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

For educational purposes only; not investment advice.

Direct answer

Section titled “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.

Definitions and conventions

Section titled “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.

Finite-difference unit check

Section titled “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.

Practical workflow

Section titled “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.

Common misconceptions

Section titled “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.
Section titled “Related topics”

Authoritative sources

Section titled “Authoritative sources”