Hydrometeorological Disaster Insurance Modeling Based on Fractional Differential Equations for Climate Change Mitigation Within the Framework of SDG 13

Rainfall-index-based disaster insurance is an efficient approach to mitigating hydrometeorological losses. However, conventional premium pricing models generally assume memoryless stochastic dynamics that do not fully capture the long-range dependence inherent in rainfall data. This study develops a hydrometeorological disaster insurance model within a fractional Black–Scholes framework to incorporate long-memory effects. The model is formulated using fractional differential equations and solved semi-analytically by integrating the Daftardar–Jafari Method (DJM) with the Kashuri–Fundo (KF) transform, yielding a closed-form solution expressed in terms of the Mittag–Leffler function. The proposed contract is structured as parametric rainfall insurance with a multi-layer payout mechanism based on percentiles corresponding to minor, moderate, and severe housing damage. The results show that variations in the fractional-order parameter significantly affect premium estimation. In particular,  δ = 0.5 recovers the classical model and tends to generate higher premiums than the fractional model with δ = 0.23153, whereas the model with δ = 0.73153 yields lower premiums. These findings indicate that fractional-order parameterization can accommodate diverse risk characteristics and policyholders’ economic capacities, enabling more adaptive, risk-sensitive premium structures. In line with SDG 13 (Climate Action), the proposed framework offers a climate-responsive disaster-mitigation strategy through accessible, actuarially relevant insurance design.  recovers the classical model and tends to generate higher premiums than the fractional model with , whereas the model with  yields lower premiums. These findings indicate that fractional-order parameterization can accommodate diverse risk characteristics and policyholders’ economic capacities, enabling more adaptive, risk-sensitive premium structures. In line with SDG 13 (Climate Action), the proposed framework offers a climate-responsive disaster-mitigation strategy through accessible, actuarially relevant insurance design.