Vanna, Charm and Vomma

The familiar Greeks change as market inputs change. Vanna, charm and vomma describe some of those changes. Their numbers only make sense with explicit volatility and time units. Different systems can display different scales or opposite charm signs for the same contract.

Vanna: delta changes with volatility

In a smooth pricing model with option value V, underlying S and decimal annualized volatility σ, vanna is ∂²V/∂S∂σ, equivalently ∂delta/∂σ. It is also the change in unscaled vega with spot when the derivatives exist. These definitions are mathematical extensions of the pricing sensitivities described in the model reference.

For a numerical estimate, suppose modeled call delta is 0.42 at 24% IV and 0.44 at 26% IV, with all other inputs fixed. Across that interval, delta changes 0.02 for a 0.02 decimal-volatility change: vanna is approximately 1.00 per unit decimal volatility, or 0.01 delta per one percentage point. For ten standard calls, a two-point move implies approximately 20 additional shares of delta under this local estimate.

Charm: delta changes as time passes

Define charm here as ∂delta/∂t, where t is elapsed calendar time. Some references instead differentiate with respect to remaining time T, which reverses the sign. If delta falls from 0.50 to 0.48 after one modeled day with spot, IV and other inputs fixed, elapsed-time charm is −0.02 delta per day over that interval.

For ten 100-multiplier calls, that change removes 20 shares of delta even without a stock move. If charm is quoted per year, convert before applying a one-day change. Near expiration, the change can be highly nonlinear; a yesterday-to-today difference is not a reliable forecast for every subsequent day.

Vomma: vega changes with volatility

Vomma, also called volga, is ∂²V/∂σ². If vega displayed per one volatility point increases from $0.10 to $0.12 per share as IV rises from 20% to 22%, the interval estimate is $0.01 per share per squared volatility point. For a standard contract, that is $1 per squared point.

Using starting vega of $10 per contract and that curvature, a two-point IV rise gives an approximate price change of $10 × 2 + ½ × $1 × 2² = $22. This is a local second-order approximation; full repricing can differ. Mixing raw-decimal vomma with percentage-point shocks creates a scaling error.

Use them as sensitivities, not signals

Hold other inputs constant when calculating finite differences and state the model, bump size and units. Multiply by signed contract quantity and quote multiplier once. These Greeks can have different signs across strikes and maturities. They do not prove the direction or size of future dealer trades, which also depend on actual positions, other hedges and execution decisions.