An Empirical Comparison of Black-Scholes and Binomial Tree Models in Pricing US Technology Stock Options

Abstract

Option pricing is an important problem in quantitative finance because an option's value depends on the underlying stock price, volatility, interest rates, and time to maturity. For European call options, the Black–Scholes (B–S) closed-form formula and binomial tree model represent two mainstream pricing approaches, yet their pricing accuracy diverges substantially when calibrated against real-world market data. This paper compares the two models by pricing at-the-money European call options with about six months to expiration on four US technology stocks: Apple (AAPL), Microsoft (MSFT), NVIDIA (NVDA), and Alphabet (GOOGL). Using daily stock prices from Yahoo Finance and risk-free rate data from FRED, this paper estimates historical volatility from daily log returns, calculates theoretical prices under both models, and compares those prices with observed market premiums. This paper also tests how quickly binomial tree prices approach B-S prices as the number of time steps increases. The results show that the binomial tree price converges to the B-S price with a convergence rate of O(1/N). This research further finds that both models produce similar pricing errors for more liquid options, while pricing errors increase when historical volatility differs from market-implied volatility.