Comprehensive Analysis of Option Pricing Methodologies

This paper conducts a rigorous mathematical and empirical analysis of three foundational option pricing methodologies: Black-Scholes, Binomial Tree, and Monte Carlo simulation. We develop a unified framework to elucidate their theoretical underpinnings, computational complexities, convergence properties, and practical performance across diverse option types and market conditions. Through detailed derivations, complexity assessments, and validations using historical market data, we delineate each model's strengths and limitations. The Black-Scholes model offers closed-form solutions for European options but struggles with volatile markets. The Binomial Tree model excels with American options despite quadratic complexity, while Monte Carlo simulation handles path-dependent options with slower convergence. We extend the analysis to recent innovations, including stochastic volatility, jump-diffusion, and machine learning integrations, providing practitioners with actionable criteria for model selection based on option characteristics, accuracy needs, and computational constraints. GitHub: https://github.com/brandonyee-cs/Option-Pricing-Model