The main motivation behind this thesis is the lack of belief subjectivity in problems of contracting, and especially in problems of demand for insurance. The idea that an underlying uncertainty in contracting problems (e.g. an insurable loss in problems of insurance demand) is a given random variable on some exogenously determined probability space is so engrained in the literature that one can easily forget that the notion of an objective uncertainty is only one possible approach to the formulation of uncertainty in economic theory. On the other hand, the subjectivist school led by De Finetti and Ramsey challenged the idea that uncertainty is totally objective, and advocated a personal view of probability (subjective probability). This ultimately led to Savage's approach to the theory of choice under uncertainty, where uncertainty is entirely subjective and it is only one's preferences that determine one's probabilistic assessment. It is the purpose of this thesis to revisit the "classical" insurance demand problem from a purely subjectivist perspective on uncertainty. To do so, we will first examine a general problem of contracting under heterogeneous subjective beliefs and provide conditions under which we can show the existence of a solution and then characterize that solution. One such condition will be called "vigilance". We will then specialize the study to the insurance framework, and characterize the solution in terms of what we will call a "generalized deductible contract". Subsequently, we will study some mathematical properties of collections of vigilant beliefs, in preparation for future work on the idea of vigilance. This and other envisaged future work will be discussed in the concluding chapter of this thesis. In the chapter preceding the concluding chapter, we will examine a model of contracting for innovation under heterogeneity and ambiguity, simply to demonstrate how the ideas and techniques developed in the first chapter can be used beyond problems of insurance demand.

This dissertation includes two topics. The first topic focuses on the problem of investor optimization of dynamic asset allocation to maximize expected utility under the value at risk (VaR) constraint. Different to previous researches, this study considers a common realistic case where the VaR horizon is equal to the whole investment horizon without a complete market constraint. Since the problem cannot be solved using the standard dynamic programming method or the martingale method, this study particularly provides an algorithm to solve this difficult problem. Similar to the mean-variance frontier suggested by Markowitz (1952), this study draws the frontiers of dynamic and static asset allocations under the VaR constraint. The analytical results clearly show that the dynamic asset allocations are more efficient than the static asset allocations. The second topic designs an optimal insurance policy form endogenously, assuming the objective of the insured is to maximize expected final wealth under the VaR constraint. The optimal insurance policy can be replicated using three options, including a long call option with a small strike price, a short call option with a large strike price, and a short cash-or-nothing call option. Moreover, expected wealth is increasing and concave in VaR and in significance level. Finally, Mean-VaR Frontiers are drawn, and reveal that the optimal insurance is more efficient than alternative insurance forms.

The main motivation behind this thesis is the lack of belief subjectivity in problems of contracting, and especially in problems of demand for insurance. The idea that an underlying uncertainty in contracting problems (e.g. an insurable loss in problems of insurance demand) is a given random variable on some exogenously determined probability space is so engrained in the literature that one can easily forget that the notion of an objective uncertainty is only one possible approach to the formulation of uncertainty in economic theory. On the other hand, the subjectivist school led by De Finetti and Ramsey challenged the idea that uncertainty is totally objective, and advocated a personal view of probability (subjective probability). This ultimately led to Savage's approach to the theory of choice under uncertainty, where uncertainty is entirely subjective and it is only one's preferences that determine one's probabilistic assessment. It is the purpose of this thesis to revisit the "classical" insurance demand problem from a purely subjectivist perspective on uncertainty. To do so, we will first examine a general problem of contracting under heterogeneous subjective beliefs and provide conditions under which we can show the existence of a solution and then characterize that solution. One such condition will be called "vigilance". We will then specialize the study to the insurance framework, and characterize the solution in terms of what we will call a "generalized deductible contract". Subsequently, we will study some mathematical properties of collections of vigilant beliefs, in preparation for future work on the idea of vigilance. This and other envisaged future work will be discussed in the concluding chapter of this thesis. In the chapter preceding the concluding chapter, we will examine a model of contracting for innovation under heterogeneity and ambiguity, simply to demonstrate how the ideas and techniques developed in the first chapter can be used beyond problems of insurance demand.

Cette thèse traite de la théorie du risque en finance et en assurance. La mise en pratique du concept de comonotonie, la dépendance du risque au sens fort, est décrite pour identifier l’optimum de Pareto et les allocations individuellement rationnelles Pareto optimales, la tarification des options et la quantification des risques. De plus, il est démontré que l’aversion au risque monotone à gauche, un raffinement pertinent de l’aversion forte au risque, caractérise tout décideur à la Yaari, pour qui, l’assurance avec franchise est optimale. Le concept de comonotonie est introduit et discuté dans le chapitre 1. Dans le cas de risques multiples, on adopte l’idée qu’une forme naturelle pour les compagnies d’assurance de partager les risques est la Pareto optimalité risque par risque. De plus, l’optimum de Pareto et les allocations individuelles Pareto optimales sont caractérisées. Le chapitre 2 étudie l’application du concept de comonotonie dans la tarification des options et la quantification des risques. Une nouvelle variable de contrôle de la méthode de Monte Carlo est introduite et appliquée aux “basket options”, aux options asiatiques et à la TVaR. Finalement dans le chapitre 3, l’aversion au risque au sens fort est raffinée par l’introduction de l’aversion au risque monotone à gauche qui caractérise l’optimalité de l’assurance avec franchise dans le modèle de Yaari. De plus, il est montré que le calcul de la franchise s’effectue aisément. / This thesis deals with the risk theory in Finance and Insurance. Application of the Comonotonicity concept, the strongest risk dependence, is described for identifying the Pareto optima and Individually Rational Pareto optima allocations, option pricing and quantification of risk. Furthermore it is shown that the left monotone risk aversion, a meaningful refinement of strong risk aversion, characterizes Yaari’s decision makers for whom deductible insurance is optimal. The concept of Comonotonicity is introduced and discussed in Chapter 1. In case of multiple risks, the idea that a natural way for insurance companies to optimally share risks is risk by risk Pareto-optimality is adopted. Moreover, the Pareto optimal and individually Pareto optimal allocations are characterized. The Chapter 2 investigates the application of the Comonotonicity concept in option pricing and quantification of risk. A novel control variate Monte Carlo method is introduced and its application is explained for basket options, Asian options and TVaR. Finally in Chapter 3 the strong risk aversion is refined by introducing the left-monotone risk aversion which characterizes the optimality of deductible insurance within the Yaari’s model. More importantly, it is shown that the computation of the deductible is tractable.