higher-order greeks

Beyond standard primary Greeks (Delta, Gamma, Vega, Theta, Rho), modern options market microstructure and dynamic dealer hedging models rely on higher-order cross-derivatives.

Standard Black-Scholes Definitions

d1 = [ln(S / K) + (r + 0.5 * σ²) * T] / (σ * √T)
d2 = d1 - σ * √T
ϕ(x) = (1 / √(2π)) * exp(-0.5 * x²)
Vega (V) = S * √T * ϕ(d1)
Gamma (Γ) = ϕ(d1) / (S * σ * √T)

Secondary Greeks (Second-Order Cross-Derivatives)

1. Vomma (Volga)

• Unit: (Δ Vega / 1% vol)
• Definition: Rate of change of Vega with respect to implied volatility:

  Vomma = ∂V/∂σ = ∂²V/∂σ² = V * (d1 * d2 / σ)

• Intuition: Measures the convexity of Vega. Long Vomma positions expand Vega as volatility rises, generating convex returns during volatility spikes.

2. Vanna (DvegaDspot)

• Unit: (Δ Delta / 1% vol) or (Δ Vega / Δ $1 spot)
• Definition: Cross-derivative of Delta with respect to volatility (or Vega with respect to spot):

  Vanna = ∂Δ/∂σ = ∂V/∂S = -ϕ(d1) * (d2 / σ) = -V * d2 / (S * σ * √T)

• Intuition: Quantifies how dealer delta-hedging requirements shift as implied volatility expands or collapses. Vanna is positive above spot (K > S) and negative below spot (K < S).

3. Charm (Delta Decay)

• Unit: (Δ Delta / day)
• Definition: Rate of change of Delta over the passage of time (where τ = T - t):

  Charm_call = -∂Δ_call/∂τ = -ϕ(d1) * [(2 * r * T - d2 * σ * √T) / (2 * T * σ * √T)]

• Intuition: Quantifies passive Delta drift as expiration approaches. For OTM options, Delta bleeds toward 0; for ITM options, Delta pulls toward 1.0 (calls) or -1.0 (puts).

4. Vera (Rhova)

• Unit: (Δ Rho / 1% vol) or (Δ Vega / 1% rate)
• Definition: Sensitivity of Vega to interest rate changes:

  Vera = ∂V/∂r = ∂ρ/∂σ = -K * T * exp(-r * T) * ϕ(d2) * (d1 / σ)

• Intuition: Important for long-dated options (LEAPS) where interest rate shifts impact volatility sensitivity.

Tertiary Greeks (Third-Order Derivatives)

1. Zomma

• Unit: (Δ Gamma / 1% vol)
• Definition: Sensitivity of Gamma with respect to implied volatility:

  Zomma = ∂Γ/∂σ = Γ * [(d1 * d2 - 1) / σ]

• Intuition: Predicts whether a market maker's gamma profile will flatten out across strikes or concentrate tightly around ATM strikes as volatility changes.

2. Speed

• Unit: (Δ Gamma / Δ $1 spot)
• Definition: Rate of change of Gamma with respect to changes in underlying spot price:

  Speed = ∂Γ/∂S = ∂³V/∂S³ = -(Γ / S) * [d1 / (σ * √T) + 1]

• Intuition: Measures the asymmetry and rate of change of Gamma acceleration. Essential for assessing tail risk and gap vulnerability.

3. Color (Gamma Decay)

• Unit: (Δ Gamma / day)
• Definition: Rate of change of Gamma over the passage of time:

  Color = -∂Γ/∂τ = -[ϕ(d1) / (2 * S * T * σ * √T)] * [2 * r * T - d2 * σ * √T * (1 - d1 / (σ * √T))]

• Intuition: Measures how rapidly gamma concentrates into an ultra-sharp spike around ATM strikes as time expires (especially critical in 0DTE trading).

4. Ultima

• Unit: (Δ Vomma / 1% vol)
• Definition: Third derivative of option price with respect to implied volatility:

  Ultima = ∂Vomma/∂σ = ∂³V/∂σ³ = -(V / σ²) * [d1 * d2 * (1 - d1 * d2) + d1² + d2²]

• Intuition: Measures the stability and convexity of Vomma when subject to massive volatility shocks.