This project analyzes American option pricing through analytical solutions, numerical methods and empirical verifications. Most equity options in the US are listed and traded as American options, which allow early exercising, and incurs an early exercise premium not captured by European-styled Black-Scholes formula. The project first validates the closed-form solution for perpetual American options (American option with infinite maturity) through Monte-Carlo simulations, then implements and benchmarks multiple binomial tree models (Cox-Ross-Rubinstein, Jarrow-Rudd, and Tian) alongside the Longstaff-Schwartz Monte Carlo method for finite American options. Finally, using OptionMetrics equity option data, we extract implied-volatility smiles and incorporate the term structure of risk-free rates across multiple securities to quantify the American premium, documenting notable deviations from Black–Scholes predictions, especially for deep in-the-money puts.