Option Pricing Models

This report by Sefirot Financial Research offers a rigorous yet accessible survey of the mathematical frameworks that underpin modern derivative pricing. From the foundational logic of Black-Scholes-Merton to the stochastic volatility of Heston and Merton's jump-diffusion extension, the paper maps the full evolution of option pricing theory and examines how practitioners actually deploy these models in live markets. Rather than a textbook treatment, the focus is on the tension between theoretical elegance and empirical reality - and what that means for pricing, hedging, and risk management in practice.

Key topics covered:

• The mechanics of European and American options: payoff structures, put-call parity, and the Greeks

• Black-Scholes-Merton: assumptions, closed-form solution, and the volatility smile problem

• Extensions: Cox-Ross-Rubinstein binomial trees, Heston stochastic volatility, and Merton jump-diffusion

• Dupire's local volatility model and the Stochastic Local Volatility (SLV) framework

• How traders use models in practice: calibration, implied volatility surfaces, and Greek-based risk management

• Key takeaway: in derivatives pricing, a model doesn't need to be true to be useful - it needs to be consistently wrong in ways you can hedge