Part VII · Theory · Chapter 7B

Second-order Greeks in context

What the higher-order Greeks do to a hedged book

Gamma told you how delta changes when spot moves. But delta also changes when implied volatility moves, with no spot move required, and when time passes, with no spot move and no volatility move. These two derivatives, vanna and charm, generate predictable hedging flows that are largely invisible to retail and relentless on a calendar-time schedule. Understanding them is the difference between knowing that dealers hedge and knowing when they hedge.

This chapter works the two most important of those flows with full dollar arithmetic: the post-FOMC vanna unwind and the charm-driven Monday gap. It then places the rest of the higher-order Greeks in the hedged book. Chapter 5D defines and derives each Greek, so this chapter recaps them only as far as the arithmetic needs.

1 · Three ways a hedge goes stale

Over a short interval, an option's delta changes for three reasons at once:

Here $dS$ is the move in spot, $d\sigma$ the change in implied volatility as a decimal, and $dt$ the calendar time that passes, in years. Each coefficient is a Greek: gamma from Chapter 5C, then vanna, $\partial\Delta/\partial\sigma$, and charm, $\partial\Delta/\partial t$ with $t$ running forward, both from Chapter 5D.

Now put the option in a dealer's book: a dealer short $n$ put contracts, each on 100 units of the underlying, carries $-100n\Delta$ units of delta, or share-equivalents, through them. A put's delta is negative, so that is a long position. The delta hedging of Chapter 6C offsets it with a short position of the same size. When the put's delta changes by $d\Delta$, the dealer must trade

units of the underlying: a purchase when the put's delta rises toward zero, and a sale when it falls further below zero.

The first term of (7B.1) is the gamma chase of Chapter 6D. The other two fire with spot standing still. The vanna term fires when implied volatility moves, as it does around scheduled events. The charm term fires as the calendar turns, whether or not the market is open.

What the first-order bet leaves out

The same Greeks also change the hedged book's profit and loss, of which Chapter 7A's formula kept only the gamma term. Expanding the option's value in spot and implied volatility together, one order further, gives the hedged book's gain above cash over an instant:

Here $\sigma$ is the implied volatility at the start of the instant. The gamma term on the first two lines is Chapter 7A's bet on realized against implied. The third line holds the vega caveat and vanna's cross term, which pays or charges when spot and volatility move together. The last line is volga, vega's own curvature. When a fall in spot comes with a rise in implied volatility, as in the vanna feedback of Chapter 5D, the vega and vanna terms can outweigh the gamma term.

2 · Vanna: the post-FOMC unwind

Vanna, briefly

Chapter 5D derived vanna's closed form, the rate at which delta changes with volatility, which equals the rate at which vega changes with spot:

The intuition is that implied volatility sets how spread out the option's pricing distribution is, and so how much probability mass sits in its tails. For an out-of-the-money (OTM) call, that means more chance of finishing in the money, which raises its delta. For an OTM put, whose delta is negative, it does the same: the delta becomes more negative, further from zero.

Vanna is large and negative for OTM puts struck below spot, where $d_2$ is positive (Chapter 5D). When the VIX spikes, OTM put deltas become more negative, larger in magnitude. A dealer short these puts, because customers were buying protection, becomes effectively longer the underlying through the option leg. To remain delta-neutral, the dealer must sell more of the underlying.

Conversely, when IV crashes, as it typically does after an event, OTM put deltas decay toward zero. The dealer's effective long exposure from the puts evaporates. The dealer was holding an extra short hedge against the puts. It now holds more of that hedge than it needs, so it buys some back.

The unwind, with full arithmetic

The FOMC is the Federal Reserve committee that sets US interest rates. Suppose SPX is at 4,500. A dealer is short 5,000 SPX OTM puts at strike 4,400, sold to an institutional hedger who bought protection ahead of the FOMC meeting. The contracts have 30 days to expiration and, before the meeting, an IV of 25% at their strike, with a 4% rate and a 1.5% dividend yield.

Treat these numbers as orders of magnitude. They come from a standard BSM calculator at the stated parameters. Vanna and delta vary continuously with all inputs, so verify them with your own calculator. The load-bearing point is the scale: tens of thousands of share-equivalents per 5,000-contract book.

Running that check teaches something: repricing at the stated inputs gives a delta of −0.352 at 25% and −0.308 at 18%, a change of 0.044 rather than 0.035. The linear estimate falls short because vanna itself grows as IV falls, from about −0.0045 per point at 25% to −0.0088 at 18%. The repriced trade is about 22,200 share-equivalents, so the vanna estimate understates the unwind by about a fifth, though the order of magnitude stands.

Across the entire dealer community, the aggregate flow scales with the size of the book. A 500,000-contract aggregate put book produces several hundred million to over a billion dollars of mechanical buying flow in the hour after the announcement. Scaled one for one from the example, 500,000 contracts would buy 1.75 million share-equivalents, about $7.9 billion at SPX 4,500. The range is well below that, as it would be for an aggregate book spread over strikes and expiries whose average put carries far less vanna than a 30-day put 2% out of the money.

The aggregate is the post-FOMC ramp that has nothing to do with anyone reading the dot plot favorably. (The dot plot is the Fed's chart of its policymakers' projections for rates.) It is dealers reversing their pre-event hedges into the vol crush. The rally is mechanical, not interpretive.

This is the standard FOMC pattern: vol bids up into the event, because hedgers position for it. IV crushes after the event regardless of its direction, because the event is now in the past. Dealers then buy back hedge stock as their put deltas decay. The most this typically contributes to the SPX move after an FOMC decision is 0.3% to 0.5%. It operates, though, with high reliability across years.

3 · Charm: the Monday gap

Charm, briefly

Charm is the rate of change of delta with respect to time. Even with spot and IV frozen, delta drifts, because the option's effective probability of finishing in the money (ITM) evolves as the expiration window shrinks. Chapter 5D gives the Black-Scholes formula, which a call and a put share when there are no dividends:

The formula gives charm per year, so divide by 365 for the drift per calendar day. For a put struck well below spot, charm measured with time running forward is positive: the put's delta rises toward zero as the days pass. Charm is small far from expiration and grows like $1/T$ as expiration approaches. It also operates on calendar time, including weekends, even though markets are closed. This is the technical reason for the empirical regularity of Monday gaps and the "charm bid" into OPEX Fridays, the Fridays on which listed options expire.

The weekend, with full arithmetic

Charm scales with $1/T$ and runs on calendar time, not trading time. Take a 30-day OTM put with a delta of −0.25 at Friday's 4:00 PM ET close, its remaining $T$ measured in calendar days. Friday's close to Monday's 9:30 AM open is 65.5 hours of calendar time, 2.73 days and roughly 9% of the option's remaining life, although markets were open for none of those hours.

The example does not state the implied volatility, and the result depends on it. With no rates or dividends, a put struck at 97% of spot has a delta of −0.25 at an IV of about 16%. Black-Scholes then gives a charm of about 0.0033 a day, a Monday delta of about −0.240 and a trade of about 97,000 share-equivalents. That is the same order of magnitude as the example's 0.004 a day and 110,000.

The drift is small per contract. But a dealer carrying 100,000 such contracts opens Monday over-hedged by enough share-equivalents to push the SPX morning open up by a few basis points to a tenth of a percent. Aggregated across the entire dealer community, the buying flow becomes meaningful. These flows of mechanical buying, about 100,000 share-equivalents per book, hit the tape in the first 30 minutes of Monday trading. They explain a meaningful chunk of the "Monday gap-up" anomaly that has been documented in SPX since the 1990s. The bull does not show up on Monday; the hedger does.

Calendar time against trading time

There is a subtle technical issue here: some option pricing models use calendar time for charm, with $T$ measured in 365-day years. Others use trading time, with $T$ measured in 252-day years of trading days. The choice affects the size of the weekend drift: calendar-time models predict larger weekend charm drift than trading-time models. On a pure trading clock, Saturday and Sunday add no time at all, so the delta at Monday's open is still Friday's. The drift then arrives only as Monday's session runs.

Empirically, the Monday-gap-up anomaly is more consistent with calendar-time charm decay than with trading-time decay. That is why most dealer risk systems use calendar-time conventions for short-dated positions. It is a pedantic detail, but the kind of pedantic detail that separates dealers who correctly anticipate Monday morning flows from those who do not.

4 · The rest of the family, in the book

Chapter 5D defines the rest of the Greek menagerie, which keeps going beyond vanna and charm. Here is the operational summary of when each matters in a hedged book:

• Volga, also called vomma, is the second derivative of value with respect to volatility: vega's convexity. It matters most for vol-of-vol risk and for VIX options. In the Dealer Mechanics Lab's reading it played a major role in Volmageddon (Chapter 8H), when short positions in the VIX call wing had massively negative vomma. As the VIX spiked, their convex losses overwhelmed the linear vega losses. The published accounts of that day document a different channel, the rebalancing of leveraged and inverse VIX products in VIX futures.
• Speed, $\partial\Gamma/\partial S$, is the third derivative of value with respect to spot. In 0DTE options, speed is enormous near the strike. It explains why a 1% move can flip a book of options from heavily long to heavily short gamma within minutes, as Chapter 5D works through. That is the source of the "gamma flipping" intraday character that makes 0DTE risk management qualitatively different from longer-dated risk management (Chapter 9B).
• Color, $\partial\Gamma/\partial t$, is the rate at which gamma changes over time. For at-the-money (ATM) options, color is positive near expiration: gamma compresses into the ATM strike as expiration approaches. By the final hour of OPEX Friday, virtually all the dealer gamma is concentrated within $5 to $10 of spot either way. This is the mathematical source of pin risk (Chapter 5F): the gamma concentration is what creates the gravitational pull toward strikes.
• Veta, $\partial\nu/\partial t$, is the rate at which vega decays over time. It is mostly relevant for managing the time decay of long-dated positions where vega is the dominant exposure. A vol-targeting fund running long VIX call positions as crash insurance worries about veta in a way that retail traders rarely do.
• Zomma, $\partial\Gamma/\partial\sigma$, is the rate at which gamma changes with volatility. It shows up in the risk management of large vega positions and in the dynamics of the volatility surface.

For understanding aggregate dealer flow, vanna and charm cover roughly 95% of the dealer-relevant higher-order dynamics. The rest get bookkeeping attention from risk managers but rarely move the broad tape on their own. If you understand vanna and charm, you understand most of what is published about "dealer flows" with any analytical rigor, and the rest is mostly noise.

Chapter 5D puts the same roughly 95% on five Greeks rather than two: vanna, charm, volga, speed and color. Its reason is that all five matter operationally for understanding dealer flow, since volga drives losses in a volatility spike while speed and color reshape gamma on expiry day. The reason for two here is that the other three rarely move the broad tape on their own. The two counts disagree, and both are rough estimates.

5 · The five Greeks, drawn for the hedged book

This chapter reads Chapter 5D's figure for its own two books. The figure draws the chosen Greek against spot from 70 to 130 for an option struck at 100, with a 4% rate and no dividends. Buttons pick the Greek, grouped by what it differentiates: delta (vanna, charm), vega (volga) and gamma (speed, color). Two sliders set the days to expiry (5 to 120, starting at 30) and the implied vol (10% to 60%, starting at 25%). A heatmap view draws the chosen Greek over spot and over 1 to 90 days to expiry at once, amber where positive and red where negative, and switches the days slider off. Charm and color are measured per year.

Vanna and charm depend on spot only through its ratio to the strike, so the figure's strike of 100 stands in for both of this chapter's puts. The FOMC put, strike 4,400 with SPX at 4,500, sits near spot 102 on the figure's axis. The weekend put, struck at 97% of spot, sits near 103. Both are to the right of the strike, where spot is above it. Pick vanna and read it near 102, then lower the IV to 18%. Pick charm at the starting settings and drag the days to expiry from 30 down to 5. Back at 30 days, set the IV to 17% and read charm near 103. Finish on color, then switch to the heatmap.

∂Δ vanna charm

∂ν volga

∂Γ speed color

view line heatmap (spot × DTE)

days to expiry

30d

implied vol σ

25%

Greek	Vanna
Definition	∂Δ/∂σ = ∂ν/∂S
What it means	How delta changes when IV moves. Drives the post-FOMC vol-crush rally.
Where it peaks	Largest for moderately OTM options. Sign asymmetric across the strike.

Near spot 102 at the starting settings, vanna reads about −0.48 per unit of volatility, or −0.0048 per volatility point, close to the FOMC example's −0.005. At 18% it deepens to about −0.91, which is why the linear estimate fell short.

Here is what to look for as you slide. Vanna changes sign at the strike: positive on the OTM-call side, where spot is below the strike, and negative on the OTM-put side, where spot is above it. This sign asymmetry is what makes vanna flow direction-dependent. Rising IV makes a dealer short downside puts sell, but makes a dealer short upside calls buy, because an OTM call's delta rises with IV.

Charm is largest a short way either side of the strike rather than at it, about −1.7 a year near spot 93 and +1.2 near 107 at the starting settings. Drag the days to expiry down to 5 and both peaks grow five to seven times larger and crowd within 3 points of the strike: charm is the mathematics of "pinning". Back at 30 days with the IV at 17%, charm near spot 103 reads about +1.0 a year, about 0.003 a day, the same order as the weekend example's 0.004.

Although the figure does not draw gamma itself, color shows gamma's growth into the strike. Pick color: at the strike it is positive, about 0.34 a year at 30 days and about 5.0 at 5 days, so the gamma peak is growing, and faster as expiry nears. Beyond spots of about 93 and 107, color turns negative, as gamma drains out of the wings into the peak. Speed changes sign at the top of the gamma hump, just below the strike, positive below it and negative above.

The joint surface view, the heatmap, is what risk managers actually look at. A position is not risky just because of any single Greek's value. It is risky because of how all the Greeks evolve jointly as conditions change. A short straddle two days before expiration is not dangerous because of gamma alone. It is dangerous because gamma, vanna, charm and vomma are all simultaneously at their peak magnitudes. Any combination of a spot move, an IV move and time passing then activates all of them against the position at once. Chapter 5D calls the gamma, vanna and charm part of this the convexity catastrophe. Its worked straddle puts spot a point above the strike, an offset that matters: exactly at the strike, vanna and volga fade to almost nothing.

6 · Check yourself