Pricing of catastrophe insurance options written on a loss index with reestimation

We propose a valuation model for catastrophe insurance options written on a loss index. This kind of options distinguishes between a loss period [0,T1], during which the catastrophes may happen, and a development period [T1,T2], during which losses entered before T1 are reestimated. Here we suppose that the underlying loss index is given by a time inhomogeneous compound Poisson process before T1 and that losses are reestimated by a common factor given by an exponential time inhomogeneous Lévy process after T1. In this setting, using Fourier transform techniques, we are able to provide analytical pricing formulas for catastrophe options written on this kind of index.