A Complete Taxonomy of Black-Scholes Greeks: Closed-Form Solutions for First, Second, and Third-Order Sensitivities

Abstract

We present the most complete unified taxonomy of Black-Scholes option price sensitivities (Greeks) available in the literature, encompassing 21 distinct measures through third order: first-order (Delta, Vega, Theta, Rho), second-order (Gamma, Vanna, Charm, Vomma, Veta, Vera, Dual Delta, Dual Gamma), third-order (Speed, Zomma, Color, Ultima, DvannaDvol, DvommaDspot), and portfolio-level (Dollar Delta, Dollar Gamma, Lambda). For each Greek we provide: (i) a fully explicit derivation from the dividend-adjusted Black-Scholes formula showing every application of the chain rule and product rule, (ii) alternative derivation paths including risk-neutral expectation differentiation and heat equation Green’s function representations, (iii) closed-form expressions for both European calls and puts, (iv) identification of put-call parity equivalences, (v) complete asymptotic analysis (deep ITM/OTM, short/long-dated, zero/high vol limits), (vi) monotonicity and convexity properties with extrema locations, (vii) dimensional analysis for practitioner interpretation, and (viii) practical trading context. We derive the Black-Scholes PDE from first principles via Itô’s lemma and the replicating-portfolio argument, establish the risk-neutral pricing connection, and prove the key symmetry lemma that simplifies every Greek derivation. The Gamma-Theta tradeoff is proved directly from the PDE with trading implications. The Vanna-Volga pricing method is derived from smile replication principles. We provide a complete treatment of sticky-strike versus stickydelta hedging conventions, delta hedging theory with continuous and discrete P&L analysis, Greeks under stochastic volatility (Heston model), numerical methods for Greeks computation (finite differences, pathwise, likelihood ratio, and adjoint algorithmic differentiation), and the behavior of Greeks near expiry including pin risk. A differential-geometric interpretation frames the Greeks as gradient, Hessian, and third-order tensor components on the six-dimensional Black-Scholes parameter manifold, with the PDE as a constraint surface. Fourth-order Greeks are derived and a convergence analysis of the Taylor price expansion justifies the third-order truncation for perturbations up to 10% of spot. Publication-quality three-dimensional surface visualizations for all 21 Greeks reveal the topology of each sensitivity across its natural parameter space. Formal proofs of all 12 put-call parity equivalences, a complete 21 Greek formula reference card, and a comprehensive monotonicity and extrema table are provided as appendices. A Python implementation with numerical verification against finite differences accompanies the paper, with over 50 figures. To our knowledge, this constitutes the most comprehensive rigorously derived reference for Black-Scholes Greeks in a single document.