# Charm, Vanna, and Vomma: How Delta and Vega Change

Use Charm, Vanna, and Vomma to understand time-driven Delta drift, volatility-driven Delta change, and Vega curvature while controlling for sign and unit conventions.

Canonical: https://wiki.fcontext.com/options/charm-vanna-vomma/
Fact checked: 2026-07-22

> For educational purposes only; not investment advice.

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## Direct answer

**Charm, Vanna, and Vomma** describe how familiar first-order option sensitivities change:

- **Charm** measures how Delta changes as time passes, holding other model inputs fixed.
- **Vanna** measures how Delta changes when implied volatility changes; equivalently, how Vega changes when spot changes under smooth-model assumptions.
- **Vomma**, also called Volga, measures how Vega changes when implied volatility changes.

They are local model derivatives, not forecasts. Their greatest practical value is revealing that a position's Delta and Vega will not remain fixed after time, spot, or volatility changes.

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## Definitions and conventions

Let option value be `V(S,σ,τ,...)`, where `S` is spot, `σ` is volatility, and `τ` is time to expiration. One explicit convention is:

`Vanna = ∂²V/(∂S∂σ) = ∂Delta/∂σ = ∂Vega/∂S`

`Vomma = ∂²V/∂σ² = ∂Vega/∂σ`

For Charm, signs depend on the time variable. If defined with time remaining:

`Charm_τ = ∂Delta/∂τ`

One calendar day passing reduces `τ`, so a screen that reports daily calendar-time Delta decay may show approximately `−Charm_τ/365`. Vendors may instead define Charm directly with calendar time, use trading days, or reverse the sign.

Volatility units also differ. A derivative per `1.00` absolute volatility is 100 times the change per one percentage point. Vega may be displayed per 1 vol point while Vanna or Vomma is stored per unit volatility. Contract multipliers and position signs add more scaling. Always inspect documentation or verify by bumping inputs and repricing.

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## Finite-difference unit check

Consider hypothetical model outputs for one option. Current Delta is `0.40`. After advancing one day with spot and IV fixed, repriced Delta is `0.385`. Under this screen's daily convention, observed Delta drift is `−0.015`. With a 100-share multiplier, directional equivalent changes from `40` to `38.5` shares before other market moves.

Now hold spot and time fixed. Increasing IV from `25%` to `30%` changes Delta from `0.40` to `0.44`:

`Vanna ≈ (0.44 − 0.40) / 5 = 0.008 Delta per vol point`

If displayed Vega is `0.12` price units per vol point at `25%` IV and `0.10` at `30%` IV:

`Vomma ≈ (0.10 − 0.12) / 5 = −0.004 price units per vol-point²`

These are illustrative finite differences, not universal values. Large shocks mix higher-order effects, and changing IV while keeping the entire skew surface fixed may not resemble a real market move.

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## Practical workflow

- Save valuation timestamp, spot, forward inputs, rates, dividends, borrow, IV surface, time convention, exercise style, and model.
- Aggregate each Greek as `per-contract Greek × signed quantity × multiplier`, using the same unit convention.
- Build one-factor checks: pass one day, move spot, and move IV separately; compare reported Greeks with full repricing.
- Then combine spot, time, skew, and term-structure shocks. Cross-effects mean the sum of isolated changes is only an approximation.
- Use bid and ask values and liquidity scenarios; Greeks computed at a midpoint do not guarantee an executable hedge.
- Recalculate near expiration and around events, where Gamma, skew, jumps, and discrete exercise behavior can dominate smooth derivatives.
- For American options and discrete dividends, use a model that supports those features rather than applying European closed-form sensitivities blindly.

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## Common misconceptions

- “Charm predicts tomorrow's Delta.” It is a local time derivative under fixed inputs.
- “Vanna is always positive.” Its sign depends on option type, moneyness, time, and conventions.
- “Vomma is just another name for Vega.” Vega is first-order volatility exposure; Vomma is its curvature.
- “Two platforms should show identical values.” Models, surfaces, clocks, signs, and scaling can differ.
- “Second-order Greeks make full repricing unnecessary.” Finite moves, jumps, and surface changes require scenario repricing.
- “A precise Greek is a precise hedge.” Quotes, discrete contracts, transaction costs, and model error limit execution.

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## Related topics

- [Delta and Gamma](/options/delta-gamma/)
- [Theta](/options/theta/)
- [Vega](/options/vega/)
- [Implied Volatility](/options/implied-volatility/)

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## Authoritative sources

- [The Pricing of Options and Corporate Liabilities](https://doi.org/10.1086/260062) - Fischer Black and Myron Scholes
- [Characteristics and Risks of Standardized Options](https://www.theocc.com/company-information/documents-and-archives/options-disclosure-document) - Options Clearing Corporation
- [Options Institute](https://www.cboe.com/optionsinstitute/) - Cboe