Interest rate and longevity risk: dynamic model and applications to derivative products and life insurance.

Summary In French: This thesis is divided into three parts. The first part consists of chapters 2 and 3 in which we consider models that describe the evolution of an underlying asset in the world of stocks as well as the evolution of interest rates. These models, which use Wishart processes, belong to the affine class and generalize the multi-dimensional Heston models. We study the intrinsic properties of these models and focus on the valuation of vanilla options. After recalling some valuation methods, we introduce approximation methods providing closed formulas of the asymptotic smile. These methods facilitate the calibration procedure and allow an interesting analysis of the parameters. The second part, from chapter 4 to chapter 6, studies mortality and longevity risks. We first recall the general concepts of longevity risk and a set of issues underlying this risk. We then present a model of individual mortality that takes into account age and other characteristics of the individual that are explanatory of mortality. We calibrate the mortality model and analyze the influence of certain individual characteristics. Finally, we introduce a microscopic population dynamics model that allows us to model the evolution over time of a population structured by age and traits. Each individual evolves over time and is likely to give birth to a child, change characteristics and die. This model accounts for the possibly stochastic evolution of individual demographic rates over time. We also describe a micro/macro link that provides this microscopic model with good macroscopic properties. The third part, concerning chapters 7 and 8, focuses on applications of the previous modeling. The first application is a demographic one since the microscopic population dynamics model allows us to make demographic projections of the French population. We also set up a demographic study of the pension problem by analyzing the solutions of an immigration policy and a reform of the retirement age. The second application concerns the study of life insurance products combining longevity and interest rate risks that have been studied in detail in the first two parts of the thesis. First, we study the basic risk generated by the heterogeneity of annuity portfolios. In addition, we introduce the Life Nominal Chooser Swaption (LNCS) which is a risk transfer product for life insurance products: this product has a very interesting structure and allows an insurance company holding an annuity portfolio to transfer its entire interest rate risk to a bank.