An Empirical Comparison of BSM and Binomial Tree Models in Delta Hedging for S&P 500 ETF (SPY) Forward Options

This paper focuses on S&P 500 stocks and their options, aiming to compare the actual effects of the BSM model and the binomial tree model in delta hedging strategies. The study selects 5 call options and 5 put options (short-term options) expiring in December 2025, as well as 1 long-term call option (Option A) expiring in June 2026. Data on the closing prices of the underlying stocks and the average trading prices of the options over 20 trading days are collected from Yahoo Finance, along with the historical volatility of S&P 500 stocks calculated from the standard deviation of returns over the past 120 trading days. After calibrating the parameters for short-term options using the BSM and binomial tree models, a dynamic delta hedging strategy for Option A is constructed based on the calibration results: adjusting stock positions daily over 5 consecutive trading days, recording hedging costs, position changes, daily profits and losses, and cumulative profits and losses. The differences between the two models in hedging errors, transaction cost control, and profit and loss stability are analyzed and the impact of implied volatility and historical volatility on practical effects is discussed. The study finds that the binomial tree model exhibits smaller hedging errors in discrete trading scenarios, providing empirical evidence for model selection in options hedging.