Arbitrage-free catastrophe reinsurance valuation for compound dynamic contagion claims

In this paper, we consider catastrophe stop-loss reinsurance valuation for a reinsurance company with dynamic contagion claims. To deal with conventional and emerging catastrophic events, we propose the use of a compound dynamic contagion process for the catastrophic component of the liability. Under the premise that there is an absence of arbitrage opportunity in the market, we obtain arbitrage-free premiums for these contracts. To this end, the Esscher transform is adopted to specify an equivalent martingale probability measure. We show that reinsurers have various ways of levying the security loading on the net premiums to quantify the catastrophic liability in light of the growing challenges posed by emerging risks arising from climate change, cyberattacks, and pandemics. We numerically compare arbitrage-free catastrophe stop-loss reinsurance premiums via the Monte Carlo simulation method. We also compare them with those from generalized compound Hawkes/compound Cox cases. Sensitivity analyses are performed by changing the retention level, the Esscher parameters, and the intensity parameters.