<p> We address the problem of valuation of life insurance risks of different nature, market independent or equity-linked, under various assumptions regarding policyholders'mortality and the financial market. Given the incomplete nature of life insurance markets, an indifference valuation approach tailored to different models of the insurer's liability is applied. To be more specific, we propose three models for the insurer's liability: a single life insurance model, the individual risk model and the collective risk model. The last two models are generalizations of the aggregate loss models with the same name from actuarial mathematics. </p> <p> First, we investigate the pricing problem of market independent life insurance risks under the assumption of random mortality, focussing on the effects of this latter assumption on the premium. We find that random mortality is an essential assumption especially when pricing in aggregate loss models. Then, we consider life insurance products with a more complex structure of the benefit, as equity-linked term life insurances. We price them via utility indifference in all liability models mentioned above, assuming deterministic mortality and a Black-Scholes market model. Comparing the results obtained, we observe that the collective risk model is computationally more efficient than the others, but at the cost of higher premium. Finally, we conclude by extend-ing our pricing results for equity-linked term life insurance to a one factor stochastic volatility market model. We obtain that in a fast-mean-reverting volatility regime, the indifference premium can be well approximated by adjusted constant volatility results, previously derived. </p> / Thesis / Doctor of Philosophy (PhD)

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While the market for credit instruments grew continuously in the decade before 2008, its liquidity has dried up significantly in the current crisis, and investors have become aware of the possible consequences of being exposed to credit risk. In this thesis we address these issues by pricing credit instruments using utility indifference pricing, a method that takes into account the investor's personal risk aversion and which is not affected by the lack of liquidity. Through stochastic optimal control methods, we use indifference pricing with exponential utility to determine corporate bond prices and CDS spreads. In the first part we examine how these quantities are affected by risk aversion under different models of default. The emphasis lies on a hybrid model, in which a regime switch of the reference entity is triggered by a creditworthiness index correlated to its stock price. The second part generalizes this setup by introducing uncertainty in the model parameters. Robust optimal control has been used independently in the literature to address model uncertainty for portfolio selection problems. Here, we incorporate this approach with utility indifference and derive some analytical and numerical results on how model uncertainty affects credit spreads.

While the market for credit instruments grew continuously in the decade before 2008, its liquidity has dried up significantly in the current crisis, and investors have become aware of the possible consequences of being exposed to credit risk. In this thesis we address these issues by pricing credit instruments using utility indifference pricing, a method that takes into account the investor's personal risk aversion and which is not affected by the lack of liquidity. Through stochastic optimal control methods, we use indifference pricing with exponential utility to determine corporate bond prices and CDS spreads. In the first part we examine how these quantities are affected by risk aversion under different models of default. The emphasis lies on a hybrid model, in which a regime switch of the reference entity is triggered by a creditworthiness index correlated to its stock price. The second part generalizes this setup by introducing uncertainty in the model parameters. Robust optimal control has been used independently in the literature to address model uncertainty for portfolio selection problems. Here, we incorporate this approach with utility indifference and derive some analytical and numerical results on how model uncertainty affects credit spreads.

博士（経営） / 85 p. / 一橋大学

We propose a model for an insurance loss index and the claims process of a single insurance company holding a fraction of the total number of contracts that captures both ordinary losses and losses due to catastrophes. In this model we price a catastrophe derivative by the method of utility indifference pricing. The associated stochastic optimization problem is treated by techniques for piecewise deterministic Markov processes. A numerical study illustrates our results.

Dans cette thèse nous considérons le problème de la maximisation d’utilité et de la formation des prix d’indifférence pour les modèles semimartingales exponentiels dépendant d’un facteur aléatoire ξ. L’enjeu est de résoudre le problème des prix d’indifférence en utilisant le grossissement de l’espace et de la filtration. Nous réduisons le problème de maximisation dans la filtration élargie au problème conditionnel, sachant {ξ = v}, que nous résolvons en utilisant une approche duale. Pour HARA-utilités nous introduisons les informations telles que les entropies relatives et les intégrales de type Hellinger, ainsi que les processus d’information correspondants, enfin d’exprimer, via ces processus, l’utilité maximal. En particulier, nous étudions les modèles de Lévy exponentiels, où les processus d’information sont déterministes ce que simplifie considèrablement les calculs des prix d’indiffrence. Enfin, nous appliquons les rèsultats au modèle du mouvement brownien géométrique et au modèle de diffusion-sauts qui inclut le mouvement brownien et les processus de Poisson. Dans les cas d’utilité logarithmique, de puissance et exponentielle, nous fournissons les formules explicites des informations, et puis, en utilisant les méthodes numériques, nous résolvons les équations pour obtenir les prix d’indifférence en cas de vente d’une option européenne. / This thesis explores the utility maximisation problem and indifference pricing for exponential semimartingale models depending on a random factor ξ. The main idea to solve indifference pricing problem consists in the enlargement of the space and filtration. We reduce the maximization problem on the enlarged filtration to the conditional one, given {ξ = v}, which we solve using dual approach. For HARA-utilities we introduce the information quantities such that the relative entropies, Hellinger type integrals, and the corresponding information processes, and we express the maximal utility via these processes. As a particular case, we study exponential Levy models, where the information processes are deterministic and this fact simplify very much indifference price calculus. Finally, we apply the results to Geometric Brownian motion model and jump-diffusion model which incorporates Brownian motion and Poisson process. In the cases of logarithmic, power and exponential utilities, we provide the explicit formulae of information quantities and using the numerical methods we solve the equations for the seller’s and buyer’s indifference prices of European put option.

在不完全市場下，衍生性金融商品可利用上套利和下套利價格來訂出價格區間。我們運用效用無差異定價於此篇論文中，此定價方式為尋找一個初始交易價，會使在起始時交易商品和無交易商品於商品到期日之最大期望效用相等。利用主要的對偶結果，我們證明在指數效用函數下，效用無差異定價區間會比上套利和下套利定價區間小。 / In incomplete markets, prices of a contingent claim can be obtained between the upper and lower hedging prices. In this thesis, we will use utility indifference pricing to nd an initial payment for which the maximal expected utility of trading the claim is indierent to the maximal expected utility of no trading. From the central duality result, we show that the gap between the seller's and the buyer's utility indierence prices is always smaller than the gap between the upper and lower hedging prices under the exponential utility function.

Dans cette thèse, nous nous intéressons à des sujets différents en mathématiques financières, tous liés aux imperfections de marché et à la technique fondamentale de la maximisation d'utilité. Elle comporte trois parties. Dans la première, qui se base sur deux papiers, nous considérons le problème d'investissement optimal sur un marché financier avec coûts de transaction proportionnels. On commence par étudier le problème d'investissement dans le cas où la fonction d'utilité est multivariée (ce qui s'adapte particulièrement bien aux marchés des devises) et l'agent a une dotation initiale aléatoire, qui peut s'interpréter comme une option ou un autre contrat dérivé. Après avoir analysé les propriétés du problème et de son dual, nous utilisons ces résultats pour examiner, dans ce contexte, certains aspects d'une technique de pricing devenue populaire dans le cadre des marchés incomplets, l'évaluation par indifférence d'utilité. Dans le deuxième chapitre, nous étudions le problème d'existence d'un ensemble de prix (appelés "prix fictifs" ou "shadow prices") qui offrirait la même utilité maximale à l'agent si le marché n'avait pas de frictions. Ces résultats sont utiles pour clarifier le lien entre la théorie classique des marchés sans frictions et la littérature en croissance rapide sur les coûts de transaction. Dans la deuxième partie de cette thèse, nous considérons le problème d'évaluation de produits dérivés par indifférence d'utilité dans des marchés incomplets, où la source d'incomplétude provient du fait que certains actifs ne peuvent pas être échangés sur le marché, ce qui est le cas par exemple dans le cadre des modèles structurels pour le prix de l'électricité. Sous certaines hypothèses, nous dérivons une caractérisation en terme d'équations différentielles stochastiques rétrogrades (EDSR) pour le prix, et nous nous concentrons ensuite sur les options européennes en établissant en particulier l'existence d'une stratégie de couverture optimale, même lorsque le payoff présente des discontinuités et est éventuellement non borné. Dans la dernière partie, nous analysons un simple problème de principal-agent à horizon fini, où le principal est essentiellement interprété comme un régulateur et l'agent comme une entreprise qui produit certaines émissions polluantes. Nous traitons séparément les problèmes du principal et de l'agent et nous utilisons la théorie des EDSR pour fournir des conditions nécessaires et suffisantes d'optimalité. Nous effectuons également des analyses de sensibilité et nous montrons des résultats numériques dans le but de fournir une meilleure compréhension du comportement des agents. / In this thesis we deal with different topics in financial mathematics, that are all related to market imperfections and to the fundamental technique of utility maximization. The work consists of three parts. In the first one, which is based on two papers, we consider the problem of optimal investment on a financial market with proportional transaction costs. We initially study the investment problem in the case where the utility function is multivariate (which is particularly suitable on currency markets) and the agent is endowed with a random claim, which can be interpreted as an option or another derivative contract. After analyzing the properties of the primal and dual problems, we apply those results to investigate, in this context, some aspects of a popular pricing technique in incomplete markets, i.e. utility indifference evaluation. In the second contribution to the transaction costs literature, we investigate the existence problem for a set of prices (called shadow prices) that would provide the same maximal utility to the agent if the market did not have frictions. These results shed some light on the link between the classical theory of frictionless markets and the quickly growing literature on transaction costs. In the second part of this thesis we consider the utility indifference pricing problem in incomplete markets, where the source of incompleteness comes from the fact that some assets in the market cannot be actively traded, which is the case for example in the framework of structural models for electricity prices. We provide a BSDE characterization for the price under mild assumptions, and then focus on the case of European claims by establishing in particular the existence of an optimal hedging strategy even when the claim presents discontinuities and is possibly unbounded. In the last contribution we analyze a simple principal-agent problem in finite time horizon, where the principal is mainly interpreted as a regulator and the agent as a firm producing some kind of polluting emissions. We separately treat both the agent's and the principal's problems and use the BSDE theory for providing necessary and sufficient conditions for optimality. We also perform some sensitivity analyses and give numerical results in order to provide a better understanding of the agents' behavior.