Proofs and Derivations for the Math Journey to Black-Scholes, Greeks and Beyond

This companion volume provides complete, rigorous proofs for every result statedin "Math Journey to Black-Scholes, Greeks and Beyond: An Extended Exposition."Part I proves the measure-theoretic and stochastic-calculus foundations, fromJensen's inequality and the Radon-Nikodym theorem through the Ito integral,Ito's Lemma, and the full Black-Scholes apparatus (Feynman-Kac, Girsanov,Novikov, Martingale Representation, the Greeks, and Put-Call Parity). Part IIproves the key results of the extended models: Dupire's formula, the Hestoncharacteristic function via Riccati ODEs, the Merton pricing formula,Cox-Ross-Rubinstein convergence to Black-Scholes, the Carr-Madan FFT formula,the SABR implied-volatility expansion (leading order proved in full), and bothFundamental Theorems of Asset Pricing. Each proof is complete except in clearlymarked places where a chapter-length tool (Burkholder-Davis-Gundy, theWiener-Ito chaos decomposition, the matched-asymptotic machinery behind SABR) istaken on faith with precise references. MSC 2020: Primary 91G20; Secondary 60H30, 91G60.