In this thesis a mathematical description and analysis of the Cumulative Prospect Theory is presented. Conditions that ensure well-posedness of the problem are provided, as well as existence results concerning optimal policies for discrete-time incomplete market models and for a family of diffusion market models. A brief outline of how this work is organised follows. In Chapter 2 important results on weak convergence and discrete time finance models are described, these facts form the main background to introduce in Chapter 3 the problem of optimal investment under the CPT theorem in a discrete time setting. We describe our model, present some assumptions and main results are derived. The second part of this work comprises the description of the martingale problem formulation of diffusion processes in Chapter 4. A key result on the limits and topological properties of the set of laws of a class of Itô processes is described in Chapter 5. Finally, we introduce a factor model that includes a class of stochastic volatility models, possibly with path-depending coefficients. Under this model, the problem of optimal investment with a behavioural investor is analysed and our main results on well-posedness and existence of optimal strategies are described under the framework of weak solutions. Further research and challenges when applying the techniques developed in this work are described.

Merton’s Optimal Investment-Consumption Problem is a classic optimization problem in finance. It aims to find the optimal controls for a portfolio with both risky and risk-less assets, inorder to maximize an investor’s utility function. One of the controls is the optimal allocationof wealth invested in a risky asset and the other control is the consumption rate. The problemis solved by using Dynamic Programming and the related Hamilton-Jacobi-Bellman equation.One of the disadvantages of the original problem is the consideration of constant volatility. Inthis thesis, we extend Merton’s problem considering the Grzelak-Oosterlee-Van Veeren modelthat describes the dynamics of a risky asset with stochastic volatility and stochastic interestrate. We derive the related Hamilton-Jacobi-Bellman for Merton’s problem considering theGrzelak-Oosterlee-Van Veeren model. We simulate the controls from Merton’s problem intwo different cases, one case where the volatility and interest rate are stochastic, following theGOVV-model. In the other case, the volatility and interest rate are assumed to be constant, asin Merton’s problem. The results obtained from simulations show that the case with stochasticvolatility and interest gave the same results as the case where the volatility and the interest ratewere assumed to be constant.

Abstract This dissertation explores two interrelated aspects of banking crises and bank regulations in perspective of regulator¡¦s soft budget constraints (SBCs in brief) and bank top management compensation. First, this paper models, in a game of incomplete information, bank behavior during banking crises when asymmetric information exists between regulators and banks. Here, I show that the situation creates the incentives for banks to roll over their defaulting loans to disguise their financial statements. Although a prudential regulator may mitigate this incentive by offering a ¡§slack¡¨ rescue packages, the bank¡¦s reputational concern may cause them to reject rescue offers. In this instance, regulators may be forced to offer amounts of recapitalization that will meet the amount necessary to restore banks to solvency. Otherwise, banks may have to gamble for resurrection, or wait until the banking crises become severe, and then more banks become insolvent, regulators have to offer optimal rescue packages subject to SBCs. New findings include (1) During banking crises, the optimal regulatory policies, on the one hand, may cause regulators have to offer rescue or bailout packages subject to different SBCs, on the other hand, mitigate banker¡¦s moral hazard. The more severe the crises will be, the greater soft budget constrained to regulators. (2) The potential severity of banking crises can be measured by the ratios, getting from net worth over the total amount of recapitalization offered by regulators and recovered from nonperforming loans. (3) As banking crises become severe, the cost of rescue becomes larger than that of bailout, the best regulatory policy is to intervene; On the contrary, if a situation labeled ¡§ too-many-to-fail¡¨ arises, the regulators may offer to rescue distressed banks subject to SBC. (4)As Bayesian equilibrium cost of regulator in crises is increasing, a random creative ambiguity for regulators to offer bailout or rescue plans may be the optimal policy to mitigate the expectation of SBC for banks . Second, this paper also shows that in the circumstances of universal banking or bank holding company, concentrating bank regulation on bank capital ratios and risk-based deposit insurance may be ineffective in controlling banker¡¦s risk-taking and moral hazard. Here, this paper follows, a more direct mechanism of influencing bank risk-taking incentives, in which the insurance premium scheme incorporate features of top management compensation. In a model of universal banking with two-periods and three-subsidiaries or departments, bank owner pre-commits to regulators to pick an optimal management compensation structure that induces the first-best value-maximizing investment choices by a bank¡¦s management. Findings include (1) If insurance premium is not fairly priced, the incentives are created for banks to have a ¡§regulatory arbitrage¡¨ by segregating its nonperforming assets from the investment bank, and shift it to the commercial bank, that increases the deposit-insurer an additional risk liability, and aggravates the risk-shifting within the universal bank; and vice versa. (2) Given management contracts{ fixed salary, a bonus paid, a fraction of equity of the bank} and { fixed salary, a penalty , a fraction of equity}for bank and security investment department respectively ; and a capitalization level corresponding must exceed the lower risky investment outcome , here bonus paid larger than 0, a penalty larger than 0, a fraction of equity between 0 and 1, then the investment policies implemented by managers, is less risky than when manger¡¦s interests are fully aligned with the equity interests. (3) Given a fairly priced insurance premium, and capitalization level corresponding must exceed the lower risky investment outcome, then the optimal management compensation structure can internalize the cost of moral hazard and induce the Pareto-optimal and department-equilibrium investment policies, thus mitigate moral hazard under universal banking. Finally, the state-owned and half-state-owned banks have experienced the institution-induced ineffectiveness; and the latter suffer from poor business performance level, partially because of the issues of ownership structure. This paper shows the investment policy with moral hazard under these banks incorporated with optimal compensation structures, and given capitalization level corresponding must exceed the lower risky investment outcome, then the optimal policies induced, that will improve their business performance level. This paper also shows that as the controlling shareholders have power over banks in excess of their cash flow rights, the incentives will be created for them to expropriate the minority shareholders. And, when the incentives for expropriation exists, the investment policy will be distorted with the managerial bias induced by their private benefits, and deteriorate morale of the banks. The regulatory mandatory requirements of one-share-one-vote principle may be proposed, instead.

Incomplete markets provide many challenges for both investment decisions and valuation problems. While both problems have received extensive attention in complete markets, there remain many open areas in the theory of incomplete markets. We present the results in three parts. In the first essay we consider the Merton investment problem of optimal portfolio choice when the traded instruments are the set of zero-coupon bonds. Working within a Markovian Heath-Jarrow-Morton framework of the interest rate term structure driven by an infinite dimensional Wiener process, we give sufficient conditions for the existence and uniqueness of an optimal investment strategy. When there is uniqueness, we provide a characterization of the optimal portfolio. Furthermore, we show that a specific Gauss-Markov random field model can be treated within this framework, and explicitly calculate the optimal portfolio. We show that the optimal portfolio in this case can be identified with the discontinuities of a certain function of the market parameters. In the second essay we price a claim, using the indifference valuation methodology, in the model presented in the first section. We appeal to the indifference pricing framework instead of the classic Black-Scholes method due to the natural incompleteness in such a market model. Because we price time-sensitive interest rate claims, the units in which we price are very important. This will require us to take care in formulating the investor’s utility function in terms of the units in which we express the wealth function. This leads to new results, namely a general change-of-numeraire theorem in incomplete markets via indifference pricing. Lastly, in the third essay, we propose a method to price credit derivatives, namely collateralized debt obligations (CDOs) using indifference. We develop a numerical algorithm for pricing such CDOs. The high illiquidity of the CDO market coupled with the allowance of default in the underlying traded assets creates a very incomplete market. We explain the market-observed prices of such credit derivatives via the risk aversion of investors. In addition to a general algorithm, several approximation schemes are proposed. / text

>Magister Scientiae - MSc / Pensions are essentially the only source of income for many retired workers. It is thus critical that the pension fund manager chooses the right type of plan for his/her workers.Every pension scheme follows its own set of rules when calculating the benefits of the fund’s members at retirement. Whichever plan the manager chooses for the members,he/she will have to invest their contributions in the financial market. The manager is therefore faced with the daunting task of selecting the most appropriate investment strat-egy as to maximize the returns from the financial assets. Due to the volatile nature of stock markets, some pension companies have attached minimum guarantees to pension contracts. These guarantees come at a price, but ensure that the member does not suffer a loss due to poorly performing equities.In this thesis we study four types of mathematical problems in pension fund management,of which three are essentially optimization problems. Firstly, following Blake [5], we show in a discrete time setting how to decompose a pension benefit into a combination of Euro-pean options. We also model the pension plan preferences of workers, sponsors and fund managers. We make a number of contributions additional to the paper by Blake [5]. In particular, we contribute graphic illustrations of the expected values of the pension fund assets, liabilities and the actuarial surplus processes. In more detail than in the original source, we derive the variance of the assets of a defined benefit pension plan. Secondly,we dedicate Chapter 6 to the problem of minimizing the cost of a minimum guarantee included in defined contribution (DC) pension contracts. Here we work in discrete time and consider multi-period guarantees similar to those in Hipp [25]. This entire chapter is original work. Using a standard optimization method, we propose a strategy that cal- culates an optimal sequence of guarantees that minimizes the sum of the squares of the present value of the total price of the guarantee. Graphic illustrations are included to in-dicate the minimum value and corresponding optimal sequence of guarantees. Thirdly, we derive an optimal investment strategy for a defined contribution fund with three financial assets in the presence of a minimum guarantee. We work in a continuous time setting and in particular contribute simulations of the dynamics of the short interest rate process and the assets in the financial market of Deelstra et al. [19]. We also derive an optimal investment strategy of the surplus process introduced in Deelstra et al. [19]. The results regarding the surplus are then converted to consider the actual investment portfolio per- taining to the wealth of the fund. We note that the aforementioned paper does not use optimal control theory. In order to illustrate the method of stochastic optimal control, we study a fourth problem by including a discussion of the paper by Devolder et al. [21] in Chapter 3. We enhance the work in the latter paper by including some simulations. The specific portfolio management strategies are applicable to banking as well (and is being pursued independently).

Le premier chapitre est une présentation des principaux concepts et résultatsconcernant la finance d'entreprise à la lumière de certains développementsrécents de l'économie du travail.Le deuxième chapitre vise à établir les propriétés d'optimalité concernantl'investissement et l'embauche d'une entreprise dans le cadre de lamaximisation d'une utilité linéaire.Le troisième chapitre traite de la problématique (éventuelle) du désinvestissementet du licenciement. Nous étudions en particulier les problèmesde la prise de décision optimale du dirigeant faisant face soit à une croissancedu marché, soit au contraire à une chute de la demande pour son produit.Le quatrième chapitre reconsidère la question en prenant en compte spécifiquementd'une borne supérieure sur la quantité pouvant être réellementvendue.Le cinquième chapitre prend en compte le phénomènes possibles de retourà la moyenne du prix unitaire du produit vendu.Le sixième et dernier chapitre reconsidère les problèmes de décision optimalepour différentes formes de dette possibles. / The first chapter is an overview of the main concepts and resultson corporate finance in the light of certain developmentsrecent labor economics .The second chapter aims to establish the optimal properties forinvestment and hiring a company under themaximizing a linear utility .The third chapter deals with the problem (if any) divestmentand firing . Nosu study particular problemsthe optimal decision of the leader facing either growthmarket , on the contrary to a drop in demand for its product.The fourth chapter reconsiders the issue , taking into account specifican upper bound on the amount that can actually besold.The fifth chapter considers the possible phenomena of retuthe average unit price of the product sold .The sixth and final chapter reconsiders the problems of optimal decisionfor different possible forms of debt

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.