Fundamental Aspects of Operational Risk and Insurance Analytics: A Handbook of Operational Risk

18.2.1.2 Real-world Pricing: Benchmark and Actuarial Frameworks (Dispersion Measures and Deflators)

ILS that are constructed to transfer catastrophic risk from nature to the capital markets require a particular framework for valuation. In general, natural catastrophes can be incorporated into models for the assets and liabilities of an insurer, reinsurer, or a financial institution by considering a class of models involving jump-diffusion processes for an underlying risk index. In this context, we noted previously that financial markets are incomplete and as a consequence the methodology of replicating portfolios is not applicable. In the paper by Gerber and Shiu (1994), they consider the family of dispersion measures known as the Esscher transform, that they utilize for option pricing (as the deflator methodology equivalent of risk-neutral pricing in financial mathematics). Under the Esscher transform, they are able to show that one may obtain an efficient technique for valuing derivative securities if the logarithms of the prices of the underlying security come from a particular class that follows a stochastic processes with independent and stationary increments. Furthermore, they show that popular processes in this class include the popular families of models given by the Wiener process, the Poisson process, the Gamma process, and the inverse Gaussian process. Therefore, under this family of dispersion measures, one may select the parameter of this transform such that when ...