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

Conic finance is a brand new quantitative finance theory. The thesis is on the applications of conic finance on South African Financial Markets. Conic finance gives a new perspective on the way people should perceive financial markets. Particularly in incomplete markets, where there are non-unique prices and the residual risk is rampant, conic finance plays a crucial role in providing prices that are acceptable at a stress level. The theory assumes that price depends on the direction of trade and there are two prices, one for buying from the market called the ask price and one for selling to the market called the bid price. The bid-ask spread reects the substantial cost of the unhedgeable risk that is present in the market. The hypothesis being considered in this thesis is whether conic finance can reduce the residual risk? Conic finance models bid-ask prices of cashows by applying the theory of acceptability indices to cashows. The theory of acceptability combines elements of arbitrage pricing theory and expected utility theory. Combining the two theories, set of arbitrage opportunities are extended to the set of all opportunities that a wide range of market participants are prepared to accept. The preferences of the market participants are captured by utility functions. The utility functions lead to the concepts of acceptance sets and the associated coherent risk measures. The acceptance sets (market preferences) are modeled using sets of probability measures. The set accepted by all market participants is the intersection of all the sets, which is convex. The size of this set is characterized by an index of acceptabilty. This index of acceptability allows one to speak of cashows acceptable at a level, known as the stress level. The relevant set of probability measures that can value the cashows properly is found through the use of distortion functions. In the first chapter, we introduce the theory of conic finance and build a foundation that leads to the problem and objectives of the thesis. In chapter two, we build on the foundation built in the previous chapter, and we explain in depth the theory of acceptability indices and coherent risk measures. A brief discussion on coherent risk measures is done here since the theory of acceptability indices builds on coherent risk measures. It is also in this chapter, that some new acceptability indices are introduced. In chapter three, focus is shifted to mathematical tools for financial applications. The chapter can be seen as a prerequisite as it bridges the gap from mathematical tools in complete markets to incomplete markets, which is the market that conic finance theory is trying to exploit. As the chapter ends, models used for continuous time modeling and simulations of stochastic processes are presented. In chapter four, the attention is focussed on the numerical methods that are relevant to the thesis. Details on obtaining parameters using the maximum likelihood method and calibrating the parameters to market prices are presented. Next, option pricing by Fourier transform methods is detailed. Finally a discussion on the bid-ask formulas relevant to the thesis is done. Most of the numerical implementations were carried out in Matlab. Chapter five gives an introduction to the world of option trading strategies. Some illustrations are used to try and explain the option trading strategies. Explanations of the possible scenarios at the expiration date for the different option strategies are also included. Chapter six is the appex of the thesis, where results from possible real market scenarios are presented and discussed. Only numerical results were reported on in the thesis. Empirical experiments could not be done due to limitations of availabilty of real market data. The findings from the numerical experiments showed that the spreads from conic finance are reduced. This results in reduced residual risk and reduced low cost of entering into the trading strategies. The thesis ends with formal discussions of the findings in the thesis and some possible directions for further research in chapter seven. / Thesis (MSc (Risk Analysis))--North-West University, Potchefstroom Campus, 2013.

Conic finance is a brand new quantitative finance theory. The thesis is on the applications of conic finance on South African Financial Markets. Conic finance gives a new perspective on the way people should perceive financial markets. Particularly in incomplete markets, where there are non-unique prices and the residual risk is rampant, conic finance plays a crucial role in providing prices that are acceptable at a stress level. The theory assumes that price depends on the direction of trade and there are two prices, one for buying from the market called the ask price and one for selling to the market called the bid price. The bid-ask spread reects the substantial cost of the unhedgeable risk that is present in the market. The hypothesis being considered in this thesis is whether conic finance can reduce the residual risk? Conic finance models bid-ask prices of cashows by applying the theory of acceptability indices to cashows. The theory of acceptability combines elements of arbitrage pricing theory and expected utility theory. Combining the two theories, set of arbitrage opportunities are extended to the set of all opportunities that a wide range of market participants are prepared to accept. The preferences of the market participants are captured by utility functions. The utility functions lead to the concepts of acceptance sets and the associated coherent risk measures. The acceptance sets (market preferences) are modeled using sets of probability measures. The set accepted by all market participants is the intersection of all the sets, which is convex. The size of this set is characterized by an index of acceptabilty. This index of acceptability allows one to speak of cashows acceptable at a level, known as the stress level. The relevant set of probability measures that can value the cashows properly is found through the use of distortion functions. In the first chapter, we introduce the theory of conic finance and build a foundation that leads to the problem and objectives of the thesis. In chapter two, we build on the foundation built in the previous chapter, and we explain in depth the theory of acceptability indices and coherent risk measures. A brief discussion on coherent risk measures is done here since the theory of acceptability indices builds on coherent risk measures. It is also in this chapter, that some new acceptability indices are introduced. In chapter three, focus is shifted to mathematical tools for financial applications. The chapter can be seen as a prerequisite as it bridges the gap from mathematical tools in complete markets to incomplete markets, which is the market that conic finance theory is trying to exploit. As the chapter ends, models used for continuous time modeling and simulations of stochastic processes are presented. In chapter four, the attention is focussed on the numerical methods that are relevant to the thesis. Details on obtaining parameters using the maximum likelihood method and calibrating the parameters to market prices are presented. Next, option pricing by Fourier transform methods is detailed. Finally a discussion on the bid-ask formulas relevant to the thesis is done. Most of the numerical implementations were carried out in Matlab. Chapter five gives an introduction to the world of option trading strategies. Some illustrations are used to try and explain the option trading strategies. Explanations of the possible scenarios at the expiration date for the different option strategies are also included. Chapter six is the appex of the thesis, where results from possible real market scenarios are presented and discussed. Only numerical results were reported on in the thesis. Empirical experiments could not be done due to limitations of availabilty of real market data. The findings from the numerical experiments showed that the spreads from conic finance are reduced. This results in reduced residual risk and reduced low cost of entering into the trading strategies. The thesis ends with formal discussions of the findings in the thesis and some possible directions for further research in chapter seven. / Thesis (MSc (Risk Analysis))--North-West University, Potchefstroom Campus, 2013.

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.

Diss. Stockholm : Handelshögskolan, 2009

The thesis is divided into three chapters.1) I study how monetary policy should be optimally designed when households show financial wealth heterogeneity.Main results: thanks to its ability to affect interest payments volatility, monetary policy has real effects even in a flexible-price cashless-limit environment; second, in a setup with nominal rigidities, price stability is no longer optimal. The extent of deviation from price stability depends on the initial level of debt dispersion.2) I assess the role of housing price movements in influencing the optimal design of monetary policy. Under the optimal simple rule, housing price movements should not be a separate target variable in addition to inflation. Furthermore, the welfare loss arising from targeting housing prices becomes quantitatively more significant the higher the degree of access to the credit market.3) I analyze the effects of fiscal policy in a currency area. Results: a public spending shock in one region increases private agents demand for imports and appreciates the terms of trade; second, a countercyclical fiscal rule can restore the Taylor principle, the uniqueness of the equilibrium and reduce macro-volatility.

La tesis estudia la evolución de la estructura de propiedad de compañías y de sus efectos sobre resultados económicos reales. El capítulo 1 proporciona un análisis empírico de la privatización en Rusia en los años 90. Se utilizan datos de 530 empresas industriales rusas para estimar los factores implicados en la decisión de privatizar una empresa, en la selección entre diversas opciones de privatización, en la distribución inicial de la propiedad que resulta, y en su evolución posterior. El capítulo 2 estudia los efectos de diferencias de intereses entre accionistas sobre decisiones de inversión y el mercado de acciones en un modelo simple. Suponiendo que los mercados financieros son incompletos, la distribución inicial de la propiedad, la riqueza y las preferencias de los propietarios afectan a su nivel preferido de inversión. Las decisiones sobre la inversión son tomadas por mayoría. Dos extensiones, la protección de accionistas de minoría y ventajas privadas del control, permiten un análisis del papel de las instituciones del gobierno corporativo en las decisiones de inversión y estructura de propiedad. Las predicciones del modelo para la evolución de la estructura de propiedad son coherentes con las observaciones empíricas en el capítulo 1. / The thesis studies the evolution of the ownership structure of companies and its effects on real economic outcomes. Chapter 1 provides an empirical analysis of the large-scale privatization in Russia in the 1990s. A comprehensive data set of 530 Russian manufacturing firms is used to estimate determinants of the decision to privatize a firm, of the choice among different options of privatization, the resulting initial ownership distribution, and its further evolution. Chapter 2 studies the effects of conflicting interests of shareholders on investment decisions and share trade in a simple model. In a setting with incomplete financial markets, the wealth, initial stake and preferences of owners affect their preferred level of investment. Decisions on investment are taken by majority. Two extensions, a protection for minority shareholders and private benefits of control, allow us to analyze the impact of corporate governance arrangements on investment decisions and ownership structure. The model predictions for the evolution of ownership are in line with the empirical observations in chapter 1.

Abstract This dissertation is comprised of three independent essays with the unifying theme of how uncertainty affects the macroeconomy. The first essay studies an incomplete market economy where the firm faces a non-trivial investment decision due to capital adjustment costs. The adjustment costs make the price of capital endogenous and help to explain the observed volatility of the returns to physical capital. The particular form of market incompleteness that is assumed in the essay is however not enough to match the observed price of risk. The essay contains also a technical contribution in showing how Arrow prices of contingent commodities can be used in computing the equilibrium in this class of models. The second essay studies the effect of population aging on asset prices. The modeling framework features deterministic transition paths for demographic structure and level of government expenditures along with aggregate uncertainty at business cycle frequency. The demographic transition leads to a projected increase of in tax rates that are needed to finance the government expenditures. This requires higher savings rates from households which reduces volatility of consumption growth and reduces the price of aggregate risk. The third essay is an empirical study which uses betting market data from the Swedish harness horse racing in conjunction with economic confidence indices. The main finding is that the risk attitudes of bettors that are reflected by the betting market data covary with the more traditional confidence measures in a reasonable way. The essay also contains a simple forecasting exercise which shows that the novel risk measure may also be useful in forecasting the industrial production. The results of the study are interpreted in terms of behavioral macroeconomics. / Tiivistelmä Tämä väitöskirja koostuu kolmesta erillisestä esseestä, joiden yhdistävä tekijä on epävarmuus ja sen vaikutukset makrotalouden ilmiöihin. Ensimmäisessä esseessä tarkastellaan taloutta, jossa markkinat ovat epätäydelliset ja fyysisen pääoman sopeuttamiskustannukset vaikuttavat yrityksen investointipäätökseen. Pääoman sopeuttamiskustannukset tekevät pääoman hinnasta endogeenisen muuttujan ja auttavat selittämään havaittua pääoman tuottojen volatiliteettia. Tutkimuksessa käytetyt markkinoiden epätäydellisyyteen johtavat oletukset eivät kuitenkaan riitä selittämään historiallisesti havaittua riskin hintaa. Essee sisältää myös teknisen kontribuution. Siinä osoitetaan, miten talouden tilasta riippuvien hyödykkeiden Arrow-hintoja voidaan hyödyntää tämän tyyppisten talouksien tasapainon numeerisessa ratkaisemisessa. Toinen essee tarkastelee väestön ikääntymisen vaikutuksia varallisuushyödykkeiden hintoihin. Malli yhdistää väestörakenteen ja julkisten kulutusmenojen deterministisen muutoksen sekä suhdannevaihtelua kuvaavan kokonaistaloudellisen epävarmuuden. Väestörakenteen odotettu muutos johtaa julkisten kulutusmenojen kasvun myötä veroasteiden nousuun. Kotitaloudet joutuvat säästämään enemmän, mikä vähentää kulutuksen kasvun volatiliteettia ja kokonaistaloudellisen riskin hintaa. Kolmas essee on empiirinen tutkimus, jossa käytetään havaintoaineistoa Ruotsin ravivedonlyöntimarkkinoilta sekä taloudellisia luottamusindikaattoreita. Tärkein tulos on että vedonlyöntiaineiston heijastama suhtautuminen riskiin näyttää olevan vuorovaikutuksessa perinteisten luottamusindikaattoreiden kanssa. Esseessä käytetään myös yksinkertaista aikasarjamallia, joka viittaa siihen, että vedonlyöntiaineiston perusteella laskettu riskiin suhtautumisen mitta voi olla hyödyllinen teollisuustuotannon ennustamisessa. Tuloksia tulkitaan behavioraalisen makrotaloustieteen valossa.

Cette thèse porte sur la valorisation et les stratégies financières de couverture des risques dans les marchés de l'énergie. Ces marchés présentent des particularités qui les distinguent des marchés financiers standards, notamment l'illiquidité et l'incomplétude. L'illiquidité se reflète par des coûts de transactions importants et des contraintes sur les volumes échangés. L'incomplétude est l'incapacité de pouvoir répliquer parfaitement des produits dérivés. Nous nous intéressons à différents aspects de l'incomplétude de marché. La première partie porte sur la valorisation dans les modèles de Lévy. Nous obtenons une formule approximative du prix d'indifférence et nous mesurons la prime minimale à apporter par rapport au modèle de Black-Scholes. La deuxième partie concerne la valorisation d'options spread en présence de corrélation stochastique. Les options spread portent sur la différence de prix entre deux sous-jacents -- par exemple gaz et électricité -- et sont très utilisées sur les marchés de l'énergie. Nous proposons une procédure numérique efficace pour calculer le prix de ces options. Enfin, la troisième partie traite de la valorisation d'un produit comportant un risque exogène dont il existe des prévisions. Nous proposons une stratégie dynamique optimale en présence de risque de volume, et l'appliquons à la valorisation des fermes éoliennes. De plus, une partie est consacrée aux stratégies optimales asymptotiques en présence de coûts de transactions. / This thesis tackles three issues on pricing and hedging in energy markets. Energy markets differ from financial markets mainly in two ways: illiquidity and incompletness. Illiquidity (or lack of liquidity) translates into transaction costs and volume constraints. Incompletness means incapacity to perfectly hedge derivatives. We study different aspects of incomplete markets. First, we focus on indifference pricing in exponential Lévy models. We obtained an approximate formula by considering a Lévy process as a perturbed Brownian motion. That way we obtain the minimal correction from Black-Scholes price. Second, we present a numerical procedure to price spread options when underlyings are stochastically correlated. These options are very popular in energy markets, underlyings being for instance gas and electricity. Third, we derive optimal strategies using exogeneous factors forecasts. We exhibit an explicit pricing formula and an optimal strategy handling volume risk and apply it to wind farms valuation. Finally, a short review of optimal strategies taking into account transaction costs is made

Diese Arbeit beginnt mit einer Analyse von stochastischen Rückwärtsdifferentialgleichungen (BSDEs) mit Sprüngen, getragen von zufälligen Maßen mit ggf. unendlicher Aktivität und zeitlich inhomogenem Kompensator. Unter konkreten, in Anwendungen leicht verifizierbaren Bedingungen liefern wir Existenz-, Eindeutigkeits- und Vergleichsergebnisse beschränkter Lösungen für eine Klasse von Generatorfunktionen, die nicht global Lipschitz-stetig im Sprungintegranden sein brauchen. Der übrige Teil der Arbeit behandelt robuste Bewertung und Hedging in unvollständigen Märkten. Wir verfolgen den No-Good-Deal-Ansatz, der Good-Deal-Grenzen liefert, indem nur eine Teilmenge der risikoneutralen Maße mit ökonomischer Bedeutung betrachtet wird (z.B. Grenzen für instantanen Sharpe-Ratio, optimale Wachstumsrate oder erwarteten Nutzen). Durchweg untersuchen wir ein Konzept des Good-Deal-Hedgings für welches Hedgingstrategien als Minimierer geeigneter dynamischer Risikomaße auftreten, was optimale Risikoteilung mit der Markt erlaubt. Wir zeigen, dass Hedging mindestens im-Mittel-selbstfinanzierend ist, also, dass Hedgefehler unter geeigneten A-priori-Bewertungsmaßen eine Supermartingaleigenschaft haben. Wir leiten konstruktive Ergebnisse zu Good-Deal-Bewertung und -Hedging im Rahmen von Prozessen mit Sprüngen durch BSDEs mit Sprüngen, sowie im Brown''schen Fall mit Driftunsicherheit durch klassische BSDEs und mit Volatilitätsunsicherheit durch BSDEs zweiter Ordnung her. Wir liefern neue Beispiele, die insbesondere für versicherungs- und finanzmathematische Anwendungen von Bedeutung sind. Bei Ungewissheit des Real-World-Maßes führt ein Worst-Case-Ansatz bei Annahme mehrerer Referenzmaße zu Good-Deal-Hedging, welches robust bzgl. Unsicherheit, im Sinne von gleichmäßig über alle Referenzmaße mindestens im-Mittel-selbstfinanzierend, ist. Daher ist bei hinreichend großer Driftunsicherheit Good-Deal-Hedging zur Risikominimierung äquivalent. / This thesis starts by an analysis of backward stochastic differential equations (BSDEs) with jumps driven by random measures possibly of infinite activity with time-inhomogeneous compensators. Under concrete conditions that are easy to verify in applications, we prove existence, uniqueness and comparison results for bounded solutions for a class of generators that are not required to be globally Lipschitz in the jump integrand. The rest of the thesis deals with robust valuation and hedging in incomplete markets. The focus is on the no-good-deal approach, which computes good-deal valuation bounds by using only a subset of the risk-neutral measures with economic meaning (e.g. bounds on instantaneous Sharpe ratios, optimal growth rates, or expected utilities). Throughout we study a notion of good-deal hedging consisting in minimizing some dynamic risk measures that allow for optimal risk sharing with the market. Hedging is shown to be at least mean-self-financing in that hedging errors satisfy a supermartingale property under suitable valuation measures. We derive constructive results on good-deal valuation and hedging in a jump framework using BSDEs with jumps, as well as in a Brownian setting with drift uncertainty using classical BSDEs and with volatility uncertainty using second-order BSDEs. We provide new examples which are particularly relevant for actuarial and financial applications. Under ambiguity about the real-world measure, a worst-case approach under multiple reference priors leads to good-deal hedging that is robust w.r.t. uncertainty in that it is at least mean-self-financing uniformly over all priors. This yields that good-deal hedging is equivalent to risk-minimization if drift uncertainty is sufficiently large.