Arbitrage-free premium calculation for extreme losses using the shot noise process and the Esscher transform

We consider the classical compound Poisson model of insurance risk, with the additional economic assumption of a positive interest rate. Insurance premiums, that are the present values of the aggregate claims, are priced by enforcing a no-arbitrage condition between the insurance and reinsurance markets. We note a duality result relating the aggregate accumulated claims and the shot noise process and so we apply the piecewise deterministic Markov processes theory. It is assumed that the claim sizes are Loggamma, Fréchet and truncated Gumbel to deal with heavy-tail losses in practice. We also use an exponential distribution for the case of non-heavy-tail losses. In order to obtain an arbitrage-free premium, we use an equivalent martingale probability measure obtained via the Esscher transform. In case of Loggamma and Fréchet distribution for claim sizes, which allow us to derive the explicit forms for the insurance premium calculations, we find that the arbitrage-free premiums can only be obtained by levying the loading in terms of claim arrival rate, not in terms of claim size measure. It is due to the non-existence of the Laplace transforms of Loggamma and Fréchet distribution for claim sizes after changing measure. We find that if claim size follows truncated Gumbel distribution, the security loading can be levied either in terms of claim arrival rate or in terms of claim size measure (or both). However, it is not possible for us to obtain the explicit form for the insurance premium calculation. Using the analytical/explicit forms for four different claim size distributions, we also provide a several numerical examples.