American options are priced using the Cox-Ross-Rubinstein (CRR) binomial tree model. To verify numerical consistency, the model output is compared against analytical solutions from the Black-Scholes-Merton (BSM) formula for European options and the Barone-Adesi & Whaley (BAW) approximation for American options. As the step count increases, the CRR tree converges to the BSM price for European contracts. For American contracts, tree prices agree with the BAW approximation to within a 0.5% relative error, which represents the threshold where the BAW approximation degrades. The CRR model is then used to calculate implied volatility smiles for SPY, QQQ, and GLD. For GLD, which has no dividends, the calculated volatility smile matches reference market implied volatilities. For SPY and QQQ, the calculated smiles show numerical issues near at-the-money puts, where discrepancies occur because dividends are modeled as continuous yields rather than discrete cash flows.