Part IX · Volatility in practice · Chapter 9A

The VIX

The same mechanism, seen from above

The other two chapters of Part IX build on this one. Chapter 9B turns to same-day options, with the 2024 and 2025 academic findings that they have, on net, damped SPX volatility. It also sets out the study that finds they raised it. Chapter 9C closes the part with seven practical heuristics for reading the tape through the dealer's hedging, formalized into a regime classifier.

1 · What the VIX measures

The VIX is widely called the "fear gauge", a label that is qualitatively useful but mathematically imprecise. The VIX is the model-free 30-day implied volatility of the S&P 500, expressed in annualized percentage points. Its construction starts from the implied variance of the 30-day SPX log return, computed from a weighted strip of out-of-the-money option prices. It then takes the square root and multiplies by 100.

So a VIX of 18 means that the model-free 30-day implied volatility of SPX is 18% a year, not 18 units of variance. "Model-free" means that no pricing model stands between the option prices and the number. The implied volatility of Chapter 5E runs one option's price backward through the Black-Scholes formula, while the VIX reads variance directly off a whole strip of prices.

The distinction matters because variance and volatility move on different scales. Variance is volatility squared, so doubling implied volatility from 18 to 36 quadruples implied variance, from 324 to 1,296 in squared percentage points. The published VIX uses the volatility scale, the one on which traders think about how far the index can move. By the rule of 16, a VIX of 18 prices a typical daily move of about $18/16 = 1.1\%$.

2 · Building variance out of options

The formula

Cboe computes the 30-day implied variance $\sigma^2$, annualized, with this formula:

The published index is then $\text{VIX} = 100\sqrt{\sigma^2}$: the square root returns to the volatility scale, and the factor 100 turns 0.18 into 18. The symbols in (9A.1) mean the following:

• Time $T$ is the time to expiry in years, $30/365$ for 30 days.
• Each strike in the strip is $K_i$. The width $\Delta K_i$ it stands for is half the distance between its two neighbors, or the one-sided distance at either end of the strip.
• The price $Q(K_i)$ is the mid-quote of the out-of-the-money option at $K_i$, which is a put below the forward and a call above it.
• The forward price $F$ is the price at which you could agree today to buy the index on the expiry date.
• The strike $K_0$ is the listed strike immediately below $F$.
• The rate $r$ is the risk-free rate. The factor $e^{rT}$ grows each option's price forward to expiry, for a reason the derivation below gives.

The first line adds up the strip's weighted option prices and scales them to an annual variance. The second line is a small correction for the forward not sitting exactly on a listed strike.

The $1/K^2$ weighting is not arbitrary: it is the coefficient that makes a portfolio of options replicate a variance swap. Demeterfi, Derman, Kamal and Zou showed this in 1999, in the Goldman Sachs Quantitative Strategies Research Notes. Holding $\Delta K/K^2$ of each strike across all out-of-the-money strikes, delta-hedged, gives a position whose P&L is, exactly, total realized variance minus a fixed strike, which is a variance swap. The VIX is the discrete version of the fair strike of that swap, scaled and annualized.

Every term connects back to dealer hedging through the gamma-theta decomposition of Chapter 7A. A delta-hedged option earns $\tfrac12\Gamma S^2[(\Delta S/S)^2 - \sigma^2\Delta t]$ over each interval. Weighted by $1/K^2$ and summed over strikes, the gammas combine into a roughly constant exposure per percent move wherever spot is. That flat dollar gamma is exactly the property a variance swap requires, and Chapter 7A works it through. The VIX is, fundamentally, the cost of replicating variance through delta-hedged options.

Where the formula comes from

The derivation needs only algebra, the log series and one integral read as accumulation. Its heart is a single identity: over a small move, a squared return is twice the gap between the simple return and the log return.

Deriving the VIX formula

Step 1: a squared return from logs. For a small daily return $R$, the log return is $\ln(1+R) = R - R^2/2 + R^3/3 - \cdots$, a series whose terms shrink fast when $R$ is small. Dropping the cube and everything after it, and rearranging, gives

$$R^2 \approx 2\,[\,R - \ln(1+R)\,]$$ (9A.2)

For a 2% day the left side is 0.000400 and the right side is $2 \times (0.02 - 0.019803) = 0.000395$.

Step 2: add up the days. The daily log returns add up to $\ln(S_T/S_0)$, the log of the final price over the first (Chapter 1A). Summing (9A.2) over the days to expiry therefore gives

$$\sum_t R_t^2 \approx 2\sum_t R_t - 2\ln\frac{S_T}{S_0}$$ (9A.3)

The left side is the realized variance over the period, which is what a variance swap pays. The right side is two trades. The first, $\sum_t R_t$, is what you earn by holding $1 of the index and resetting it to $1 every day, a dynamic trade like a delta hedge. The second, $-\ln(S_T/S_0)$, depends on the final price alone. It is a short position in a log contract, which pays the logarithm of the final price, and options can build it.

Step 3: value the daily trade. Take no dividends, to keep the algebra short. Under the risk-neutral probabilities that price options (Chapter 5C), each day's expected return is the risk-free rate for that day, so the daily trade is worth $rT$ in expectation.

Step 4: measure the log contract from the forward. With no dividends the forward is $F = S_0e^{rT}$, so $\ln(F/S_0) = rT$ and

$$-\ln\frac{S_T}{S_0} = -\ln\frac{S_T}{F} - rT$$ (9A.4)

Now define the payoff $g(S) = S/F - 1 - \ln(S/F)$. It is zero at $S = F$, its slope there is zero, and its curvature, the slope of its slope, is $1/S^2$.

Step 5: build g from options. A payoff that is zero with zero slope at $F$ is a pile of options, puts below $F$ and calls above it, holding $g''(K)\,dK = dK/K^2$ of each strike:

$$\begin{aligned} g(S_T) = &\int_0^F \frac{(K-S_T)^+}{K^2}\,dK \\ &+ \int_F^\infty \frac{(S_T-K)^+}{K^2}\,dK \end{aligned}$$ (9A.5)

Here $x^+$ means $x$ when it is positive and zero otherwise, which is an option's payoff. To check it for a final price below the forward, note that only the puts struck above $S_T$ pay. They add up to $\int_{S_T}^F (K - S_T)/K^2\,dK = \ln(F/S_T) + S_T/F - 1$, which is $g(S_T)$. This is where the $1/K^2$ comes from: it is the curvature of the short log contract.

Step 6: price the strip. Under those probabilities, an option's price is its expected payoff discounted at the risk-free rate, so its expected payoff is its price grown by $e^{rT}$. With $Q(K)$ the out-of-the-money price at each strike,

$$\mathbb{E}[g(S_T)] = e^{rT}\!\int_0^\infty \frac{Q(K)}{K^2}\,dK$$ (9A.6)

Under the same probabilities the forward is the expected final price, so $\mathbb{E}[S_T/F - 1] = 0$ and the short log contract's expected payoff is $\mathbb{E}[-\ln(S_T/F)] = e^{rT}\int_0^\infty Q(K)/K^2\,dK$.

Step 7: assemble. Take expectations in Step 2 and substitute Steps 3, 4 and 6, and the two $rT$ terms cancel:

$$\mathbb{E}\Big[\sum_t R_t^2\Big] \approx 2e^{rT}\!\int_0^\infty \frac{Q(K)}{K^2}\,dK$$ (9A.7)

Divide by $T$ to annualize, and replace the integral by a sum over the listed strikes, each carrying its width $\Delta K_i$. That is the first line of (9A.1).

Step 8: the correction. Cboe switches from puts to calls at $K_0$, the strike just below $F$, rather than at $F$ itself. Redoing Steps 4 to 7 with the switch at $K_0$ leaves an extra $-2[x - \ln(1+x)]$, with $x = F/K_0 - 1$. By the log series that is about $-x^2$. Dividing by $T$ gives the second line of (9A.1).

The toy strip lands at 11.5 because its quotes correspond to an implied volatility of around 12%, a calm-regime strip. The same chain with every quote roughly doubled gives a VIX near 16 to 17. With every quote roughly tripled, typical of quotes after 2020, it gives one near 20.

The VIX grows with the square root of the strip's premium level, not in proportion to it. Doubling every quote doubles the variance and multiplies the VIX by $\sqrt{2} = 1.41$, from 11.5 to 16.3, while tripling multiplies it by $\sqrt{3} = 1.73$, to 19.9. The geometry of the formula does not change, which is the payoff of doing it by hand once.

Two caveats separate the toy from the real index. The real VIX is computed from about 150 strikes under Cboe's methodology, not six. What changes is the precision, not the chain of steps. The published VIX also interpolates between two expirations, the near and the next, to reach exactly 30 days, where the toy uses one.

The figure below computes a VIX from a toy strip you control. Its top panel draws a 30-day implied volatility smile over strikes 70 to 130, one apart, around a forward of 100. Three sliders shape it: the ATM IV, from 8% to 45% and starting at 18%; the put skew steepness, from 0 to 0.40 and starting at 0.18; and the call skew steepness, from 0 to 0.20 and starting at 0.04. The bottom panel draws each strike's term, $(\Delta K/K^2)\,e^{rT}Q(K)$, with $T = 30/365$ and $r = 4\%$, using the put below the forward and the call at or above it.

The forward always sits on a strike, so the figure needs no correction term. Readouts give the computed VIX, the forward, and the shares of the variance from the put wing and the call wing. Drag the ATM IV from 18% to 30%, then drag each skew slider across its range, and press the three SPX shock buttons, −1%, 0% and +1%.

ATM IV

18.0%

put skew steepness

0.18

call skew steepness

0.040

SPX shock −1% 0% +1%

computed VIX = 18.51

forward = 100.0

put wing share = 41%

call wing share = 59%

At its defaults the computed VIX reads 18.51, with 41% of the variance from the put wing and 59% from the calls, counted from the strike at the forward. The index follows the ATM IV almost point for point, reading about 30.7 at an ATM IV of 30%. The skews move it far less: taking the put skew from 0 to 0.40 lifts it only from about 18.1 to 19.1. The shock buttons barely move it at all, to 18.507 after a 1% fall and 18.503 after a 1% rise, against 18.505 unshocked. The figure carries its whole smile along with the forward, so after a shock each option keeps its implied volatility relative to the new forward, and the strip's variance hardly changes.

3 · Why the VIX rises when SPX falls

The $1/K^2$ weighting puts disproportionate weight on low-strike options: at equal spacing, a put at 4,400 counts $(4{,}700/4{,}400)^2 = 1.14$ times as much as a call at 4,700 of the same price. Combine that with the equity volatility skew, in which out-of-the-money puts trade at higher implied volatility than out-of-the-money calls. By the Dealer Mechanics Lab's account, a 1% downside move in SPX then re-prices the put wing far more than a 1% upside move re-prices the call wing.

On that account, this is why the VIX is structurally inverse to SPX. It is not "fear" as such, but the mechanical asymmetry of how variance replication weights the strikes where the market is hedged. The correlation between daily SPX returns and daily VIX changes is approximately −0.80 across decades of data. The Dealer Mechanics Lab attributes that correlation not to fear and stock prices moving together, but to the maths of variance replication. That maths requires a put-side reweighting that responds asymmetrically to direction.

The mechanism needs implied volatility that stays attached to its strikes as spot moves. Take the figure's default smile, in which implied volatility rises by 0.18 for each unit of $\ln(K/F)$ below the forward and falls by only 0.04 for each unit above it. If each strike keeps its implied volatility while spot falls 1%, the new at-the-money strike sits at $\ln(0.99) = -0.010$ on the old smile. Its implied volatility is higher by $0.18 \times 0.010 = 0.0018$, 0.18 of a volatility point. After a 1% rise, the new at-the-money strike sits on the flat call side, where implied volatility is lower by only $0.04 \times 0.010$, 0.04 of a point. The fall re-prices the options around the money four and a half times as much as the rise.

The strip builder cannot show this asymmetry. Its smile moves with the forward, so a shock leaves every option's implied volatility unchanged relative to the forward. If you hold the same default smile fixed to its strikes instead, a calculation the figure does not make, the asymmetry appears and grows with the size of the move. A 5% fall then lifts the computed index by about 0.7 points, while a 5% rise lowers it by about 0.4.

4 · The vanna loop

Recall vanna from Chapters 5D and 7B: $\partial\Delta/\partial\sigma$, the change in an option's delta as implied volatility moves. When the VIX rises, the deltas of out-of-the-money puts become more negative. A dealer who is short those puts, the dealer of record for most institutional hedging demand, sees its effective long exposure increase. To stay delta-neutral, it must sell stock.

Selling stock pushes SPX further down. A further fall lifts the VIX further, which deepens the put deltas again. Then the loop goes round once more. Vanna is the technical name for the variable that makes VIX-driven sell-offs so violent.

The figure below walks one pass of the loop, a step at a time. A dealer is short 1,000 out-of-the-money SPX puts struck at 4,400 against an institutional hedger, with spot at 4,500, the VIX at 20 and the put's implied volatility at 23%. Move the step slider from 0 to 5, or press play. Four readouts follow the loop: SPX, the VIX, the shift in the dealer's delta, and the stock it sells, in share-equivalents for 1,000 contracts of 100.

step

0 · setup

play pause reset

SPX = 4,500.0

VIX = 20.00

dealer Δ shift = 0

hedge flow = —

Step 0 · setup. Dealer is short 1,000 SPX OTM puts (K = 4400) against an institutional hedger. The short puts are long delta, so the initial delta hedge is a short stock position. Spot 4500, VIX 20, IV at this strike 23%.

At step 1 SPX drops 1% to 4,455. The spot move alone makes the put delta 0.04 more negative through gamma, so the dealer's delta shifts 4,000 longer. At step 2 the VIX rises 8%, from 20 to 21.6, while the put's implied volatility rises from 23% to roughly 24.8%. At step 3 vanna fires: the volatility move alone deepens the put delta by about 0.07 more, leaving the dealer 11,000 longer than it should be. At step 4 it sells those 11,000 share-equivalents, which is 0.11 of delta on 1,000 contracts of 100. By step 5 the selling has pushed SPX 2.5% below the start and the VIX 18% above it. The dealer's delta shift reaches 20,000, with a hedge flow of 19,500. In normal markets each pass is smaller than the last, while in stress the system can become locally unstable.

The loop's arithmetic starts from a hedge that is easy to get backward.

The figure's numbers are illustrative inputs, as its caption says, not the output of a model. For scale, Black-Scholes gives the 30-day put of the example a gamma step of about 0.06 for the 1% fall. Its vanna step for the rise in implied volatility to 24.8% is only about 0.004, not 0.07. Vanna is larger further out in the downside wing, the typical institutional hedge book (Chapter 5D). Its push also grows with the size of the volatility jump and of the book, which in a real spike is the whole street's.

In the Dealer Mechanics Lab's view, that minute, 16:08 EST on the day called Volmageddon, is the cleanest empirical example of the feedback the figure walks through. Chapter 8H tells the whole episode.

5 · Trading the VIX: futures, ETPs and options

The VIX is computed from a strip of SPX options, and it is not directly tradable. What you can trade comes in three kinds.

VIX futures are monthly contracts. Each expires on a Wednesday, 30 days before the SPX monthly options expiration (OPEX), and settles to a special opening quotation (SOQ) of the VIX on its expiry morning (Chapter 2A).

VIX options are cash-settled options on the VIX index level, with notional based on the index level at $100 per VIX point (Chapter 5A).

VIX exchange-traded products (ETPs) hold VIX futures, and four are common:

Contango and the roll

Under normal conditions the VIX futures curve is in contango: the front-month future sits above spot VIX, with the second month above the front. The Dealer Mechanics Lab gives the economic reason as the variance risk premium, since buyers pay extra for protection over each forward window.

For long-volatility ETPs, contango is a cost. In the Dealer Mechanics Lab's description, every month they sell the front contract at the lower price and buy the second at the higher one, locking in a loss. Strictly, the swap itself costs nothing, because selling one contract to buy a dearer one only leaves you holding fewer contracts. The loss comes as each contract ages: in contango a future bought above spot drifts down toward spot as its expiry nears, even if spot VIX holds still. That drift is the roll yield, which is negative for a long position in contango.

That is the size the Dealer Mechanics Lab reports: the accumulated drag is approximately −10% annualized for VXX, and UVXY suffers roughly twice this drag. For inverse-volatility ETPs, contango is a benefit, and SVXY profits from the roll. The roll was the dominant return source for XIV and SVXY before 2018, until an eight-month low-volatility regime made their positioning catastrophically large and triggered Volmageddon.

In stress the curve inverts into backwardation, with the front month above the second. The roll then reverses, so long-volatility ETPs profit and inverse-volatility ETPs lose. Backwardation is a reliable stress indicator: a ratio of the VIX to the VIX3M crossing 1.0 preceded major SPX drawdowns in nearly every multi-year cycle.

The ETP issuers, Barclays for VXX and ProShares for UVXY and SVXY, are themselves significant participants in VIX futures. Their daily rebalancing is mechanical and well understood. The 16:08 EST volume spike on February 5, 2018 was the cumulative effect of XIV's forced rebalancing in the closing auction, interacting with dealers' hedging of volatility of volatility (vol-of-vol).

The operational implication: VIX futures and VIX ETPs are not the same instrument as the VIX index. The roll structure adds a multi-percent annualized drag in contango, or a boost in backwardation. The futures' basis to spot VIX can also diverge in stress. Use the VIX as a regime indicator. Trade VIX futures and options as separate instruments, with their own dynamics.

The Edge Book goes further and rejects VIX ETPs, long and short, for a retail account (Chapter 10I). It puts the roll decay of long VIX ETPs at 5% to 15% a year in normal times, which brackets the Dealer Mechanics Lab's 10%. It also treats the VIX term structure as a regime filter that sizes other trades, never as a trade in itself (Chapter 10H). The two sources agree that the index is a gauge, and differ on whether its derivatives belong in a retail account.

6 · The VIX family

Cboe launched VIX1D in April 2023 to measure one-day expected volatility. It did so explicitly because the 30-day VIX no longer captured the short-horizon volatility regime driven by same-day options (Chapter 9B). VIX1D applies the same model-free variance-replication formula to the very-near-dated SPX option strip. The Cboe VIX family includes six indexes:

The relationships among these indexes form the VIX term structure. In normal markets it is in contango, short below long, reflecting the variance risk premium: implied volatility is bid up further out, because more can happen over longer horizons. In stress it inverts into backwardation, short above long, as panic concentrates demand into near-dated puts.

In the Dealer Mechanics Lab's words, the sign of gamma exposure (GEX) combined with the shape of the VIX term structure is the most common composite signal that serious traders use to characterize the regime. Its standard combinations are these:

This five-regime classification underlies the composite scorecard of Chapter 11G and the conviction score of Chapter 13B. Chapter 9C turns a four-regime version of it into a classifier you can drive, and sets it beside the Edge Book's lower opinion of GEX.

7 · Check yourself