https://doi.org/10.65770/EANN8830
ABSTRACT
Although the Black-Scholes model provides a closed-form analytical solution for European option pricing, numerical methods remain essential when discrete dividend adjustments and integral based pricing formulations are involved. This study investigates the application of the Adaptive Hybrid Quadrature Scheme (AHQS) to the valuation of European call and put options under the Black-Scholes framework with discrete dividend payments. The method combines Simpson’s 1/3 rule and two-point Gauss-Legendre quadrature within an adaptive integration framework, where local error estimates are obtained from the difference between the two numerical approximations. Numerical experiments were conducted using empirical market data consisting of five strike prices representing in-the-money (ITM), at-the-money (ATM), and out-of-the-money (OTM) option contracts. The performance of AHQS was evaluated against the composite Simpson’s 1/3 rule with subintervals using absolute error (AE), absolute percentage error (APE), and the number of function evaluations. For call options, AHQS reduced the average number of function evaluations by 61.7%, while the average differences in AE and APE were only +0.0019 and −0.0129, respectively. For put options, the average reduction in function evaluations reached 59.7%, with average differences in AE and APE of −0.0007 and +0.0010, respectively. These results indicate that the changes in pricing accuracy are negligible despite a substantial reduction in computational effort. Overall, the findings demonstrate that AHQS successfully preserves the accuracy of the composite Simpson method while significantly improving computational efficiency. Therefore, AHQS represents a promising deterministic alternative for integral-based European option pricing under the Black–Scholes model with discrete dividends.
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