Financial and actuarial valuation of insurance derivatives
This dissertation looks into the interplay of financial and insurance markets that is created by securitization of insurance related risks. It comprises four chapters on both the common ground and different nature of actuarial and financial risk valuation. The first chapter investigates the market for catastrophe insurance derivatives that has been established at the Chicago Board of Trade in 1992. Modeling the underlying index as a compound Poisson process the set of financial derivative prices that exclude arbitrage opportunities is characterized by the market prices of frequency and jump size risk. Fourier analysis leads to a representation of price processes that separates the underlying stochastic structure from the contract's payoff and allows derivation of the inverse Fourier transform of price processes in closed form. In a market with a representative investor, market prices of frequency and jump size risk are uniquely determined by the agent's coefficient of absolute risk aversion which consequently fixes the price process on the basis of excluding arbitrage strategies. The second chapter analyzes a model for a price index of insurance stocks that is based on the Cramer-Lundberg model used in classical risk theory. It is shown that price processes of basic securities and derivatives can be expressed in terms of the market prices of risk. This parameterization leads to formulae in closed form for the inverse Fourier transform of prices and the conditional probability distribution. Financial spreads are examined in more detail as their structure resembles the characteristics of stop loss reinsurance treaties. The equivalence between a representative agent approach and the Esscher transform is shown and the financial price process that is robust to these two selection criteria is determined. Finally, the analysis is generalized to allow for risk processes that are perturbed by diffusion. In the third chapter an integrated market is introduced containing both insurance and financial contracts. The calculation of insurance premia and financial derivative prices is presented assuming the absence of arbitrage opportunities. It is shown that in contrast to financial contracts, there exist infinitely many market prices of risk that lead to the same premium process. Thereafter a link between financial and actuarial prices is established based on the requirement that financial prices should be consistent with actuarial valuation. This connection is investigated in more detail under certain premium calculation principles. The starting point of the final chapter is the Fourier technique developed in Chapters 1 and 2. It is the aim of this chapter to generalize the analysis to underlying Levy processes. Expressions for the conditional moments and probabilities based on these processes are derived and their inverse Fourier transforms are obtained in closed form. The representation of conditional moments and probabilities separates the stochastic structure from the deterministic dependence on the underlying Levy processes.