Mathematical Models for Insurance Risk

The aim of the project is to improve upon the available mathematical tools to analyze risk associated with insurance. This includes the extension of currently used stochastic models for the surplus process of an insurance portfolio that adequately address the presence of economic factors and dependence among risks. In particular, we want to refine the existing analytical techniques for the study of measures related to the event of ruin. Another main theme of our research will be the mathematical analysis of reinsurance treaties. We will focus on optimality criteria and appropriate risk measures to decide what type of reinsurance should be taken and for what premium. Furthermore we intend to work out variance reduction techniques for the stochastic simulation of sums of dependent and heavy-tailed risks. These shall in turn be used to analyze complex situations that do not allow for an analytical treatment. We also plan to develop semi-static hedging strategies for certain classes of exotic options. The results should lead to effective strategies in competing markets, to improvements in dealing with the rare-event risk of natural catastrophes and to new insights in coping with insurance risk.

In this project, mathematical tools for the analysis of insurance risk were further developed and applied to concrete situations for the measurement of risk in insurance portfolios. For specific dependence structures of the insurance risks the resulting behaviour of the ruin probability was characterized and consequences for the risk management of insurance policies on natural catastrophes were investigated. Correction terms for some approximations that are popular in insurance practice were identified. These results can lead to a better understanding of aggregation of extreme risks in the insurance industry. In the area of optimal profit participation schemes, stochastic control techniques were developed further to identify dividend strategies that maximize certain combinations of profitability and solvency objectives. In particular, effects of interest and transaction costs could be included in the studies. Furthermore, explicit formulas for ruin probabilities and related quantities under predefined surplus- dependent premium strategies could be derived. In the classical collective risk model a surprisingly simple identity for the ruin probability in the insurance portfolio was found, if taxes have to be paid on profits according to a loss- carried-forward tax scheme. For the proof of this identity, connections of risk theory and queuing theory were extended and applied. New results were also established in the structural analysis of parallel multiple reinsurance contracts on insurance portfolios. For the pricing of CDO tranches a generic Levy model was developed that provides a general framework for a number of models that up to now had been investigated separately. For a so- called local Levy model, which improves upon some disadvantages of other equity models in financial practice, regularization techniques were developed to address the inverse problem of model calibration, if market prices of traded vanilla options are available. For the Heston volatility model, it was known in practice for some time that pricing options with long maturity can sometimes lead to implausible results. This problem could now be identified mathematically with tools from complex analysis and an algorithmic procedure was worked out to solve this problem.