Cluster B: The Greeks Deep Dive
How option prices change with underlying price, time, volatility, and interest rates. From intuitive understanding to mathematical rigor.
Delta: Directional Sensitivity
Delta as rate of change, delta as probability proxy (with caveats), delta hedging intuition
Gamma: The Rate of Delta Change
Gamma as acceleration, gamma near ATM vs wings, gamma and time to expiration, gamma risk on 0DTE
Theta: Time Decay
Non-linear decay curves, theta acceleration near expiration, theta as the cost of holding optionality
Vega: Volatility Sensitivity
Vega and IV changes, vega term structure, why longer-dated options have more vega, vega vs actual vol moves
Rho: Interest Rate Sensitivity
Rho mechanics, when rho matters (LEAPS, high-rate environments), carry cost effects
Higher-Order Greeks
Vanna, charm, vomma, speed. Second and third derivatives, charm as delta decay, vanna as delta-vol cross-sensitivity
Greeks in Practice: Dynamic Hedging
Discrete hedging in real markets, hedging frequency vs transaction costs, gamma scalping mechanics
Portfolio Greeks and Net Exposure
Aggregating Greeks across positions, beta-weighted delta, portfolio-level gamma and vega exposure
Mathematical Derivation of the Greeks
Partial derivatives of BSM, closed-form Greek formulas, numerical differentiation for non-analytic models
Lambda: Option Elasticity and Leverage
Option elasticity in percent terms. How much an option return exceeds the underlying stock return, used for position sizing and leverage analysis.
Epsilon: Dividend Yield Sensitivity
Epsilon (also called Psi) measures how option prices respond to changes in dividend yield. When it matters (single-name equities, ex-dividend), when it is negligible (indices), and how it behaves as Rho's structural twin.
Veta: The Decay of Vega Over Time
Veta as the time-derivative of Vega, why LEAPS lose vol exposure faster than the calendar suggests, Veta for ATM vs wings, Veta's role in calendar spreads and VIX-targeting strategies, closed-form derivation from BSM
Vera: The Vega-Rate Cross-Sensitivity
Vera as the second-order Vega-Rate cross partial. How Vega shifts when rates move, equivalently how Rho shifts when vol moves. When Vera matters (long-dated fixed-income derivatives, multi-year LEAPS) and why it is nearly zero for short-dated options.
Zomma: Gamma's Volatility Sensitivity
Zomma as the derivative of gamma with respect to implied vol. ATM vs wing sign flip, stochastic-vol desk applications, why short-gamma books are fragile in vol spikes, closed-form Gamma*(d1*d2-1)/sigma.
Color: Gamma Decay
Color as the time-derivative of Gamma. How Gamma-hedged books evolve overnight, daily Color normalization, and why Color explodes for ATM options near expiry
Ultima: Third-Order Volatility Curvature
The third derivative of option value with respect to volatility. How Vomma changes when IV moves. Exotic pricing and extreme-scenario stress tests only.