Numerical Pricing of Discrete-Dividend European Options: An Empirical Case Study

Abstract

This applied quantitative study implements Simpson's 1/3 numerical integration to systematically evaluate the pricing of discrete-dividend-adjusted European options. Using an anonymized technology stock, XYZ Tech Corp., as an empirical case study, this research pursues two primary objectives: validating the computational convergence of the numerical algorithm against the exact Black-Scholes analytical benchmark using Relative Error, and analyzing the model's empirical pricing deviation against real market observations across various moneyness zones. Computational tests demonstrate that the Simpson's 1/3 method, which is theoretically bounded by a fourth-order truncation error, achieves optimal and rapid convergence. By establishing the grid partition at N = 200, the algorithm successfully suppresses the relative error strictly below a 0.001% threshold compared to the analytical solution, while executing efficiently in under 0.005 seconds. Empirically, while the theoretical model exhibits high accuracy for In-The-Money (ITM) options with minimal deviation, it consistently reveals a significant overvaluation bias for Out-Of-The-Money (OTM) Call contracts, whereas OTM Put valuations exhibit a structurally different deviation pattern. This valuation asymmetry suggests a potential limitation of the constant historical volatility assumption, which appears inadequate to fully capture the implied volatility skew and shifting risk perceptions prevalent in actual market microstructures.