Doubly stochastic point processes in reinsurance and the pricing of catastrophe insurance derivatives
This dissertation presents pricing models for stop-loss reinsurance contracts for catastrophic events and for catastrophe insurance derivatives. We use doubly stochastic Poisson process or the Cox process for the claim arrival process for catastrophic events. The shot noise process is able to measure the frequency, magnitude and time period needed to determine the effect of the catastrophe. This process is used for the claim intensity function within the Cox process. The Cox process with shot noise intensity is examined by piecewise deterministic Markov process theory. We apply the Cox process incorporating the shot noise process as its intensity to price stop-loss catastrophe reinsurance contracts and catastrophe insurance derivatives. In order to calculate fair prices for reinsurance contracts and catastrophe insurance derivatives we need to assume that there is an absence of arbitrage opportunities in the market. This can be achieved by using an equivalent martingale probability measure in our pricing models. The Esscher transform is used to change probability measure. The dissertation also shows how to estimate the parameters of claim intensity using the likelihood function. In order to estimate the distribution of claim intensity, state estimation is employed as well. Since the claim intensity is not observable we filter it out on the basis of the number of claims, i.e. we employ the Kalman-Bucy filter. We also derive pricing formulae for stop-loss reinsurance contracts for catastrophic events using the distribution of claim intensity that is obtained by the Kalman-Bucy filter. Both estimations are essential in pricing stop-loss reinsurance contracts and catastrophe insurance derivatives.