Advanced Greeks

Advanced Greeks are refinements of the basic Greeks, offering insights into the subtle behaviors of options prices and risks. These metrics are often used in quantitative strategies to manage complex portfolios. Below are key advanced Greeks and their roles:

Sensitivity to Volatility and Higher-Order Risks

  1. Vomma (Volga)

    • Definition: Measures the rate of change of Vega with respect to implied volatility.
    • Importance: Indicates how sensitive an option's Vega is to changes in implied volatility.
    • Use Case: Evaluating risk in options with high Vega exposure.
    • Formula: ∂Vega∂σ\frac{\partial \text{Vega}}{\partial \sigma}

  2. Vanna

    • Definition: Measures the sensitivity of Delta to changes in implied volatility or the sensitivity of Vega to changes in the underlying price.
    • Importance: Useful in dynamic hedging when both price and volatility are volatile.
    • Formula: ∂Δ∂σ\frac{\partial \Delta}{\partial \sigma} or ∂Vega∂S\frac{\partial \text{Vega}}{\partial S}

  3. Zomma

    • Definition: Measures the rate of change of Gamma with respect to implied volatility.
    • Importance: Analyzes convexity risks in highly volatile markets.
    • Formula: ∂Γ∂σ\frac{\partial \Gamma}{\partial \sigma}

  4. Charm

    • Definition: Measures the rate of change of Delta with respect to the passage of time (Theta of Delta).
    • Importance: Highlights how Delta changes as expiration approaches.
    • Formula: ∂Δ∂t\frac{\partial \Delta}{\partial t}

Rate of Change of Greeks (Meta-Risks)

  1. Speed

    • Definition: Measures the rate of change of Gamma with respect to the underlying price.
    • Importance: Assesses Gamma sensitivity, crucial for dynamic hedging.
    • Formula: ∂Γ∂S\frac{\partial \Gamma}{\partial S}

  2. Color

    • Definition: Measures the rate of change of Gamma with respect to time.
    • Importance: Useful for predicting Gamma shifts as expiration nears.
    • Formula: ∂Γ∂t\frac{\partial \Gamma}{\partial t}

  3. DvegaDtime

    • Definition: Measures the rate of change of Vega with respect to time.
    • Importance: Tracks how volatility risk decays over time.
    • Formula: ∂Vega∂t\frac{\partial \text{Vega}}{\partial t}

Tail Risk Sensitivities

  1. Ultima

    • Definition: Measures the sensitivity of Vomma to changes in implied volatility.
    • Importance: Relevant for extreme moves in volatility, such as during market shocks.
    • Formula: ∂Vomma∂σ\frac{\partial \text{Vomma}}{\partial \sigma}

  2. Vera

    • Definition: Measures the sensitivity of Rho (interest rate risk) to changes in volatility.
    • Importance: Useful for understanding the interaction between interest rate changes and volatility.
    • Formula: ∂ρ∂σ\frac{\partial \rho}{\partial \sigma}

  3. Theta of Vega (Veta)

  • Definition: Measures the rate of change of Vega with respect to time.
  • Importance: Tracks how volatility sensitivity decays over time.

Applications

  • Portfolio Hedging: These Greeks help in understanding how to balance a portfolio dynamically under changing conditions.
  • Volatility Arbitrage: Advanced Greeks are essential in identifying opportunities when implied volatilities deviate from expected levels.
  • Stress Testing: By analyzing higher-order sensitivities, traders can anticipate how positions behave during extreme market conditions.