Part VIII · The whole market · Chapter 8F

Beyond gamma

Vanna, charm and the expiration calendar

Chapter 7B split the change in an option's delta into three parts, (7B.1). They are gamma times the move in spot, vanna times the change in implied volatility, and charm times the time that passes. It worked the last two for one dealer's book, a post-FOMC vol crush and a weekend. This chapter adds them up across the market, as Chapter 8A added up gamma.

1 · Vanna and charm across the market

Vanna exposure and charm exposure

Chapter 8A summed each option line's dollar gamma, signed by the dealers' side, into gamma exposure. The same sum over vanna gives vanna exposure, or VEX, the change in the dealers' delta, in dollars, when implied volatility rises one point:

Here the sign $s_i$ is +1 if dealers are long line $i$ and −1 if they are short it. The line's vanna per unit of volatility is $\text{vanna}_i$, and 0.01 turns that into one volatility point. The product $S M N_i$ turns units of delta into dollars, with $M$ the multiplier and $N_i$ the open interest. The heatmap of Chapter 8C draws VEX strike by strike in its VEX view. When implied volatility changes by $\Delta\sigma$ points, dealers trade $-\text{VEX} \times \Delta\sigma$ to stay flat: they sell when VEX is positive and volatility rises.

The same sum over charm gives charm exposure, the change in the dealers' delta, in dollars, as one calendar day passes with spot and volatility fixed:

with charm measured per year, time running forward. Each day, dealers trade $-\text{CHEX}$.

The signs, wing by wing

Vanna changes sign at the strike (Chapter 5D). For an out-of-the-money put it is negative: a rise in volatility makes the put's delta more negative. For an out-of-the-money call it is positive: a rise makes the call's delta larger. So a dealer short options on both wings carries VEX of opposite signs on each:

• Short out-of-the-money puts: positive VEX. A rise in volatility makes the puts' deltas more negative, so the dealer, short them, is longer and sells. A fall in volatility makes it buy.
• Short out-of-the-money calls: negative VEX. A rise in volatility makes the dealer shorter, so it buys. A fall makes it sell.

Skylit's education pages say that "A dealer who is short options carries positive vanna". That holds on the put wing only. Its own worked example, a dealer short puts who buys back stock as volatility falls, is the put case.

Charm pulls every out-of-the-money delta toward zero as time passes. A dealer short out-of-the-money puts holds a short hedge against them and buys it back as their deltas shrink. A dealer short out-of-the-money calls holds a long hedge and sells it. The two wings trade in opposite directions again.

Time and volatility, one variable

With no interest rate and no dividends, an option's delta depends on volatility and time only through their combination $\sigma\sqrt{T}$, the standard deviation of the log price to expiry. A day passing shrinks $\sigma\sqrt{T}$, and so does a fall in implied volatility. To delta they are the same thing, and charm is vanna in another guise:

with charm per year of calendar time and $T$ in years. One day with $T$ days left does to delta what a fall in implied volatility of $\sigma/(2T)$ does, with $T$ in days. With $\sigma$ in percent, the fall is in volatility points.

Why time acts like a fall in volatility

Step 1: delta's argument. With $r = q = 0$, a call's delta is $N(d_1)$, and

$$d_1 = \frac{\ln(S/K)}{\sigma\sqrt{T}} + \frac{\sigma\sqrt{T}}{2}$$ (8F.4)

so $\sigma$ and $T$ enter only through $\sigma\sqrt{T}$. A put's delta is $N(d_1) - 1$, which changes by the same amounts.

Step 2: vanna. Differentiating with respect to $\sigma$ gives $\partial d_1/\partial\sigma = -d_2/\sigma$, so $\text{vanna} = -\varphi(d_1)\,d_2/\sigma$, the formula of Chapter 5D.

Step 3: charm. Differentiating with respect to $T$ gives $\partial d_1/\partial T = -d_2/(2T)$. Time running forward shortens $T$, so $\text{charm} = \varphi(d_1)\,d_2/(2T)$.

Step 4: the ratio. Dividing Step 3 by Step 2 gives $\text{charm}/\text{vanna} = -\sigma/(2T)$, which is (8F.3). With a rate, charm carries an extra term, $-r\varphi(d_1)/(\sigma\sqrt{T})$, so the identity holds only nearly.

Two consequences follow for dealers short out-of-the-money puts. A vol crush and the passage of time push their hedges the same way, toward buying. And over a window that ends at expiry, the two decide only the timing. At expiry every out-of-the-money delta is zero, whatever path volatility took, so the day-by-day trades add up to the hedge held at the start. Volland's user guide says, in a vendor's words, that charm "is the opposite sign of vanna on each strike", because it is computed for "+1 day passing". The identity (8F.3) is the reason, exactly with no rate and nearly with one.

The figure below draws a put's delta against $\sigma\sqrt{T}$ for a strike 3% below spot, with no rate, so that the identity is exact. Two sliders set the days to expiry, from 1 to 45 and starting at 28, and the implied volatility, from 5% to 40% and starting at 18%. Both move one dot along the same curve. The lower panel draws $\sigma/(2T)$, the volatility points one calendar day is worth, against the days left.

days to expiry T

28 days

implied volatility σ

18%

σ√T = 0.0499

put delta Δ = −0.262

one day = IV fall of 0.32 pts

At the start, $\sigma\sqrt{T} = 0.0499$, the put's delta is −0.262, and one day is worth a fall of 0.32 volatility points, Step 1 of the worked example. Drag the days down and the dot slides toward zero delta. Drag the volatility down instead and it slides the same way, along the same curve: 28 days at 18% and 7 days at 36% give the same delta. The lower panel reads 1.8 points with 5 days left and 9.0 with 1, Step 2's numbers. Near expiry each day is worth whole volatility points.

2 · Charm into expiration week

The figure follows the made-up book of the worked example through the last 20 trading days before a monthly SPX expiration. That is four Monday-to-Friday weeks, with the options expiring at the close of week 4. Spot stays at 6,000 and time runs on the calendar, so each Monday's bar carries a weekend of decay. Each bar is one day's change in the dealers' futures hedge, in billions of dollars, up for buying. A toggle adds out-of-the-money calls that customers are long (6,100, 6,200 and 6,300; 15,000, 10,000 and 5,000 contracts), drawn as a second leg with the net as a tick. The second control lowers implied volatility by 0 to 5 points in equal daily steps, with outlines for the same book with volatility unchanged. Start with puts only, then add the IV fall, then add the calls.

customers long puts puts and calls

IV fall

0.0 pts

total: dealers buy $10.40B

largest day: +$1.82B, Mon, week 4

The total is the starting hedge. Every option here expires out of the money with no delta, so over the 20 days dealers unwind all of the hedge they held at the start: $10.40B short in futures.

With puts only and no change in volatility, the dealers buy $10.40 billion over the 20 days. The bars grow into expiry: $0.17 billion to $0.20 billion a day in week 1 outside Monday, and $0.72 billion to $0.85 billion in week 4 before Friday. The Mondays stand out, at $0.46 billion, $0.66 billion, $1.04 billion and $1.82 billion, the largest day of all, in week 4. That is Chapter 7B's weekend effect. The final Friday falls back to $0.43 billion, because the puts are so far out of the money on their last day that little delta is left to decay.

Now lower volatility by 5 points. The total stays at $10.40 billion. The window ends at expiration, where every out-of-the-money delta is zero, so the daily flows always add up to the starting hedge. What the fall in volatility changes is the timing: it moves $1.63 billion of the buying out of week 4 and into weeks 1 to 3, and the largest day falls to $1.38 billion. The fall in volatility adds to each day's charm in the same direction, which is (8F.3) at work, and so the buying arrives earlier.

Set the IV fall back to 0 and add the calls. The dealers' long hedge against them is sold down by $5.89 billion over the same weeks. The net is $4.51 billion of buying, with the largest day $0.79 billion, again a Monday in week 4. A book with both wings short gives a smaller net flow than its larger wing alone, and its sign depends on which wing is larger.

What the evidence says

The calendar is real arithmetic, but its effect on prices is less certain. A Goldman Sachs strategist's teaching note, known through republication, describes the mechanics the figure draws: into expiration "near-the-money options' gamma rises, while far out-of-the-money options' gamma falls". The academic evidence is thin:

• Stivers and Sun (2013), in the Journal of Banking and Finance, study the weekly returns of S&P 100 stocks over option-expiration weeks, the weeks ending on a month's third Friday. Those returns "tend to be high" against less option-active stocks, against the same stocks' other weeks and against risk-based alphas. The following weeks "underperform modestly". The authors suggest two partial contributors. One is "delta-hedge rebalancing by option market makers, with a reduction in short-stock hedge positions over the OE week", which is charm. The other is "declining risk perceptions over the OE week, as measured by option-derived implied volatilities", which works through vanna. The sample is single stocks, not the index.
• Wang (2013), a doctoral thesis, finds that "weekly return reversals are significantly stronger surrounding option expiration days". The hedging pressure "mainly comes from index options rather than individual stock options", and "index option prices tend to be high before option expiration". It is a thesis chapter, not peer-reviewed.
• Golez and Jackwerth (2012) name the time decay of hedges as one channel of the expiration effects they find in S&P 500 futures (Chapter 8E).

No study found tests charm flows into weekly or same-day SPX expirations. For the monthly expiration, one working paper tests exactly the channel the figure draws, at the moment the options settle.

Into the monthly a.m. settlement

Monthly SPX options settle on the Friday-morning special opening quotation, built from the first trade in each component stock, not on a closing price (Chapters 2A and 12H). Baltussen, Terstegge and Whelan, in an SSRN working paper first circulated in 2023 as "The Equity Derivative Payoff Bias", compare that settlement value with Thursday's close.

In the 2023 version, the settlement value is close to the prior close on ordinary days, but on settlement days it "exceeds Thursdays close by a significant margin". The overnight return "raises (lowers) S&P 500 call (put) option payoffs, inducing a wealth transfer of around $4 billion per year in SPX options alone". The authors find the bias "most consistent with inventory management by option market makers and/or market manipulation by sophisticated investors", both relying on an illiquid period before settlement.

A later version, circulated as "The Derivative Payoff Bias", reports that opening prices on third Fridays "are systematically biased upward and fully reverse by midday". The transfer is put at about $2.5 billion a year. The aggregate delta of dealers' option positions "drifts down into" the third-Friday expiry, "generating an incentive to buy equities". That drift predicts overnight equity returns into third-Friday expirations, and on no other days. In summaries of that version, SPX calls paid $10.46 billion on the average third-Friday morning against $10.23 billion at Thursday's close, $230 million a month. Retail investors, net short delta, lose at least $3.8 billion. One of the authors, Julian Terstegge, summarizes the channel in his 2025 doctoral thesis. In his words, "Charm spikes immediately before derivative payoffs are calculated and thus creates a derivative payoff bias, which exceeds $3bn a year".

Three cautions go with it. It is a working paper, and its headline numbers changed between versions, so quote the version. The a.m.-settled monthly is one contract design: PM-settled SPXW options and same-day expiries settle differently, and the paper says nothing about them. And the effect reverses by midday, the signature of price pressure rather than information.

Index expiration effects have a history. Stoll and Whaley (1987) studied the quarterly expirations of 1984 and 1985, when index futures and options expired together on the close, the day traders call triple witching. Last-hour volume and volatility were higher and the market tended to fall. The average price effect, measured by the next morning's reversal, was "about 0.4 per cent of the closing index value". S&P index option expirations on their own produced no abnormal movements. Their 1991 follow-up found smaller effects after the move to settlement on the open in 1987. The classic expiration effect came from unwinding futures arbitrage, not from option hedging.

3 · After expiration: the "window of weakness"

Practitioners have a name for the weeks after a monthly expiration. Cem Karsan of Kai Volatility Advisors describes the flows into expiration in talks and interviews. In his account, dealers "are often buying back stock as risk premia decays (charm) and buying back stock as implied volatility compresses (vanna)". After an expiration, in his words, comes "a window of non-strength some people call it a window of weakness", "a period where flows have subsided". A secondary account says that "the one to two weeks following the day of the monthly options expiration" are "a short period whereby supportive Vanna and Charm flows ... are much less prominent". Karsan also calls expiration week "the best time to sell volatility". Who coined the phrase is not established.

The window of weakness has a mechanical reading in this chapter's terms. After expiration the puts that were decaying are gone. The next monthly puts have about four weeks left, and by (8F.3) a calendar day is worth far fewer volatility points to them. In the figure, week 1's ordinary days bring $0.17 billion to $0.20 billion of buying, against $0.72 billion to $0.85 billion in week 4 before Friday. If dealers are short the puts, the supportive flow is weakest right after an expiry. Bank desks use a similar image: after one expiration, a Nomura note, as a vendor quoted it, described the aggregate "$Gamma" of QQQ dropping by over 50%, "unclenching" the index.

Three things keep it lore rather than knowledge:

1. No test on SPX. No published test by the practitioners who use the phrase was found, and none on the S&P 500 index. The closest academic result is Stivers and Sun's modest underperformance of S&P 100 stocks in the week after expiration.
2. Its direction is a sign convention. The support into expiration needs dealers short the decaying puts. If customers are net sellers of those puts, dealers are long them, and the same decay makes dealers sell into expiration and stop selling after it.
3. Calendar effects overlap. The weeks after the monthly expiration also hold month-end flows, the VIX expiration, which practitioners pair with the SPX expiration as "OPEX and VOLEX", and in some months a Fed meeting. Nothing found separates them. No source on the VIX expiration's own effect was found.

Some dated flows are known. The JPM collar resets every quarter, about 35,000 SPX contracts per leg by vendors' accounts, and its put strike is one of the few whose holder is public (Chapters 8B and 8E). Vendors claim the market "sells off meaningfully after the quarterly roll", a claim no published test supports. Defined-outcome ETFs reset their FLEX options on fixed dates (Chapter 4A), another calendar no heatmap of listed open interest shows.

4 · The vol-crush rally

Chapter 7B worked one dealer's post-FOMC unwind. Implied volatility falls 7 points after the announcement, the deltas of out-of-the-money puts shrink, and a dealer short 5,000 of them buys back tens of thousands of share-equivalents. Summed across the market, that is the vol-crush rally practitioners describe, the vanna half of Karsan's account, after the IV crush that follows a scheduled event (Chapter 12A). The Dealer Mechanics Lab puts its typical contribution after an FOMC decision at 0.3% to 0.5% of SPX at most (Chapter 7B). By (8F.1), the flow is VEX times the fall in volatility. The figure's book, at +$0.53 billion per point, would buy about $3.7 billion on a 7-point crush, by the linear estimate that Chapter 7B found runs low.

What the research has tested is narrower:

• No direct test. The practitioner claim that falling implied volatility after events forces dealer buying of the index has no direct test in what the research found.
• A tested channel from volatility to the index. Bangsgaard and Kokholm (2025), in the Journal of Banking and Finance, study the end-of-day demand of VIX exchange-traded products for VIX futures. The market makers on the other side "can hedge their volatility exposure by trading SPX options and futures". The products' demand "is a strong predictor of the end-of-day SPX futures return in-sample and out-of-sample". The move partly reverses afterward, "suggesting that VIX futures hedging activities can move the SPX futures market for reasons unrelated to price discovery". It is a volatility-to-index hedging channel, measured, though not the option dealers' vanna itself.
• The falling volatility of expiration week, one of Stivers and Sun's two suggested contributors.

Two findings qualify every vanna number on a map. Hull and White (2017), in the Journal of Banking and Finance, show that "the normally calculated delta does not minimize the variance of changes in the value of a trader's position". The reason is that price and volatility move together. Their minimum-variance delta "takes account of both price changes and the expected change in volatility conditional on a price change". It reduces hedging variance by about 42% for the highest-strike calls and about 17% for in-the-money ones, in search summaries of their S&P 500 results. A dealer who hedges that way has already folded the typical vanna of a price move into its delta. Vanna then adds a separate flow only when volatility moves without the price, as after an event, or when the VIX jumps with the index still.

Willeboordse (2026), in Finance Research Letters, compares Black-Scholes gamma with the convexity option positions actually show, across five underlyings from 2020 to 2025. At a daily horizon the correlation turns "significantly negative (−0.25 to −0.47 across symbols)", more so as vega exposure grows relative to gamma exposure. The author reads it as "substantial vega contamination through negative spot–IV comovement". A map that adds a GEX flow and a VEX flow computed separately can count the same hedge twice.

The one practitioner test of vanna exposure found points the same way. FlashAlpha, testing its own product on 1,972 SPY days from 2018 to 2026, reports that VEX correlates +0.72 with the VIX and +0.76 with at-the-money implied volatility. Once those are controlled for, its correlation with next-day realized volatility is −0.01 (p = 0.77): VEX "is mostly a VIX proxy". Chapter 8G gives the study.

5 · Vendors' rules, checked against Black-Scholes

Skylit builds two rules on these Greeks. In its words, "GEX is the hand brake; VEX is the slope of the hill", and "GEX governs the range; VEX governs the drift". And on tenor: "For options 0–5 DTE, gamma is dominant; for 7–180 DTE, vanna is dominant". Both can be checked against the Black-Scholes Greeks of Chapter 5D, though neither has been tested against prices.

"GEX governs the range; VEX governs the drift"

The rule has a basis in (7B.1). The gamma term needs a move in price, so it shapes how each move develops: it damps or amplifies moves, which sets the range. The vanna and charm terms fire with the price standing still, so they can push the same way day after day, which is a drift. Three things qualify it:

• Charm is the steadier drift. Vanna pushes only while implied volatility moves, and its direction follows the direction of the move in volatility. The calendar-driven drift is charm's.
• Both drifts carry a sign convention. A drift upward into expiration needs dealers short the puts. Under the other sign, the drift reverses.
• The flows are not separate. When volatility moves with the price, as it usually does in the index, a minimum-variance hedger has folded part of the vanna into the gamma hedge. And Willeboordse's result says the two are hard to separate over a day.

"Gamma for 0 to 5 days, vanna for 7 to 180"

Compare the two hedges per contract on a common footing: the change in the dealer's delta, in dollars, for a 1% move in spot and for a one-point rise in implied volatility. Black-Scholes gives their ratio in one line:

so a one-point rise in volatility moves the delta as much as a spot move of $-d_2\sqrt{T}$ percent. The table takes a put with a delta of −0.25 at each expiry, SPX at 6,000, 18% implied volatility and a 4% rate. The last column is the move in implied volatility, per 1% move in spot, at which the vanna hedge would equal the gamma hedge.

Four things follow, and none of them is a threshold at 5 or 7 days:

• For options of the same delta, vanna's weight grows smoothly with $\sqrt{T}$. The vanna hedge per contract barely changes with expiry, while the gamma hedge falls like $1/\sqrt{T}$. That is the grain of truth in the rule.
• At a fixed strike it hardly changes. For a put struck 5% below spot, the break-even is about 3.5 points per 1% at 5 days, 3.4 at 30 and 2.9 at 180.
• At the money, vanna is close to zero at every expiry. Same-day gamma sits almost entirely at the money, which is why gamma dominates same-day books, whatever the calendar says.
• Dominance depends on a market fact. How far implied volatility moves for each 1% of spot is not in Black-Scholes, which holds volatility constant. On an event day, when volatility falls with the price still, vanna is the whole of the change at any expiry.

Skylit also maps VEX by its position against spot, in four cells:

• "+VEX below spot (support in calm, rug-pull risk in stress)";
• "–VEX below spot (fragile floor in calm, fast bounces in stress)";
• "+VEX above spot (breakouts in calm, caps in rising vol)";
• "–VEX above spot (soft ceiling in calm, squeeze fuel during vol spike)".

Read with the signs above, the matrix is consistent with the mechanics. Positive VEX below spot is dealers short puts, who buy as volatility falls in calm and sell as it rises in stress. It is untested, and it inherits the sign convention of every cell. So does Skylit's warning that when volatility spikes, back-end positive VEX "creates significant sell flow" that is "entirely invisible in a front-month gamma map". That is the vanna loop of Chapter 9A, seen on a map.

Volland infers each trade's dealer side from the options order feed, in a 2023 working paper by DeLorenzo for the vendor. The paper claims that "option dealers are predominantly more sensitive to changes in implied volatility than to the movement in the underlying equity". It also claims that charm "yields a significant edge when applied in the right time frame relative to expiration". Neither claim comes with a test a reader can check.

6 · The link to the vanna loop

Chapter 9A followed the vanna loop in a sell-off: the index falls, the VIX rises, the deltas of out-of-the-money puts deepen, and dealers short them sell, which pushes the index lower. The vol-crush rally is the same loop run backward. With positive VEX from short puts, a rise in volatility makes dealers sell and a fall makes them buy. The exposure is the same, and the sign of the move in volatility decides the direction.

The measured channel sits between the two: Bangsgaard and Kokholm's VIX-futures hedging moves S&P 500 futures at the close and then partly reverses. Chapter 8H works the same arithmetic for March 2020, with the index held still while volatility rises.

7 · Check yourself