Simulating and Pricing CAT Bonds Using the Spectral Method Based on Chebyshev Basis

The intensity, hardness, and extent of catastrophic accidents in recent decades have led insurance companies to seek resources to raise the capital to deal with the caused damage by these accidents. One of the most effective tools to cover the risk of catastrophic events such as earthquakes, floods, etc., which is widely used in the world, is catastrophic bonds. The purpose of this study is to provide a model for the pricing of catastrophic bonds. Because earthquake catastrophic damage causes major changes to corporate assets as well as investment trends, a jump sentence is added to the study model to indicate the severity and probability of damage. To produce the time discretization, the suggested methodology employs a one-order correct expression in the first process. To generate the full-discretization in the second level, the spectral collocation method approach that relies on the Chebyshev basis of the second kind is presented. Adding the jump sentence causes the original model to become an integral differential model, which approximates using the spectral method. Also, based on the derivatives in the obtained model, we approximated the derivative operator by using this type of base. The numerical investigation confirms the temporal discretized formulation’s stability and convergence. We also approximated the integral sentence using the expansion of Gauss-Laguerre quadrature and presented numerical results.