Principle:Avhz RustQuant Option Greeks

Knowledge Sources	Black-Scholes 1973 Haug - The Complete Guide to Option Pricing Formulas Rouah - The Heston Model and Its Extensions
Domains	Derivatives, Risk_Management, Option_Pricing
Last Updated	2026-02-07 20:00 GMT

Overview

Partial derivatives of the option pricing function with respect to model parameters, measuring sensitivity of an option's value to changes in underlying factors.

Description

Option Greeks are the partial derivatives of the option price with respect to various inputs. They are essential for hedging, risk management, and understanding how option values change as market conditions evolve.

The Greeks hierarchy:

• First-order: Delta (spot), Theta (time), Vega (volatility), Rho (rate)
• Second-order: Gamma (delta-spot), Vanna (delta-vol), Charm (delta-time), Vomma (vega-vol)
• Third-order: Speed (gamma-spot), Zomma (gamma-vol), Color (gamma-time), Ultima (vomma-vol)

Greeks enable construction of delta-neutral, gamma-neutral, or vega-neutral portfolios for hedging purposes.

Usage

Use this principle after computing an option price when you need to understand risk exposures. Greeks are computed from the same pricing engine, using the same option contract and model parameters.

Theoretical Basis

For the generalized BSM model, the key first-order Greeks are:

$\mathrm{\Delta }=\frac{\partial C}{\partial S}=e^{(b-r)T}N(d_1)$

$\mathrm{\Gamma }=\frac{{\partial }^{2}C}{\partial S^2}=\frac{e^{(b-r)T}n(d_1)}{S\sigma \sqrt{T}}$

$\mathrm{\Theta }=\frac{\partial C}{\partial t}$

${𝒱}=\frac{\partial C}{\partial \sigma }=S{e}^{(b-r)T}n(d_1)\sqrt{T}$

$\rho =\frac{\partial C}{\partial r}=KT{e}^{-rT}N(d_2)$

Higher-order Greeks are partial derivatives of these first-order sensitivities:

• Vanna = d(Delta)/d(sigma) = d(Vega)/d(S)
• Charm = d(Delta)/d(t)
• Lambda = Delta * S / C (leverage/elasticity)
• Zomma = d(Gamma)/d(sigma)
• Speed = d(Gamma)/d(S)
• Color = d(Gamma)/d(t)
• Vomma = d(Vega)/d(sigma)
• Ultima = d(Vomma)/d(sigma)

Related Pages

Implemented By

• Implementation:Avhz_RustQuant_AnalyticOptionPricer_Greeks

Uses Heuristic

• Heuristic:Avhz_RustQuant_Finite_Difference_Grid_Sizing