The Development of Fractional Black–Scholes Model Solution Using the Daftardar-Gejji Laplace Method for Determining Rainfall Index-Based Agricultural Insurance Premiums

The Black–Scholes model is a fundamental concept in modern financial theory. It is designed to estimate the theoretical value of derivatives, particularly option prices, by considering time and risk factors. In the context of agricultural insurance, this model can be applied to premium determination due to the similar characteristics shared with the option pricing mechanism. The primary challenge in its implementation is determining a fair premium by considering the potential financial losses due to crop failure. Therefore, this study aimed to analyze the determination of rainfall index-based agricultural insurance premiums using the standard and fractional Black–Scholes models. The results showed that a solution to the fractional model could be obtained through the Daftardar-Gejji Laplace method. The premium was subsequently calculated using the Black–Scholes model applied throughout the growing season and paid at the beginning of the season. Meanwhile, the fractional Black–Scholes model incorporated the fractional order parameter to provide greater flexibility in the premium payment mechanism. The novelty of this study was in the application of the fractional Black–Scholes model for agricultural insurance premium determination, with due consideration for the long-term effects to ensure more dynamism and flexibility. The results could serve as a reference for governments, agricultural departments, and insurance companies in designing agricultural insurance programs to mitigate risks caused by rainfall fluctuations.