Vanna, Charm, and Vomma Exposure

What These Higher-Order Greeks Are

First-order Greeks (delta, vega, theta) measure sensitivity to one input. Second-order Greeks measure how those sensitivities themselves change as inputs move. Three matter most for dealer-hedging analytics:

• Vanna = d delta / d sigma = d vega / d S. How much option delta changes when implied vol moves by one vol point. Cross-derivative between spot and vol. Vanna is positive for OTM calls (rising IV pulls delta toward 0.5) and negative for OTM puts (rising IV pulls delta toward -0.5).
• Charm = d delta / d t. How much option delta changes per unit of time decay. ITM options drift toward delta = 1 as expiration approaches; OTM options drift toward delta = 0. Charm is the structural force behind end-of-week and end-of-month delta-rebalancing flows.
• Vomma = d vega / d sigma. How much option vega changes when implied vol moves. Vomma is positive for OTM options (further OTM = more sensitivity to vol changes). Vomma drives the convexity of vega: a 5-vol-point move impacts position vega differently than a 1-vol-point move would suggest.

Aggregating each across all listed strikes, weighted by open interest, yields the dealer's vanna exposure (VEX), charm exposure (CEX), and vomma exposure. These are the higher-order analogs of GEX and DEX.

Why These Matter

• Charm-driven end-of-week flows. ITM SPX calls held by dealers see delta drift toward 1 over each day of decay. The accumulated drift across a billion-dollar option book produces a daily net delta change that dealers neutralize with underlying trades. The sum of these flows over Wednesday-Thursday-Friday is the structural reason equity indexes have positive Friday-bias.
• Vanna-driven post-event flows. An IV crush event (post-earnings, post-FOMC) shifts every option's delta through vanna. If dealer book is net long vanna, the IV drop produces a positive delta shift that requires dealers to sell underlying. If net short vanna, the IV drop forces buying. This is the structural reason post-vol-crush spot moves often appear "free" in either direction.
• Vomma-driven vol-of-vol pricing. Vomma measures how vega itself responds to vol. When market makers face concentrated vega risk (high-OI vol-sensitive positions), vomma exposure tells you how second-order vega risk concentrates. This is how dealer books position into VVIX events.

Worked Example

SPY 30-day option, spot 510, strike 530 (3.9% OTM call), IV 14.5%, r=4.5%, q=1.3%:

• Delta (BSM): ~0.20
• Vanna: ~0.017 per 1-vol-point IV change (equivalently, ~1.7 per unit of sigma). A 1-point IV rise lifts delta by about 0.017, taking it from ~0.20 to ~0.217. A 5-point IV swing would shift delta by ~0.085.
• Charm: ~-0.0028/day. The contract loses about 0.28% of delta per day from time decay alone, before any spot move.
• Vomma: ~80 (per unit sigma). A 1-point IV rise raises vega by ~0.80; vega convexity matters for sized positions but is a small per-1-vol-point increment.

Now scale across the open interest. If 80,000 contracts of this strike are open and dealers are net short half (40,000 contracts = 4M shares of underlying notional):

• Aggregate vanna at this strike: 4M × 0.017 = 68K share-equivalents per vol point. A 5-point IV crush would shift dealer delta by ~340K shares - all hedged on the spot side as the IV moves.
• Aggregate charm at this strike: 4M × -0.0028 = -11.2K shares per day. Dealers see their option-book delta drift down by ~11.2K shares each calendar day, requiring continuous spot rebalancing.

How Each Pricing Model Computes These Greeks

• Black-Scholes: closed-form vanna, charm, vomma. The standard retail provider numbers come from BSM. They are reasonable approximations near ATM; they degrade further OTM where the model's flat-vol assumption fails.
• Heston: Heston vanna and vomma differ from BSM because Heston has a structural relationship between spot and vol. The Heston-implied vanna can be 20-40% larger or smaller than BSM-vanna for OTM puts in equity markets due to the negative correlation parameter.
• SABR: the SABR-implied higher-order Greeks reflect the per-expiration smile dynamics. Particularly relevant for short-tenor options where SABR captures smile better than BSM.
• Local volatility: LV-vanna and LV-charm are well-defined but differ from stochastic-vol models because LV's deterministic spot-vol relationship implies different dynamics than Heston/SABR.

The OPEX Pattern

Monthly options expiration day (third Friday) features a recurring structural pattern that vanna+charm exposure explains:

• Wednesday/Thursday before OPEX: charm acceleration on near-expiration ITM calls produces a positive delta drift on dealer books. Dealers sell underlying to rebalance. Mild downward pressure.
• OPEX Friday: charm goes vertical for at-the-money strikes (delta drifts violently toward 0 or 1 as time-to-expiry collapses). Dealers face the largest single-day delta rebalancing of the month. The post-AM-settlement period (around 11 AM ET) is when most accumulated delta unwinds.
• Post-OPEX Monday: the aggregate option book contracts as expired contracts settle. Vanna and charm exposure step-change. Many institutional vol strategies reset positions, producing flow that retail GEX-only models do not anticipate.

VEX in Conjunction With GEX/DEX

Reading vanna exposure alongside gamma and delta exposure resolves cases where each metric alone is ambiguous:

• Negative DEX + Negative VEX: dealer book is net short delta and net short vanna. An IV rise would force dealers to buy underlying (via the negative vanna mechanism) AND face widening short-delta exposure. Vulnerable to vol-up + spot-up combos.
• Positive DEX + Negative VEX: dealer book is net long delta but vol moves still produce destabilizing hedging because vanna is short. This combination often appears pre-events where put-side hedging is concentrated.
• Negative GEX + Positive Vomma: dealer hedging amplifies spot moves AND vol moves. Combined exposure of both first- and second-order vega means a vol-spike event pulls vega back convex - the regime where short-vol strategies can blow up.

Related Concepts

Vanna (Greek) · Charm (Greek) · Vomma (Greek) · Dealer Gamma Exposure · Gamma Exposure (GEX) · Dealer Delta Exposure (DEX) · IV Crush · Pricing Model Landscape

References & Further Reading

• Hull, J. C. (2018). Options, Futures, and Other Derivatives (10th ed.). Pearson. Standard reference on Greek letters in the BSM framework.
• Haug, E. G. (2007). The Complete Guide to Option Pricing Formulas (2nd ed.). McGraw-Hill. Comprehensive closed-form Greek references.
• Sinclair, E. (2013). Volatility Trading, 2nd ed. Wiley. Practitioner-oriented treatment of higher-order Greeks for vol-trading desks.
• Gatheral, J. (2006). The Volatility Surface: A Practitioner's Guide. Wiley. Cross-model treatment of vanna and vomma in stochastic-vol models.

View live SPY dealer Greek exposure surface (GEX, DEX, vanna, charm, vomma) ->

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