Pricing European Call Options Under Constant Volatility Using the Black–Scholes Model

Author

Chukwudi Anderson Ugomma

Abstract

This study empirically evaluates the pricing performance of the calibrated Black–Scholes option pricing model using European call options written on the S&P500 Weekly Index (SPXW). Market data comprising SPXW option contracts were obtained from the Chicago Board Options Exchange (CBOE). Following comprehensive data screening and cleaning, 49 complete option contracts were retained for empirical analysis. The observed market price of each option was computed as the midpoint of the bid and ask quotations, while theoretical prices were generated using the Black–Scholes model calibrated with market-derived implied volatilities and the adopted valuation inputs. Model performance was assessed using the Root Mean Square Error (RMSE), Mean Absolute Error (MAE), Mean Pricing Bias, Pearson correlation coefficient, and coefficient of determination.The empirical results showed that the calibrated Black–Scholes model achieved an RMSE of 128.60, an MAE of 42.2 and a mean pricing bias of −3.33, indicating relatively small average pricing errors and only a slight tendency to underprice observed market values. The model further exhibited a strong positive relationship between theoretical and observed prices, with a Pearson correlation coefficient of 0.9532 and a  value of 0.8982, implying that approximately 89.82% of the variation in observed option prices was explained by the calibrated model. Contract-level analysis showed that pricing accuracy was generally high across the retained contracts, although relatively larger deviations occurred for deep in-the-money and deep out-of-the-money options. The study concludes that the calibrated Black–Scholes model remains a reliable benchmark for pricing highly liquid European equity index options, while highlighting the importance of calibration using market-derived inputs to improve pricing accuracy under actual market conditions.

Keywords

Black–Scholes model; option pricing; SPXW options; calibration; implied volatility; European call options; pricing error; CBOE.

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