Monte Carlo Simulation of Geometric Brownian Motion for American Option Pricing

Geometric Brownian Motion (GBM) is a fundamental stochastic process widely used for modeling stock price fluctuations in financial markets. The Monte Carlo simulation serves as a powerful computational technique for pricing financial derivatives, particularly useful when analytical solutions are unavailable. American options represent a significant category of financial derivatives that offer greater flexibility than European options due to their early exercise feature, making them applicable across broader financial scenarios. Implementing GBM-based Monte Carlo simulation for American option pricing involves generating multiple price paths using the discretized equation: S(t+Δt) = S(t) × exp((μ - σ²/2)Δt + σ√Δt × Z), where Z follows a standard normal distribution. For American options, the algorithm incorporates dynamic programming to check exercise opportunities at each time step, typically using Least Squares Monte Carlo (LSMC) with regression-based continuation value estimation. Key implementation aspects include path generation, payoff calculation, and optimal stopping decision logic using basis functions for value approximation.