Comparison of Option Price from Black-Scholes Model to Actual Values

The Black-Scholes model is a widely used method for pricing European-style options in a straightforward way, through the use of calculations and ideal market assumptions. Due to certain unrealistic ideal conditions exercised by the model, The Black-Scholes technique of pricing options may not be entirely accurate in implementation. This paper addresses these problems due to the model limitations, determining how The Black-Scholes method compares to the results when using the actual data. Using a mix of historical S&P500 data and generated normal distributions, we first calculated and graphed option prices through the Black-Scholes formulas. With the help of R, we then calculated the real option prices by using pricing formulas and the real data, with no simulated numbers. Once both sets of results were computed, the Black-Scholes method was compared to the real prices through data and graphical comparisons. The Black-Scholes model under-priced the option at the expiration date, due to the following ideal conditions the method used. The model supposes that the volatility of the option is constant over time, thus failing to realize spikes in the stock data. Another model assumption is the use of a known and constant risk-free interest rate: interest rates change rapidly over time, thus using one to analyze prices leads to incorrect results. We concluded that the model does not accurately follow the distribution of the actual stocks, and thus does not appropriately calculate option prices because of the Black-Scholes’ reliance on unrealistic market assumptions