Essays on modeling, hedging and pricing of insurance and financial products

Cette thèse est composée de trois articles abordant différentes problématiques en relation avec la modélisation, la couverture et la tarification des risques d’assurance et financiers. “A general class of distortion operators for pricing contingent claims with applications to CAT bonds” est un projet présentant une méthode générale pour dériver des opérateurs de distorsion compatibles avec la valorisation sans arbitrage. Ce travail offre également une nouvelle classe simple d’opérateurs de distorsion afin d’expliquer les primes observées dans le marché des obligations catastrophes. “Local hedging of variable annuities in the presence of basis risk” est un travail dans lequel une méthode de couverture des rentes variables en présence de risque de base est développée. La méthode de couverture proposée bénéficie d’une exposition plus élevée au risque de marché et d’une diversification temporelle du risque pour obtenir un rendement excédentaire et faciliter l’accumulation de capital. “Option pricing under regime-switching models : Novel approaches removing path-dependence” est un projet dans lequel diverses mesures neutres au risque sont construites pour les modèles à changement de régime de manière à générer des processus de prix d’option qui ne présentent pas de dépendance au chemin, en plus de satisfaire d’autres propriétés jugées intuitives et souhaitables.This thesis is composed of three papers addressing different issues in relation to the modeling, hedging and pricing of insurance and financial risks. “A general class of distortion operators for pricing contingent claims with applications to CAT bonds” is a project presenting a general method for deriving probability distortion operators consistent with arbitrage-free pricing. This work also offers a simple novel class of distortions operators for explaining catastrophe (CAT) bond spreads. “Local hedging of variable annuities in the presence of basis risk” is a work in which a method to hedge variable annuities in the presence of basis risk is developed. The proposed hedging scheme benefits from a higher exposure to equity risk and from time diversification of risk to earn excess return and facilitate the accumulation of capital. “Option pricing under regime-switching models: Novel approaches removing path-dependence” is a project in which various risk-neutral measures for hidden regime-switching models are constructed in such a way that they generate option price processes which do not exhibit path-dependence in addition to satisfy other properties deemed intuitive and desirable

On robust multi-period pre-commitment and time-consistent mean-variance portfolio optimization

We consider robust pre-commitment and time-consistent mean-variance optimal asset allocation strategies, that are required to perform well also in a worst-case scenario regarding the development of the asset price. We show that worst-case scenarios for both strategies can be found by solving a specific equation each time step. In the unconstrained asset allocation case, the robust pre-commitment as well as the time-consistent strategy are identical to the corresponding robust myopic strategies, by which investors perform robust portfolio control only for one time step and conduct a risk-free strategy afterwards. In the experiments, the robustness of pre-commitment and time-consi

MEAN VARIANCE OF FRACTIONAL STOCHASTIC MODEL AND LOGARITHM UTILITY OPTIMIZATION OF A PENSION FUND WITH TAX AND TRANSACTION COST

This work looked at the mean-variance of fractional continuous time stochastic model for the dynamics of a pension fund with tax and transaction cost, where the effect of tax and transaction cost charging makes on the expected logarithmic utility of the pensioner was established. The associated H-J-B equation in the optimization problem is obtained using lto’s lemma. An explicit solution to the pensioners’ problems was derived under stated condition

On Some Stochastic Optimal Control Problems in Actuarial Mathematics

The event of ruin (bankruptcy) has long been a core concept of risk management interest in the literature of actuarial science. There are two major research lines. The first one focuses on distributional studies of some crucial ruin-related variables such as the deficit at ruin or the time to ruin. The second one focuses on dynamically controlling the probability that ruin occurs by imposing controls such as investment, reinsurance, or dividend payouts. The content of the thesis will be in line with the second research direction, but under a relaxed definition of ruin, for the reason that ruin is often too harsh a criteria to be implemented in practice. Relaxation of the concept of ruin through the consideration of "exotic ruin" features, including for instance, ruin under discrete observations, Parisian ruin setup, two-sided exit framework, and drawdown setup, received considerable attention in recent years. While there has been a rich literature on the distributional studies of those new features in insurance surplus processes, comparably less contributions have been made to dynamically controlling the corresponding risk. The thesis proposes to analytically study stochastic control problems related to some "exotic ruin" features in the broad area of insurance and finance. In particular, in Chapter 3, we study an optimal investment problem by minimizing the probability that a significant drawdown occurs. In Chapter 4, we take this analysis one step further by proposing a general drawdown-based penalty structure, which include for example, the probability of drawdown considered in Chapter 3 as a special case. Subsequently, we apply it in an optimal investment problem of maximizing a fund manager's expected cumulative income. Moreover, in Chapter 5 we study an optimal investment-reinsurance problem in a two-sided exit framework. All problems mentioned above are considered in a random time horizon. Although the random time horizon is mainly determined by the nature of the problem, we point out that under suitable assumptions, a random time horizon is analytically more tractable in comparison to its finite deterministic counterpart. For each problem considered in Chapters 3--5, we will adopt the dynamic programming principle (DPP) to derive a partial differential equation (PDE), commonly referred to as a Hamilton-Jacobi-Bellman (HJB) equation in the literature, and subsequently show that the value function of each problem is equivalent to a strong solution to the associated HJB equation via a verification argument. The remaining problem is then to solve the HJB equations explicitly. We will develop a new decomposition method in Chapter 3, which decomposes a nonlinear second-order ordinary differential equation (ODE) into two solvable nonlinear first-order ODEs. In Chapters 4 and 5, we use the Legendre transform to build respectively one-to-one correspondence between the original problem and its dual problem, with the latter being a linear free boundary problem that can be solved in explicit forms. It is worth mentioning that additional difficulties arise in the drawdown related problems of Chapters 3 and 4 for the reason that the underlying problems involve the maximum process as an additional dimension. We overcome this difficulty by utilizing a dimension reduction technique. Chapter 6 will be devoted to the study of an optimal investment-reinsurance problem of maximizing the expected mean-variance utility function, which is a typical time-inconsistent problem in the sense that DPP fails. The problem is then formulated as a non-cooperative game, and a subgame perfect Nash equilibrium is subsequently solved. The thesis is finally ended with some concluding remarks and some future research directions in Chapter 7