Option Greeks: From Delta to Vomma — A Complete Guide

Greeks are the partial derivatives of the option price with respect to various inputs. They are the primary risk management tools for options traders and market makers.

First-Order Greeks

Delta (Δ)

Delta measures sensitivity to the underlying price: ∂C/∂S.

• Call delta ranges from 0 to +1; put delta from −1 to 0.
• An ATM call has Δ ≈ 0.50. Deep ITM calls approach Δ = 1.
• Interpretation: holding one call option is approximately equivalent to holding Δ shares of the underlying.
• Hedging: to be delta-neutral, short Δ shares for each long call.

Gamma (Γ)

Gamma is the rate of change of delta: ∂²C/∂S² = ∂Δ/∂S.

• Always positive for long options (calls and puts).
• Peaks at-the-money and near expiry — this is where "gamma risk" concentrates.
• High gamma means delta changes rapidly, requiring frequent re-hedging. Scalpers love gamma; sellers fear it.

Vega (ν)

Vega measures sensitivity to implied volatility: ∂C/∂σ.

• Quoted per 1% change in implied volatility.
• Positive for both calls and puts (long volatility).
• Largest for ATM options with longer time to expiry.
• Vega is the key risk for volatility traders — long straddles are long vega, short iron condors are short vega.

Theta (Θ)

Theta is time decay: ∂C/∂T, the rate at which the option loses value as time passes.

• Typically negative for long options (they lose value each day).
• Accelerates near expiry, especially for ATM options.
• Theta and gamma have an inverse relationship — high gamma comes with high theta cost.

Rho (ρ)

Rho measures sensitivity to the risk-free rate: ∂C/∂r.

• Positive for calls (higher rates increase call value), negative for puts.
• Relatively small for short-dated options; more significant for long-dated LEAPS.

Second-Order Greeks

Vanna

Vanna = ∂Δ/∂σ = ∂ν/∂S. It measures how delta changes with volatility, and how vega changes with spot price.

Key for structured products desks — large vanna exposures arise in skew-sensitive books. A dealer short OTM puts will accumulate negative vanna, which can amplify losses in a sell-off (spot falls, IV rises, delta moves unfavourably).

Vomma (Volga)

Vomma = ∂ν/∂σ — the convexity of vega with respect to volatility. Positive for long options.

Vomma is why OTM options benefit disproportionately from volatility spikes. A long strangle is long vomma.

Charm

Charm = ∂Δ/∂T. Delta's time decay — how much delta bleeds overnight. Critical for delta-hedging books that re-hedge daily. Near expiry, ITM options charm towards ±1 while OTM options charm towards 0.

Speed and Color

Speed = ∂Γ/∂S: how gamma changes with spot. Relevant for large spot moves.

Color = ∂Γ/∂T: gamma's time decay. How fast gamma accelerates as expiry approaches.

Practical Hedging Summary