a straddle priced at σ_R. So the seller’s average P&L is about “the straddle priced at IV” minus “the straddle priced at RV.”

Average realized volatility is 14.4%, 3.6 points below IV, and the seller makes 0.83 a month on average, about a fifth of the premium. But a single crisis month loses 8.43, the profit of 5.3 calm months. In reality, crisis-month losses are often bigger still, because of price gaps, hedges that can’t be placed in time and margin calls, none of which are in the table above.

For buyers, the VRP means “on average, you are paying up.” To win, buy when IV is low relative to its own history (Section 6.4), or when you have reason to believe realized volatility will clearly exceed implied. Chapter 7 turns this into concrete rules for buyers; Chapter 8 covers how sellers control position size to survive crisis months.

6.4IV Rank and IV Percentile

Whether IV is high or low in absolute terms means little; what matters is where it sits in its own history. IV Rank measures where the current value sits between the one-year high and low; IV Percentile measures how many days in the past year were lower than now.

“Normal” volatility varies widely between stocks. IV of 60% may be the calmest a biotech stock gets all year, while 25% may already be a rare high for a utility. Comparing 60% with 25% directly tells you nothing; you have to compare each stock with itself.

Why the same IV gives two very different numbers

Rank uses only two points: the high and the low. A single extreme spike in the past year stretches the denominator, and Rank looks low for a long time afterward. Percentile uses every day’s data; a spike takes up only a few days, so its effect is small.

The chart below shows a simulated year of IV: mostly between 20% and 30%, it spikes to 62.5% on day 59 after a sharp sell-off and falls back a few weeks later. The one-year low is 19.2%. With current IV at 30%:

• IV Rank = (30 − 19.2) / (62.5 − 19.2) × 100 = 24.9, which looks low
• IV Percentile: 208 of 252 days were below 30%, giving 82.5, which is actually high

How to use it

These indicators are useful because IV tends to mean-revert: high IV usually falls back toward normal, and low IV drifts back up. But mean reversion is only a tendency. When the market shifts into a new regime (say, a bear market lasting months), high IV can go higher and stay there for a long time.

6.5Term structure

Connect the ATM IVs of one underlying across different expirations and you get the term structure. In calm markets it slopes upward, which is called contango; in a panic the near end sits above the far end, which is called backwardation.

The shape comes from mean reversion. Volatility has a long-run normal level. When current volatility is below normal, the market expects it to drift back up; the longer the term, the more of that “recovery” it includes, so the curve slopes up. When current volatility is far above normal, the market expects it to calm down gradually, so the near end is high, the far end is low, and the curve is in backwardation.

Forward volatility: reading “later” from the curve

Variance adds over time. 30-day IV covers days 0 to 30 and 90-day IV covers days 0 to 90; take one from the other and you get the market’s price for days 30 to 90:

• Contango: 30-day 15%, 90-day 18%. Forward = √[(0.18² × 90 − 0.15² × 30) / 60] = 19.3%. The market thinks things will be more turbulent a month from now than they are today.
• Backwardation: 30-day 35%, 90-day 28%. Forward = √[(0.28² × 90 − 0.35² × 30) / 60] = 23.7%. The market thinks things are dangerous right now but will be noticeably calmer in a month.

How to read it and use it

• Indexes: the usual gauge is the ratio of VIX to VIX3M (the equivalent 3-month index). A ratio below 1 is normal; above 1 means near-term fear has overtaken the far end.
• Single stocks: look for humps first. When one expiration’s IV suddenly sits above the neighboring months, there is usually an earnings report or another known event in that window.
• The near end moves more: in a shock, 30-day IV might rise 10 points while 1-year IV rises only 3. So two positions with “the same Vega” but different expirations don’t actually carry the same IV risk.
• Strategy implications: in contango, a calendar spread that sells the front month and buys a back month earns the roll down the curve; in backwardation, front-month options are relatively expensive, so buyers are better off expressing views with back months. Chapter 9 goes into this.

6.6Skew and the volatility surface

For the same expiration, IV differs across strikes. For stock indexes, OTM put IV is systematically higher than OTM call IV. This is called put skew.

Black-Scholes assumes one volatility for all strikes. If the market really priced options that way, plotting IV by strike would give a horizontal line. In practice, a stock index gives a line sloping down from left to right, most single stocks give a shallow smile, and story stocks sometimes curl up even higher on the right.

Why stock indexes have put skew

• Memory of the crash: after the stock market crash of October 1987, put skew in index options became persistent and pronounced. The left tail of real returns is much fatter than a normal distribution’s, and the market priced that straight in.
• Demand for protection: institutions keep buying OTM index puts to insure their portfolios, and the market makers who sell to them demand a higher price.
• Volatility rises in a sell-off: when stocks fall, corporate leverage rises and fear spreads, so volatility tends to rise at the same time. In a decline a put gains from both the direction and the rise in IV, so naturally it is worth more.
• Supply of calls: large covered-call programs sell OTM calls on indexes and single stocks year-round, pushing down IV on the call side.

Single-stock skew is usually flatter than index skew, because a single stock can crash but can also surge on a takeover, new drug data or an earnings beat. For story stocks and stocks with a big retail following, demand for OTM calls is strong, so the curve curls up on the right, forming call skew.

How to measure skew

The most common measure is the 25 Delta risk reversal: the IV of the 25 Delta call minus the IV of the 25 Delta put. The more negative it is, the steeper the put skew. The three hypothetical curves above come out at −4.1, −1.7 and +10.2 points. Another intuitive measure is the IV difference between the 90% strike and the 110% strike. Like IV, skew only means something when compared with its own history.

What skew means for trading

• OTM index puts are expensive to buy: for the 90 put in the chart above, most of the price you pay is skew premium.
• A spread lets you “sell back” skew: buy the 95 put (0.457) and sell the 90 put (0.197), and the short leg recovers 43% of the cost. If the curve were flat, the 90 put would be worth only 0.008 and recover just 3.7%. Put spreads are especially good value when skew is steep.
• Selling OTM puts collects skew premium: it pays off over the long run, at the cost of taking the first hit in a crash.

The volatility surface

Put term structure (looking across expirations) and skew (looking across strikes) together and you get a volatility surface: one IV for every strike and every expiration.

You can read this table in two directions. Down the ATM column, 15.0, 16.8, 18.0 is a term structure in contango. Across the rows, skew is steepest in the front month, where 90 and 110 differ by 14.6 points; at 180 days the gap is only 5.9 points. The reason is that over a short horizon it takes a crash to get the price down to 90, while over a long horizon the stock has many more paths to every price level, so the differences between strikes even out.

The surface moves as a whole. When an index falls, the entire surface usually shifts up, the near end most of all. If the index drops from 100 to 95, the old 95-strike IV (19.2%) becomes the IV near the new at-the-money level, and the IV of the ATM option you hold “slides” up the skew. That is why ATM IV almost always rises when an index falls.

6.7VIX and its products

VIX is the implied volatility over the next 30 days, calculated from the prices of a basket of S&P 500 index options and expressed as an annualized percentage.

It is calculated differently from a single option’s IV. Cboe takes two sets of SPX options expiring on either side of 30 days, weights the prices of a large number of OTM calls and OTM puts by strike (inversely to the square of the strike) and sums them into a variance, interpolates to exactly 30 days, takes the square root and multiplies by 100. Because it uses many OTM puts, VIX includes skew and usually sits slightly above SPX ATM IV.

VIX / √12: the one-month expected move

To convert annualized volatility σ to T years, multiply by √T. 30 days is about 1/12 of a year, so:

Example: VIX is 16, and assume the S&P 500 is at 7,800. One standard deviation over a month is 16 / 3.46 = 4.62%, about ±360 points, so the range is roughly 7,440 to 8,160. If returns were normally distributed, the index would have about a 68% chance of ending the month inside this range, and about a 95% chance of ending within ±2 standard deviations (±9.2%). One standard deviation per day is 16 / 15.87 ≈ 1.01%, about 79 points.

VIX futures: you can’t buy “spot VIX”

VIX itself is a calculated index and can’t be bought or sold directly. What you can trade are VIX futures and options, plus ETFs / ETNs that track VIX futures (collectively, ETPs). At expiration, VIX futures settle to a special opening quotation that morning; before expiration, a future’s price reflects the market’s pricing of “what VIX will be on that day,” which can be far from spot.

In calm periods, VIX futures are usually in contango too: spot is lowest, and the further out the contract, the more expensive it is. The reason is again mean reversion, plus the willingness of insurance buyers to pay a bit extra.

Roll decay: the trap in long VIX products

To maintain a term of about 30 days, a long VIX ETP sells some front-month futures and buys second-month futures every day. In contango, that means constantly selling the cheap and buying the expensive. If the curve’s shape doesn’t change, each future’s price “slides” down toward the lower spot level as its expiration approaches.

Run the numbers with a hypothetical set of quotes: spot VIX 15, front-month future (expiring in 15 days) 16.0, second-month future (expiring in 45 days) 17.2. You buy the second month and hold it for 30 days, by which time it has become the front month with 15 days left. If spot and the curve’s shape stay the same, its price slides from 17.2 to 16.0, a drop of 7.0%, while VIX itself hasn’t moved at all. Repeat that for 12 months and the cumulative loss is 1 − (16.0 / 17.2)¹² = 58%.

In reality the curve doesn’t stay fixed, and these products soar when panic breaks out. But ETPs that hold long VIX futures have historically seen their NAV slide steadily lower over time, and many have needed reverse splits. They suit short-term trading or short-term hedging, not long-term holding.

Inverse products: earn the roll most days, give it back in one

Inverse ETPs that short VIX futures are the mirror image: they earn roll yield in calm periods. But they rebalance daily to a fixed multiple, so on a day VIX soars they are forced to buy back a large amount of futures. The size of the rebalance can be calculated:

Example: a −1x product has a NAV of 100, and futures rise 50% in a day. It loses 50, leaving a NAV of 50, but its short exposure has grown to 150. To get back to −1x it can keep only 50 of short exposure, so it must buy back 100, which is exactly (−1) × (−2) × 50% × 100. If futures rise 100% in a day, a product like this goes to zero. A −0.5x product in the same situation needs to buy back 0.375 × 100 = 37.5, much less pressure.

6.8Event volatility

Scheduled events such as earnings, FOMC meetings and CPI releases lift the IV of options that span them before the event, and IV is pushed back down once the event passes. This part of IV can be stripped out and converted into the price the market puts on the event.

Stripping the event out of IV

Again, variance adds up: for an option that spans an event, total variance = ordinary daily variance × number of days + the extra variance of the event day.

Example: before earnings, weekly options expiring in 7 days are at IV 80%, and the base IV without earnings is about 35%. m = √[(0.80² − 0.35²) × 7/365] = 9.96%. The market has priced a one-standard-deviation move of about ±10% for this report. Student C’s ±5.3% in the opening was calculated the same way: √[(0.32² − 0.26²) × 30/365].

Straddle price ≈ average move

An ATM straddle’s price has an easy-to-remember approximation:

The 7-day straddle above, priced at 80%, costs 8.83; the approximation gives 0.8 × 0.80 × √(7/365) × 100 = 8.86. The market often calls “straddle price ÷ stock price” the expected move; here it is 8.8%.

In each cell, the figure before the slash is for a drop and the one after is for a rise. The buyer’s breakevens are a rise of 8.65% or a drop of 8.83%. If the stock doesn’t move, the buyer loses 59% overnight: the event variance priced in before earnings vanishes instantly, leaving only 6 days of base time value.

How to judge whether an event is expensively priced

• Compare the expected move with the company’s actual moves (in absolute terms) after its past 8–12 earnings reports. If the historical average is clearly smaller than the expected move, history has favored sellers; if it is clearly larger, history has favored buyers.
• The sample is small: only 4 reports a year. If the business has changed or the market’s attention has shifted, discount the past numbers.
• Look at the distribution, not just the average. One huge move in 8 can eat most of a seller’s average profit (Case 3).

Macro events

The same method works for indexes. Suppose that before a CPI release, SPX options expiring in 2 days are at IV 18%, against about 13% normally. One standard deviation on the event day = √[(0.18² − 0.13²) × 2/365] = 0.92%. FOMC decision days can be broken down the same way. Macro-event humps are much smaller than earnings humps, but they affect every stock and recur throughout the year. The post-event IV compression is the IV crush from Chapter 5; Chapter 10 is devoted to trading events.

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