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The pension system has become more and more complex and structured all over Europe in the last decades. Because of the financial and social crisis, several countries implemented strong reforms in the state welfare in order to reduce the pension costs on the state budget balance. Furthermore, they allowed and encouraged the establishment of private pension facilities. We have to consider the three pillars. The first one concerns the state pension system. The second concerns the relationship between the employer and the employee. The third pillar consists in individual investment plans typically issued by insurance companies. This thesis analyzes a pension fund from the point of view of the three main actors involved: the pension plan provider, who decides the tactical allocation of the pension funds, the fund manager who takes care of the strategic investment problem, and the individual investor who faces the problem to allocate his/her savings in a retirement perspective. All these problems involve the analysis of a long-term choice and require to take into consideration some elements of uncertainty. Therefore, it is natural to face them using the instruments provided by Stochastic Programming. After a thorough review of the literature and a particular focus on the liability side of a pension fund, this thesis addresses the problem of a private pension plan sponsor who has to define the best pension funds for its members. This study is the first to face this problem from a quantitative point of view and proposes an innovative double step approach consisting of a cluster analysis of the pension fund population and of a portfolio optimization multistage stochastic program. The most important contribution relies on the introduction of the stochastic dominance in a real-life problem of an individual investor in a retirement perspective. Moreover, the proposed model considers a withdraw event allowed by the Italian pension system. The objective of the model is to minimize the Average Value at Risk Deviation measure (AV@RD) and to satisfy wealth goals. Three different wealth target formulations are considered. The first is a deterministic wealth target, i.e. a comparison between the accumulated average wealth and a fixed threshold. The second and the third are two stochastic dominance relations, first-order and second-order, which need to introduce a benchmark portfolio and then require the optimal portfolio to satisfy the stochastic dominance constraints with respect to the benchmark wealth.

At the market, we can identify various kinds of options. Some of them are traded at organized exchanges and are quite liquid. Others are traded only between particular parties. The current market practice is to obtain implied volatility of liquid options as based on Black-Scholes type (BS hereafter) models. Such volatility is subsequently used to price illiquid or even exotic options. It therefore follows that the BS model at one time moment can be related to the whole set of IVs as given by maturity/moneyness relation of tradable options. One can therefore get IV curve or surface (a so called smirk or smile). Since the moneyness and maturity of IV often do not match the data of valuated options, some sort of estimating and local smoothing is necessary. However, it can lead to arbitrage opportunity, if no-arbitrage conditions on state price density (SPD) are ignored. In this paper, using option data on DAX index, we aim on the analyses of the behavior of IV and SPD with respect to different choices of bandwidth parameter h and identification of a set of bandwidths which violates no-arbitrage conditions. We document that the change of h implies interesting changes in the violation interval of moneyness. Finally, we also show the impact of h on the total area of SPD under zero, which can be seen as a degree of no-arbitrage violation.

This thesis is focused on distortion risk measures and distortion reward-risk ratios. Firstly, we summarize the properties of these measures related to coherency axioms and stochastic dominance. We present the proofs and explain the assumptions for consistency of distortion risk measures with stochastic dominance. Furthermore, their relation to risk measures Value- at-Risk and Expected Shortfall is explained, and numerous examples of these measures are presented. Then, the basics of the theory of distortion reward-risk ratios are provided. The main theoretical result of this thesis is the proposition of the distortion reward-risk optimization model. We propose this model with the assumption of a discrete loss random variable that has realizations with equal probabilities. Lastly, we analyze and discuss the results and limitations of our implementation in the specialized optimization software GAMS on real financial data. As it turns out, the class of distortion risk measures is prospective because it allows us to re- weight probabilities in the distribution and to model the risk-aversion preferences.

This thesis covers, in detail, theoretical background of micro-level stochastic model, which includes definition and properties of non-homogeneous Poisson process. This the- ory is then applied to real data generated by MTPL portfolio. Estimates of provisions under micro-level stochastic model are calculated using ordinary Monte Carlo simula- tion method. Results obtained from micro-level stochastic model are compared to Mack Chain-ladder estimates. 1