What does this research mean for the field?

Modern option pricing, from the foundational Black-Scholes model to advanced stochastic and local volatility alternatives, can be systematically derived and understood through a rigorous, self-contained mathematical framework. Novelty: ClaimNovelty.SYNTHESIS. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To provide a self-contained, mathematically rigorous derivation of the Black-Scholes option pricing model and evaluate modern advanced alternatives that address its limitations.

September 8, 2026Open Access

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

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Authors

YDY Diawara

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Implication

Theoretical exposition demonstrates complete derivations of classical and modern option pricing frameworks, highlighting mathematical solutions to overcome Black-Scholes model limitations.

Key Points

To provide a self-contained, mathematically rigorous derivation of the Black-Scholes option pricing model and evaluate modern advanced alternatives that address its limitations.
Applies measure-theoretic probability, martingales, and stochastic calculus, including the Itô integral and Itô's Lemma, to formulate asset dynamics.
Derives analytical pricing solutions via the Feynman-Kac theorem, Girsanov's change of measure, and the Fundamental Theorems of Asset Pricing.
Evaluates extensions to Black-Scholes, including Dupire local volatility, Heston stochastic volatility, Merton jump-diffusion, and SABR models alongside numerical pricing methods.
Establishes a unified theoretical framework for the Greeks, implied volatility surfaces, exotic options, and dynamic hedging under realistic market conditions.
Identifies the empirical and theoretical shortcomings of constant volatility assumptions in classical Black-Scholes pricing.
Demonstrates how modern local-stochastic volatility and jump-diffusion models resolve volatility smiles and model risk in derivative pricing.

Cite This Study

Y Diawara (2026) studied this question.

synapsesocial.com/papers/6a9fd75858e84d0ff5b45f1a https://doi.org/10.5281/zenodo.22499298

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Implication

Theoretical exposition demonstrates complete derivations of classical and modern option pricing frameworks, highlighting mathematical solutions to overcome Black-Scholes model limitations.

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

Discussion

Member takes

Implication

Key Points

Cite This Study

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

Discussion

Member takes

Implication

Key Points

Cite This Study

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Also Consider