Lesson 8: Greeks Beyond the Basics

Promise: Understand vanna, volga, and charm: the second-order sensitivities that explain why your P&L doesn't match your Greeks.

Why More Greeks?

In Options Explainers, you learned the Big Four: delta, gamma, theta, vega. These are first-order sensitivities: how your option price changes when one variable moves.

But these Greeks themselves change. Delta changes when spot moves (that's gamma). Vega changes when vol moves. Delta changes as time passes. These second-order effects are the advanced Greeks.

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First-order Greeks tell you your exposure. Second-order Greeks tell you how that exposure will change.

The Advanced Greeks Map

Greek

What It Measures

Derivative Of

Vanna

How delta changes with vol

∂Δ/∂σ or ∂ν/∂S

Volga (Vomma)

How vega changes with vol

∂ν/∂σ

Charm

How delta changes with time

∂Δ/∂t

Veta

How vega changes with time

∂ν/∂t

Speed

How gamma changes with spot

∂Γ/∂S

Color

How gamma changes with time

∂Γ/∂t

We'll focus on the three most important: vanna, volga, and charm.

Vanna: Delta's Sensitivity to Vol

Vanna measures how your delta exposure changes when implied volatility moves.

$$\text{Vanna} = \frac{\partial \Delta}{\partial \sigma} = \frac{\partial \nu}{\partial S}$$

Intuition

Think about an OTM call with delta = 0.20. If vol increases, there's a higher probability it ends up ITM. So delta increases. That's positive vanna.

Vanna: How Delta Changes with Vol

Reference: Vol = 50%

Current: Vol = 50%

Implied Volatility: 50%

Key insight: At baseline vol. Adjust the slider to see how delta changes across strikes when vol moves.

Option Type

Vanna Sign

What It Means

OTM Call

Positive

Delta increases as vol rises

OTM Put

Negative

Delta (more negative) increases magnitude as vol rises

ATM

~Zero

Delta relatively stable around 0.50

ITM

Opposite of OTM

Delta moves toward 1 or -1

Why Vanna Matters

1. Spot-vol correlation effects: When spot drops and vol spikes (negative correlation), vanna creates additional delta exposure
2. Hedging: Your delta hedge becomes wrong when vol moves
3. Pin risk: Near expiry, vanna effects can be large

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If you're long OTM options and vol spikes, you suddenly have more delta than you thought.

Volga (Vomma): Vega's Sensitivity to Vol

Volga (also called Vomma) measures how your vega exposure changes when vol moves.

$$\text{Volga} = \frac{\partial \nu}{\partial \sigma} = \frac{\partial^2 V}{\partial \sigma^2}$$

Intuition

Volga is the "gamma of vega." Just like gamma makes your delta position bigger as spot moves in your favor, volga makes your vega position bigger as vol moves.

Volga: How Vega Changes with Vol

Low Vol (40%)

Current: 50%

High Vol (70%)

Implied Volatility: 50%

Wing convexity: Notice how OTM options (wings) gain more vega in percentage terms when vol rises. ATM vega stays relatively stable, but wing vega explodes. This is why OTM options are convex bets on volatility.

Option Type

Volga

What It Means

ATM

Low/Zero

Vega relatively stable

OTM (wings)

High Positive

Vega increases as vol rises

Deep OTM

Highest

Most convex vega profile

Why Volga Matters

1. Wing options are convex in vol: OTM options benefit disproportionately from vol spikes
2. Vol-of-vol exposure: High volga means you're exposed to volatility of volatility
3. Smile trading: Volga is why wing options command a premium

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Wing options have high volga. When vol explodes, their vega explodes too. They're convex bets on vol.

Charm: Delta's Sensitivity to Time

Charm measures how your delta changes as time passes, holding everything else constant.

$$\text{Charm} = \frac{\partial \Delta}{\partial t}$$

Intuition

As expiry approaches, OTM options become less likely to end up ITM (delta decreases toward 0), while ITM options become more certain (delta increases toward 1 or -1). Charm captures this drift.

Charm: How Delta Changes with Time

Days to Expiry: 30

Days to Expiry: 30

Medium term: Delta curve is moderately sloped. Charm effects are present but manageable. OTM options still have meaningful delta that will decay as expiry approaches.

Option Position

Charm Effect

Delta Drift

OTM Call

Negative charm

Delta drifts toward 0

ITM Call

Positive charm

Delta drifts toward 1

ATM Call

Small/variable

Delta stays near 0.5 until close to expiry

Why Charm Matters

1. Delta hedging costs: Your delta hedge needs constant adjustment as time passes
2. Weekend decay: Charm effects accumulate over weekends
3. Near-expiry dynamics: Charm accelerates dramatically as expiry approaches

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Charm is why delta hedging is not "set and forget." Your hedge drifts even if spot doesn't move.

How These Greeks Interact

The advanced Greeks don't exist in isolation. In real markets:

Vol Spike Scenario

Spot drops 5%, vol spikes 15 points:

1. Delta: Increases (you're shorter if you were long calls)
2. Vanna effect: Additional delta change from vol spike
3. Gamma effect: Delta changed from spot move
4. Vega: Your vol exposure increased (if long options)
5. Volga effect: Vega itself increased because vol is higher

Your actual P&L is the sum of all these effects.

Time Decay Scenario

Weekend passes, nothing moves:

1. Theta: Time decay (expected)
2. Charm: Delta drifted (need to re-hedge)
3. Veta: Vega exposure changed

Portfolio-Level View

For complex portfolios, you don't track each option's Greeks. You aggregate:

Greek	Portfolio Reading	Interpretation
Net Vanna	+500	Delta will increase 500 per 1% vol rise
Net Volga	+200	Vega will increase 200 per 1% vol rise
Net Charm	-300	Delta will decrease 300 per day

This tells you how your portfolio's risk profile will evolve.

Common Mistakes

Mistake	Correction
Ignoring vanna when vol spikes	Your delta hedge is wrong after vol moves. Re-hedge.
Not understanding why wings outperform in vol spikes	It's volga. Wings have convex vega.
Forgetting charm over weekends	Delta drifts even with no spot move.
Treating Greeks as static	They're all functions of spot, vol, and time.
Over-complicating	You don't need to track all 20 Greeks. Focus on vanna, volga, charm.

For Black-Scholes:

Vanna:

$$\text{Vanna} = \frac{\nu}{S} \left( 1 - \frac{d_1}{\sigma\sqrt{T}} \right) = -e^{-qT}\phi(d_1)\frac{d_2}{\sigma}$$

Volga:

$$\text{Volga} = \nu \frac{d_1 d_2}{\sigma}$$

Charm:

$$\text{Charm} = -e^{-qT}\phi(d_1)\left( \frac{2(r-q)T - d_2\sigma\sqrt{T}}{2T\sigma\sqrt{T}} \right)$$

Where $\phi$ is the standard normal PDF, and $d_1$, $d_2$ are the usual Black-Scholes terms.

Test your understanding before moving on.

Q: What does vanna measure?

Q: Why do wing (OTM) options have high volga?

Q: What is charm and when does it matter most?

💡 Tip: Try answering each question yourself before revealing the answer.

See Also

• Vanna Reference
• Volga Reference
• Charm Reference
• Lesson 6: Surface Dynamics
• Lesson 8: Reading Your Greeks →

Navigation: ← Lesson 7: Surface Dynamics | Lesson 9: Reading Your Greeks →