In this project, we study the Merton jump diffusion (MJD) model, originally proposed and analyzed by Robert C. Merton in 1976. While the Black-Scholes model assumes that the stock price follows a continuous stochastic process (geometric Brownian motion), real stock paths involve occasional discontinuous jumps caused by various reasons. On top of the geometric Brownian motion, the MJD model adds a compound Poisson process to cover the sudden jumps in stock prices. This modification captures not only the discrete jumping behaviors of real stocks, but also the heavy tails and skewness of the return distribution. As a result, MJD is able to produce the volatility smile that is often seen in real-world option markets. With MJD, we implement a delta hedging experiment, a volatility smile visualization, and an option pricing performance comparison with Black-Scholes on three stocks.