Part V · Options · Chapter 5D

Higher-order Greeks

Vanna, charm, volga, speed and color

1 · Why the Greeks themselves move

The five primary Greeks describe first-order sensitivities: how the option price changes when each of its five inputs moves by a small amount. Each of those sensitivities also changes as the inputs move. These second derivatives matter, especially for dealer hedging at scale and across timescales. The higher-order Greeks describe how the first-order Greeks evolve with spot, time and volatility.

Chapter 5C wrote delta as $\partial V/\partial S$, the rate at which the option's value $V$ changes with spot $S$ while time and volatility stay fixed. Gamma, $\partial \Delta/\partial S$, is the rate of change of that rate. Nothing stops you from asking how delta changes with volatility or with time instead. Each pairing of a first-order Greek with an input gives a new Greek:

Vanna appears twice because, for a smooth function, differentiating by spot and then by volatility gives the same result as the other order. The rate at which delta changes with volatility therefore equals the rate at which vega changes with spot.

The full menagerie has dozens of names: vanna, charm, volga (also called vomma), speed, color, veta, vera, zomma and ultima. Five of them matter operationally for understanding dealer flow: vanna, charm, volga, speed and color. The rest are bookkeeping details that risk managers track but that rarely move the broad tape.

2 · The shared toolkit

Every Greek in this chapter is built from the same few pieces of the Black-Scholes formula of Chapter 5C. Take an option with strike $K$ and time to expiry $T$ in years, on an underlying at spot $S$, with volatility $\sigma$, risk-free rate $r$ and dividend yield $q$:

Here $\varphi(x)$ is the height of the standard normal curve at $x$, while $N(x)$ is the area under that curve to the left of $x$. The derivations below also use one fact about the bell curve, that its slope at $x$ is $-x\,\varphi(x)$: the curve rises to the left of zero and falls to the right.

The chapter follows one option throughout, which it calls the example put: spot 100, strike 95, three months to expiry ($T = 0.25$) and implied volatility of 20%. Rates and dividends are set to zero, $r = q = 0$, to keep the arithmetic short, although the figure later in the chapter uses a rate of 4%.

Both $d_1$ and $d_2$ are positive, as they are for any strike below spot, apart from $d_2$ for a strike within a hair of spot. The term $\ln(S/K)$ is positive whenever the strike is below spot, which makes $d_1$ positive. It also outweighs the small $\sigma^2 T/2$ that $d_2$ subtracts, here 0.051 against 0.005.

3 · Vanna: delta's sensitivity to volatility

Vanna is the rate of change of delta with respect to implied volatility, equivalently the rate of change of vega with respect to spot:

Its closed form, which a call and a put with the same strike share because their deltas differ by a constant (Chapter 5F), is

If implied volatility rises by one volatility point, a percentage point such as 20% to 21%, delta moves by vanna $\times\ 0.01$.

The intuition for why vanna exists is that implied volatility sets how spread out the option's pricing distribution is. A higher IV means more probability mass in the tails of the distribution. For an out-of-the-money (OTM) call, that means more chance of finishing in the money, which raises its delta toward 1.

Where the vanna formula comes from

Step 1: differentiate delta. A call's delta is $e^{-qT}N(d_1)$, where the slope of $N$ is $\varphi$. The chain rule multiplies the slope of the outer function by the slope of the inner one:

$$\frac{\partial \Delta}{\partial \sigma} = e^{-qT}\varphi(d_1)\,\frac{\partial d_1}{\partial \sigma}$$ (5D.5)

Step 2: split d1 into two terms. Write $a = \ln(S/K) + (r - q)T$ for the part of the numerator that does not contain $\sigma$, so that

$$d_1 = \frac{a}{\sigma\sqrt{T}} + \frac{\sigma\sqrt{T}}{2}$$ (5D.6)

Step 3: differentiate each term by σ. The first term goes as $1/\sigma$, whose slope is $-1/\sigma^2$, while the second is a straight line in $\sigma$:

$$\begin{aligned} \frac{\partial d_1}{\partial \sigma} &= -\frac{a}{\sigma^2\sqrt{T}} + \frac{\sqrt{T}}{2} \\ &= -\frac{1}{\sigma}\Big(\frac{a}{\sigma\sqrt{T}} - \frac{\sigma\sqrt{T}}{2}\Big) \end{aligned}$$ (5D.7)

Step 4: recognize d2. The bracket is $d_1 - \sigma\sqrt{T}$, which is $d_2$. Putting $\partial d_1/\partial\sigma = -d_2/\sigma$ into Step 1 gives (5D.4).

Out-of-the-money puts: negative vanna

For OTM puts in the downside wing, the typical institutional hedge book, vanna is negative. With the strike below spot, $d_1$ and $d_2$ are both positive, as the toolkit showed. Since $\varphi(d_1) > 0$ and $\sigma > 0$, the minus sign in (5D.4) makes the product negative.

Read the sign this way: rising IV makes the put's delta more negative, so the put becomes more sensitive to spot in the bearish direction. Falling IV, as in the post-event vol crush, pulls the put's delta back toward zero.

For a dealer who is short these puts, the effective exposure is the mirror image. As IV rises and customer-put deltas become more negative, the dealer's effective long-stock exposure increases. The dealer appears "more long", which forces it to sell stock to stay delta-neutral. As IV crushes after the event, the put deltas decay toward zero and the dealer's apparent long exposure shrinks, so it buys back hedge stock. This is the mechanism behind the pre-event vol bid and the post-event ramp.

Vanna is large and negative for OTM puts. When the VIX spikes, OTM put deltas become more negative in magnitude. A dealer who is short these puts, because customers were buying protection, becomes effectively longer the underlying through the option leg and must sell more of the underlying to remain delta-neutral. This is the vanna feedback that amplifies sell-offs: rising VIX forces dealer selling, which can amplify the SPX move that caused VIX to rise in the first place.

The same mechanism run backward drives the post-FOMC rally. The FOMC is the Federal Reserve committee that sets US interest rates. After the Fed's decision lands, IV typically collapses (vol crush) regardless of the direction of the rate decision. Dealers short puts see those put deltas decay toward zero, so their effective long-stock exposure from the puts evaporates. They were holding an extra short hedge against the puts, which they buy back now that they need less of it. That mechanical flow goes against the prior fear trade, producing the familiar post-event ramp without any new bullish news: the rally is just dealers reversing fear-driven hedges.

Chapter 7B works a full vanna example with actual dollar amounts, and Chapter 9A draws the feedback loop step by step.

4 · Charm: delta's drift with time

Charm is the rate of change of delta with respect to time:

Here $t$ is calendar time running forward, so the time left, $T$, falls as $t$ rises. Even if spot and IV are frozen, delta still drifts, because the option's effective probability of finishing in the money evolves as the expiration window shrinks. OTM options have their delta drift toward zero, since less time means less chance of finishing in the money. In-the-money (ITM) options drift toward ±1, since less time leaves spot less room to cross back over the strike.

The Black-Scholes formula for the charm of a call, with no dividends, is

With no dividends, a put's delta is the call's delta minus 1, so the put has the same charm. When the rate is also zero, (5D.9) shrinks to

Both are measured per year, so divide by 365 for the drift per calendar day. The sign follows $d_2$: when the strike is well below spot, $d_2 > 0$ and charm is positive, so a call's delta climbs toward 1 and a put's delta rises toward zero.

Where the charm formula comes from, with no rates or dividends

Step 1: differentiate delta. As for vanna, $\partial\Delta/\partial t = \varphi(d_1)\,\partial d_1/\partial t$. Since $T$ falls one-for-one as $t$ rises, $\partial d_1/\partial t = -\partial d_1/\partial T$.

Step 2: write d1 in powers of T. With $r = q = 0$ and $b = \ln(S/K)/\sigma$, the definition becomes

$$d_1 = b\,T^{-1/2} + \tfrac{\sigma}{2}\,T^{1/2}$$ (5D.11)

Step 3: differentiate by T. The slope of $T^{p}$ is $p\,T^{p-1}$, so

$$\begin{aligned} \frac{\partial d_1}{\partial T} &= -\tfrac{b}{2}T^{-3/2} + \tfrac{\sigma}{4}T^{-1/2} \\ &= -\frac{1}{2T}\Big(\frac{b}{\sqrt{T}} - \frac{\sigma\sqrt{T}}{2}\Big) \end{aligned}$$ (5D.12)

Step 4: recognize d2. The bracket is $d_2$, so $\partial d_1/\partial t = d_2/(2T)$, and Step 1 gives (5D.10).

Charm is small far from expiration and grows aggressively as $1/T$ 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 (options expiration).

A concrete case: dealers carrying lots of OTM puts overnight see those put deltas decay through the weekend, because charm runs on calendar time. They come in Monday morning holding too many hedge shares, since the puts they sold are now effectively closer to zero-delta than they were on Friday afternoon. Buying back that excess hedge is Monday-morning systematic buying, which contributes to the well-documented "Monday gap-up" anomaly. Chapter 7B includes the full charm weekend gap worked example.

Charm does not peak at the strike, where $d_2$ is close to zero and so is (5D.10). It is largest a short way either side of the strike, where $d_2$ is large enough to count but $\varphi(d_1)$ has not yet faded.

5 · Volga: vega's sensitivity to volatility

Volga, also called vomma, is the second derivative of value with respect to implied volatility, equivalently the rate of change of vega with respect to vol:

Its closed form multiplies vega, $\nu = S e^{-qT}\varphi(d_1)\sqrt{T}$, by a simple factor:

The factor comes from two slopes met already. Differentiating vega by $\sigma$ replaces $\varphi(d_1)$ with its slope, $-d_1\varphi(d_1)$, times the slope of $d_1$ in $\sigma$, which the vanna derivation found to be $-d_2/\sigma$. The two minus signs cancel.

Volga measures the convexity of the option's vega: how vega itself changes when IV changes. A long option away from the strike has positive volga, because $d_1$ and $d_2$ then share a sign. Its vega rises as IV rises, so a long option position benefits non-linearly from rising vol. At the strike, $d_1$ and $d_2$ sit either side of zero, their product is slightly negative, and volga is close to zero. Volga matters most for OTM options, where it peaks at moderately OTM strikes, and for vol-of-vol exposure.

The practical use of volga is a better estimate of what a volatility move does to a position. Vega alone gives a straight-line estimate. Adding half of volga times the square of the move bends the line to follow the curve:

Here $\delta\sigma$ is the change in implied volatility as a decimal, while $\delta V$ is the change in the option's value. Gamma corrects delta for a move in spot with the same two-term shape: the slope, plus half the curvature times the square of the move.

Operationally, volga is the reason that vol-targeting strategies and variance swaps have non-trivial dynamics during stress. A vol-targeting strategy scales its exposure to hold its volatility near a target. As vol rises, the volga of long-vol positions amplifies their gain, or the loss of short-vol positions. The Dealer Mechanics Lab reads Volmageddon, in February 2018, as in part a volga story. In its account, the dealers short the VIX call wing had massively negative volga, so as the VIX rose, their losses accelerated faster than vega alone would predict. The published accounts of that day document a different channel, the rebalancing of leveraged and inverse VIX products in VIX futures (Chapter 8H).

6 · Speed: gamma's sensitivity to spot

Speed is the rate of change of gamma with respect to spot, equivalently the third derivative of option value with respect to spot:

Its closed form, found the same way as volga's, is

Speed describes how quickly gamma itself shifts as the underlying moves. Gamma has a hump near the strike (Chapter 5C), so speed is positive below the hump, where gamma rises as spot climbs, and negative above it. For the example put, speed is $-(0.0340/100)(0.563/0.10 + 1) = -0.0023$, so a one-point rise in spot would take its gamma from 0.0340 to about 0.0317.

For 0DTE options, speed is enormous near the strike. A 1% move in spot can flip the position from heavily long-gamma to heavily short-gamma (or the reverse) within minutes. A single long option's gamma never turns negative, so the flip happens in a book that holds options at more than one strike. This is the source of the "gamma flipping" intraday character of 0DTE. Dealer positioning that is stable across the morning can become unstable in the afternoon, as spot moves through high-OI strikes, those with large open interest, and speed re-shapes the gamma profile.

Speed is also why explicit 0DTE risk management is much harder than risk management on longer-dated positions. The Greeks at any given moment understate the speed at which they themselves can change.

7 · Color: gamma's drift with time

Color is the rate of change of gamma with respect to time:

As with charm, $t$ runs forward, so color is the change in gamma as time passes. With no dividends, its closed form is

When the rate is zero, $k$ is just $-d_2$, and the formula becomes

The sign is now easy to read. Color is positive when $d_1 d_2 < 1$, which happens near the money. It is negative when $d_1 d_2 > 1$, which happens far from the strike, where $d_1$ and $d_2$ are both large and share a sign.

Color describes how gamma decays, or grows, as expiration approaches. For at-the-money (ATM) options, color is positive: gamma grows as expiration approaches, which is the $1/\sqrt{T}$ growth in the gamma formula. For deep OTM or ITM options, color is negative: gamma decays as expiration approaches, because the option becomes either definitely zero or definitely the underlying.

Color is the technical reason gamma "compresses" into the ATM strike as expiration approaches. As $T$ shrinks, the gamma peak at the money grows taller while the wings of the gamma profile shrink. By the final hour of trading on OPEX Friday, virtually all the dealer gamma is concentrated within $5 to $10 of spot. This is the mathematical source of pin risk (Chapter 5F).

Color and charm are tied, because the order of differentiation does not matter. Color, the change of $\partial\Delta/\partial S$ with time, equals the change of charm with spot, so color is zero exactly where charm peaks.

8 · The five, drawn together

The figure draws each of the five against spot, for an option struck at $K = 100$ with a rate of 4% and no dividends. Its 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%). The heatmap view draws the chosen Greek over spot and over 1 to 90 days to expiry at once, amber where it is positive and red where it is negative. Its horizontal axis varies spot, with the strike held at 100.

A table under the chart gives each Greek's definition, its meaning and where it peaks. The figure measures charm and color per year. Pick vanna and find where it changes sign, then pick charm and drag the days to expiry from 30 down to 5. After that, pick volga and look at the strike before switching color 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.

Vanna crosses zero at the strike, positive where spot is below the strike (the OTM-call side) and negative above it (the OTM-put side). At the starting settings it reaches about +1.0 near spot 93 and −0.9 near spot 107. Charm is largest a short way either side of the strike rather than at it: about −1.7 per year near spot 93 and +1.2 near spot 107, against −0.2 at the strike. At 5 days to expiry the two charm peaks are five to seven times larger and sit within 3 points of the strike.

Volga has two humps, at call deltas of about 8 to 12 and 92 to 95, and almost nothing at the strike. Those are the OTM calls of about 8 to 12 delta and the OTM puts of about 5 to 8 delta. Color peaks at the strike, at about 0.34 per year at 30 days and about 5.0 at 5 days. At 30 days it turns negative beyond spots of about 93 and 107, the same spots where charm peaks. On the heatmap, both crowd in toward the strike as the days run down. Chapter 5C's Greek surface draws vanna and charm as well, as wireframes over spot and 1 to 90 days to expiry, which gives a third view of the same shapes.

9 · The rest of the family

• Veta, $\partial\nu/\partial t$, the rate of change of vega with respect to time, is mostly relevant for managing the time decay of long-dated positions where vega is the dominant exposure.
• Zomma, $\partial\Gamma/\partial\sigma$, the rate of change of gamma with respect to volatility, shows up in the risk management of large vega positions.
• Vera, $\partial\rho/\partial\sigma$, the rate of change of rho with respect to volatility, is mostly bookkeeping. Here $\rho$ stands for the Greek rho, not a correlation.
• Ultima, $\partial\,\text{volga}/\partial\sigma$, the third derivative of value with respect to volatility, is used in vol-of-vol-of-vol management.

The names start to feel made up after a point. For this book's purposes, if you know vanna, charm, volga, speed and color, you have covered roughly 95% of the higher-order dynamics that matter for understanding aggregate dealer flow. The rest can wait until you meet a specific situation that requires them.

10 · Greeks through time

A position's actual behavior comes from how the first-order and higher-order Greeks interact as spot moves and time passes. As expiration approaches, a long ATM call's gamma sharpens into a spike at the strike, which mathematicians call a delta function. At the same moment, its charm and vanna magnitudes explode just either side of the strike. Static views of one Greek mislead, because positions live or die on the joint motion of all of them.

The "convexity catastrophe", a term used in dealer-mechanics circles, is the joint amplification of negative gamma, vanna and charm against short-option positions in the final days before expiration. A short straddle two days before expiration is not dangerous because of gamma alone. It is dangerous because gamma, vanna and charm are all at their peak magnitudes at once. Any combination of a spot move, an IV move and time passing then activates all three against the position together.

11 · Check yourself