Math Journey to Black-Scholes, Greeks and Beyond: An Extended Exposition

Abstract

This is a self-contained, graduate-level exposition tracing the path fromelementary probability to modern option pricing. Every model is built from itsmathematical foundations, every formula is derived, and every assumption isstated and examined. Part I develops the complete theoretical framework for Black-Scholes pricing:measure-theoretic probability, stochastic calculus (filtrations, martingales,the Ito integral, Ito's Lemma), and the derivation of the Black-Scholes modelvia the Feynman-Kac theorem, Girsanov's change of measure, and the FundamentalTheorems of Asset Pricing. It then treats the Greeks, implied volatility and thevolatility surface, exotic options, Put-Call Parity, and hedging in realisticmarkets. Part II addresses the limitations of Black-Scholes and surveys the modernalternatives: local volatility (Dupire), stochastic volatility (Heston),jump-diffusion (Merton), the main numerical pricing methods, interest-ratevolatility (SABR), local-stochastic volatility, and model risk. The expositionincludes worked numerical examples, Python listings, 13 figures, and a notationindex. A companion volume provides complete proofs for every result stated here. MSC 2020: Primary 91G20; Secondary 60H30, 91G60.