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Essays in mathematical financeMurgoci, Agatha January 2009 (has links)
Diss. Stockholm : Handelshögskolan, 2009
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A generalized Neyman-Pearson lemma for hedge problems in incomplete marketsRudloff, Birgit 07 October 2005 (has links)
Some financial problems as minimizing the shortfall risk when hedging in incomplete markets lead to problems belonging to test theory. This paper considers
a generalization of the Neyman-Pearson lemma. With methods of convex duality
we deduce the structure of an optimal randomized test when testing a compound
hypothesis against a simple alternative. We give necessary and sufficient optimality
conditions for the problem.
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Comparing Approximations for Risk Measures Related to Sums of Correlated Lognormal Random VariablesKarniychuk, Maryna 30 November 2006 (has links)
In this thesis the performances of different approximations are compared for a standard actuarial and
financial problem: the estimation of quantiles and conditional
tail expectations of the final value of a series of discrete cash
flows.
To calculate the risk measures such as quantiles and Conditional
Tail Expectations, one needs the distribution function of the
final wealth. The final value of a series of discrete payments in
the considered model is the sum of dependent lognormal random
variables. Unfortunately, its distribution function cannot be
determined analytically. Thus usually one has to use
time-consuming Monte Carlo simulations. Computational time still
remains a serious drawback of Monte Carlo simulations, thus
several analytical techniques for approximating the distribution
function of final wealth are proposed in the frame of this thesis.
These are the widely used moment-matching approximations and
innovative comonotonic approximations.
Moment-matching methods approximate the unknown distribution
function by a given one in such a way that some characteristics
(in the present case the first two moments) coincide. The ideas of
two well-known approximations are described briefly. Analytical
formulas for valuing quantiles and Conditional Tail Expectations
are derived for both approximations.
Recently, a large group of scientists from Catholic University
Leuven in Belgium has derived comonotonic upper and comonotonic
lower bounds for sums of dependent lognormal random variables.
These bounds are bounds in the terms of "convex order". In order
to provide the theoretical background for comonotonic
approximations several fundamental ordering concepts such as
stochastic dominance, stop-loss and convex order and some
important relations between them are introduced. The last two
concepts are closely related. Both stochastic orders express which
of two random variables is the "less dangerous/more attractive"
one.
The central idea of comonotonic upper bound approximation is to
replace the original sum, presenting final wealth, by a new sum,
for which the components have the same marginal distributions as
the components in the original sum, but with "more dangerous/less
attractive" dependence structure. The upper bound, or saying
mathematically, convex largest sum is obtained when the components
of the sum are the components of comonotonic random vector.
Therefore, fundamental concepts of comonotonicity theory which are
important for the derivation of convex bounds are introduced. The
most wide-spread examples of comonotonicity which emerge in
financial context are described.
In addition to the upper bound a lower bound can be derived as
well. This provides one with a measure of the reliability of the
upper bound. The lower bound approach is based on the technique of
conditioning. It is obtained by applying Jensen's inequality for
conditional expectations to the original sum of dependent random
variables. Two slightly different version of conditioning random
variable are considered in the context of this thesis. They give
rise to two different approaches which are referred to as
comonotonic lower bound and comonotonic "maximal variance" lower
bound approaches.
Special attention is given to the class of distortion risk
measures. It is shown that the quantile risk measure as well as
Conditional Tail Expectation (under some additional conditions)
belong to this class. It is proved that both risk measures being
under consideration are additive for a sum of comonotonic random
variables, i.e. quantile and Conditional Tail Expectation for a
comonotonic upper and lower bounds can easily be obtained by
summing the corresponding risk measures of the marginals involved.
A special subclass of distortion risk measures which is referred
to as class of concave distortion risk measures is also under
consideration. It is shown that quantile risk measure is not a
concave distortion risk measure while Conditional Tail Expectation
(under some additional conditions) is a concave distortion risk
measure. A theoretical justification for the fact that "concave"
Conditional Tail Expectation preserves convex order relation
between random variables is given. It is shown that this property
does not necessarily hold for the quantile risk measure, as it is
not a concave risk measure.
Finally, the accuracy and efficiency of two moment-matching,
comonotonic upper bound, comonotonic lower bound and "maximal
variance" lower bound approximations are examined for a wide range
of parameters by comparing with the results obtained by Monte
Carlo simulation. It is justified by numerical results that,
generally, in the current situation lower bound approach
outperforms other methods. Moreover, the preservation of convex
order relation between the convex bounds for the final wealth by
Conditional Tail Expectation is demonstrated by numerical results.
It is justified numerically that this property does not
necessarily hold true for the quantile.
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FRM Financial Risk MeterAlthof, Michael Gottfried 19 September 2022 (has links)
Der Risikobegriff bezieht sich auf die Wahrscheinlichkeit eines Schadens aufgrund einer Gefährdungsexposition, in der Finanzwelt meist finanzielle Verluste. Viele Risiken der globalen Finanzwirtschaft sind unbekannt. „Wir wissen es, wenn wir es sehen“, um Potter Stewart (1964) zu paraphrasieren. Der Financial Risk Meter (FRM) soll Aufschluss über die Entstehung systemischer Risiken geben. Durch Verwendung von Quantilregressionstechniken ist der FRM nicht nur ein Maß für finanzielle Risiken. Er bietet durch seine Netzwerktopologie einen tiefen Einblick in die Spill-over-Effekte, die sich als systemische Risikoereignisse manifestieren können. Das FRM-Framework wird in verschiedenen Märkten und Regionen entwickelt. Die FRM-Daten werden für Risiko-Prognose sowie für Portfoliooptimierung genutzt. In Kapitel 1 wird der FRM vorgestellt und auf die Aktienmärkte in den USA und Europa, sowie auch auf die Zinsmärkte und Credit-Default-Swaps angewendet. Der FRM wird dann verwendet, um wirtschaftliche Rezessionen zu prognostizieren. In Kapitel 2 wird der FRM auf den Markt der Kryptowährungen angewendet, um das erste Risikomaß für diese neue Anlageklasse zu generieren. Die errechneten FRM-Daten zu Abhängigkeiten, Spillover-Effekten und Netzwerkaufbau werden dann verwendet, um Tail-Risk-optimierte Portfolios zu erstellen. Der Portfoliooptimierungsansatz wird in Kapitel 3 weitergeführt, in dem der FRM auf die sogenannten Emerging Markets (EM)-Finanzinstitute angewendet wird, mit zwei Zielen. Einerseits gibt der FRM für EM spezifische Spillover-Abhängigkeiten bei Tail-Risk-Ereignissen innerhalb von Sektoren von Finanzinstituten an, zeigt aber auch Abhängigkeiten zwischen den Ländern. Die FRM-Daten werden dann wieder mit Portfoliomanagementansätzen kombiniert. In Kapitel 4 entwickelt den FRM for China ist, eines der ersten systemischen Risikomaße in der Region, zeigt aber auch Methoden zur Erkennung von Spill-Over-Kanälen in Nachbarländer und zwischen Sektoren. / The concept of risk deals with the exposure to danger, in the world of finance the danger of financial losses. In a globalised financial economy, many risks are unknown. "We know it when we see it", to paraphrase Justice Potter Stewart (1964). The Financial Risk Meter (FRM) sheds light on the emergence of systemic risk. Using of quantile regression techniques, it is a meter for financial risk, and its network topology offers insight into the spill-over effects risking systemic risk events. In this thesis, the FRM framework in various markets and regions is developed and the FRM data is used for risk now- and forecasting, and for portfolio optimization approaches. In Chapter 1 the FRM is presented and applied to equity markets in the US and Europe, but also interest rate and credit-default swap markets. The FRM is then used to now-cast and predict economic recessions. In Chapter 2 the FRM is applied to cryptocurrencies, to generate the first risk meter in this nascent asset class. The generated FRM data concerning dependencies, spill-over effects and network set-up are then used to create tail-risk optimised portfolios. In Chapter 3 the FRM is applied to the global market Emerging Market (EM) financial institutions. The FRM for EM gives specific spill-over dependencies in tail-risk events within sectors of financial institutions, but also shows inter-country dependencies between the EM regions. The FRM data is then combined with portfolio management approaches to create tail-risk sensitive portfolios of EM Financial institutions with aim to minimize risk clusters in a portfolio context. In Chapter 4 the Financial Risk Meter for China is developed as the first systemic risk meter in the region, but also derives methods to detect spill-over channels to neighbouring countries within and between financial industry sectors.
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Dynamic convex risk measuresPenner, Irina 17 March 2008 (has links)
In dieser Arbeit werden verschiedene Eigenschaften von dynamischen konvexen Risikomaßen für beschränkte Zufallsvariablen untersucht. Dabei gehen wir vor allem der Frage nach, wie die Risikobewertungen in verschiedenen Zeitpunkten von einander abhängen, und wie sich solche Zeitkonsistenzeigenschaften in der Dynamik der Penalty-Funktionen und Risikoprozesse widerspiegeln. Im Kapitel 2 widmen wir uns zunächst der starken Zeitkonsistenz und charakterisieren diese mithilfe von Akzeptanzmengen, Penalty-Funktionen und einer gemeinsamen Supermartingaleigenschaft des Risikoprozesses und seiner Penalty-Funktion. Die Charakterisierung durch Penalty-Funktionen liefert eine explizite Form der Doob- und der Riesz-Zerlegung des Prozesses der Penalty-Funktionen. Anschließend führen wir einen schwächeren Begriff der Zeitkonsistenz ein, den wir Besonnenheit nennen. In Analogie zu dem zeitkonsistenten Fall charakterisieren wir Besonnenheit durch Akzeptanzmengen, Penalty-Funktionen und eine bestimmte Supermartingaleigenschaft. Diese Supermartingaleigenschaft gilt allgemeiner für alle beschränkten adaptierten Prozesse, die sich ohne zusätzliches Risiko aufrechterhalten lassen. Wir nennen solche Prozesse nachhaltig und beschreiben Nachhaltigkeit durch eine gemeinsame Supermartingaleigenschaft des Prozesses und der schrittweisen Penalty-Funktionen. Dieses Resultat kann als eine verallgemeinerte optionale Zerlegung unter konvexen Restriktionen gesehen werden. Mithilfe der Supermartingaleigenschaft identifizieren wir das stark zeitkonsistente dynamische Risikomaß, das aus jedem beliebigen Risikomaß rekursiv konstruiert werden kann, als den kleinsten Prozeß, der nachhaltig ist und den Endverlust minimiert. Diese Beschreibung liefert ein neues Argument für den Einsatz von zeitkonsistenten Risikomaßen. Im Kapitel 3 diskutieren wir das asymptotische Verhalten von zeitkonsistenten und von besonnenen Risikomaßen hinsichtlich der asymptotischen Sicherheit und der asymptotischen Präzision. Im Kapitel 4 werden die allgemeinen Ergebnisse aus den Kapiteln 2 und 3 anhand des entropischen Risikomaßes und des Superhedging-Preisprozesses veranschaulicht. / In this thesis we study various properties of a dynamic convex risk measure for bounded random variables. The main subject is to investigate possible interdependence of conditional risk assessments at different times and the manifestation of these time consistency properties in the dynamics of corresponding penalty functions and risk processes. In Chapter 2 we focus first on the strong notion of time consistency and characterize it in terms of penalty functions, acceptance sets and a joint supermartingale property of the risk measure and its penalty function. The characterization in terms of penalty functions provides the explicit form of the Doob and of the Riesz decomposition of the penalty function process for a time consistent risk measure. Then we introduce and study a weaker notion of time consistency, that we call prudence. Similar to the time consistent case, we characterize prudent dynamic risk measures in terms of acceptance sets, of penalty functions and by a certain supermartingale property. This supermartingale property holds more generally for any bounded adapted process that can be upheld without any additional risk. We call such processes sustainable, and we give an equivalent characterization of sustainability in terms of a combined supermartingale property of a process and one-step penalty functions. This result can be viewed as a generalized optimal decomposition under convex constraints. The supermartingale property allows us to characterize the strongly time consistent risk measure arising from any dynamic risk measure via recursive construction as the smallest process that is sustainable and covers the final loss. Thus our discussion provides a new reason for using strongly time consistent risk measures. In Chapter 3 we discuss the limit behavior of time consistent and of prudent risk measures in terms of asymptotic safety and of asymptotic precision. In the final Chapter 4 we illustrate the general results of Chapter 2 and Chapter 3 by examples. In particular we study the entropic dynamic risk measure and the superhedging price process under convex constraints.
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On some aspects of coherent risk measures and their applicationsAssa, Hirbod 07 1900 (has links)
Le sujet principal de cette thèse porte sur les mesures de risque. L'objectif général est d'investiguer certains aspects des mesures de risque dans les applications financières. Le cadre théorique de ce
travail est celui des mesures cohérentes de risque telle que définie dans Artzner et al (1999). Mais ce n'est pas la seule classe de mesure du risque que nous étudions. Par exemple, nous étudions aussi quelques aspects des "statistiques naturelles
de risque" (en anglais natural risk statistics) Kou et al (2006) et des mesures convexes du risque Follmer and Schied(2002). Les contributions principales de cette
thèse peuvent être regroupées selon trois axes: allocation de capital, évaluation des risques et capital requis et solvabilité.
Dans le chapitre 2 nous caractérisons les mesures de risque avec la propriété de Lebesgue sur l'ensemble des processus bornés càdlàg (continu à droite, limité à gauche). Cette caractérisation nous permet de présenter deux applications dans l'évaluation des risques et l'allocation de
capital. Dans le chapitre 3, nous étendons la notion de statistiques naturelles de risque à l'espace des suites infinies.
Cette généralisation nous permet de construire de façon cohérente des mesures de risque pour des bases de données de n'importe quelle taille. Dans le chapitre 4, nous discutons le concept de "bonnes affaires" (en anglais Good Deals), pour notamment caractériser les situations du marché où ces positions pathologiques
sont présentes. Finalement, dans le chapitre 5, nous essayons de relier les trois chapitres en étendant la définition de "bonnes affaires" dans un cadre plus
large qui comprendrait les mesures de risque analysées dans les chapitres 2 et 3. / The aim of this thesis is to study several aspects of risk measures particularly in the context of financial applications. The primary framework that we use is that of coherent risk measures as defined in Artzner et al (1999). But this is not the only class of risk measures that we study here. We also investigate the concepts of natural risk statistics Kou et al (2006) and convex risk measure Follmer/ and Schied (2002). The main contributions of this Thesis can be classified in three main axes: Capital allocation, risk measurement and capital requirement and solvency. In chapter 2, we characterize risk measures with the Lebesgue property on bounded càdlàg processes. This allows to present two applications in risk assessment and capital allocation. In chapter 3, we extend the concept of natural risk statistics to the space of infinite sequences. This has been done in order to introduce a consistent way of constructing risk measures for data bases of any size. In chapter 4, we discuss the concept of Good Deals and how to deal with a situation where these pathological positions are present in the market. Finally, in chapter 5, we try to relate all three chapters by extending the definition of Good Deals to a larger set of risk measures that somehow includes the discussions in chapters 2 and 3.
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Risk preferences and their robust representationDrapeau, Samuel 16 June 2010 (has links)
Ziel dieser Dissertation ist es, den Begriff des Risikos unter den Aspekten seiner Quantifizierung durch robuste Darstellungen zu untersuchen. In einem ersten Teil wird Risiko anhand Kontext-Invarianter Merkmale betrachtet: Diversifizierung und Monotonie. Wir führen die drei Schlüsselkonzepte, Risikoordnung, Risikomaß und Risikoakzeptanzfamilen ein, und studieren deren eins-zu-eins Beziehung. Unser Hauptresultat stellt eine eindeutige duale robuste Darstellung jedes unterhalbstetigen Risikomaßes auf topologischen Vektorräumen her. Wir zeigen auch automatische Stetigkeitsergebnisse und robuste Darstellungen für Risikomaße auf diversen Arten von konvexen Mengen. Diese Herangehensweise lässt bei der Wahl der konvexen Menge viel Spielraum, und erlaubt damit eine Vielfalt von Interpretationen von Risiko: Modellrisiko im Falle von Zufallsvariablen, Verteilungsrisiko im Falle von Lotterien, Abdiskontierungsrisiko im Falle von Konsumströmen... Diverse Beispiele sind dann in diesen verschiedenen Situationen explizit berechnet (Sicherheitsäquivalent, ökonomischer Risikoindex, VaR für Lotterien, "variational preferences"...). Im zweiten Teil, betrachten wir Präferenzordnungen, die möglicherweise zusätzliche Informationen benötigen, um ausgedrückt zu werden. Hierzu führen wir einen axiomatischen Rahmen in Form von bedingten Präferenzordungen ein, die lokal mit der Information kompatibel sind. Dies erlaubt die Konstruktion einer bedingten numerischen Darstellung. Wir erhalten eine bedingte Variante der von Neumann und Morgenstern Darstellung für messbare stochastische Kerne und erweitern dieses Ergebnis zur einer bedingten Version der "variational preferences". Abschließend, klären wir das Zusammenpiel zwischen Modellrisiko und Verteilungsrisiko auf der axiomatischen Ebene. / The goal of this thesis is the conceptual study of risk and its quantification via robust representations. We concentrate in a first part on context invariant features related to this notion: diversification and monotonicity. We introduce and study the general properties of three key concepts, risk order, risk measure and risk acceptance family and their one-to-one relations. Our main result is a uniquely characterized dual robust representation of lower semicontinuous risk orders on topological vector space. We also provide automatic continuity and robust representation results on specific convex sets. This approach allows multiple interpretation of risk depending on the setting: model risk in the case of random variables, distributional risk in the case of lotteries, discounting risk in the case of consumption streams... Various explicit computations in those different settings are then treated (economic index of riskiness, certainty equivalent, VaR on lotteries, variational preferences...). In the second part, we consider preferences which might require additional information in order to be expressed. We provide a mathematical framework for this idea in terms of preorders, called conditional preference orders, which are locally compatible with the available information. This allows us to construct conditional numerical representations of conditional preferences. We obtain a conditional version of the von Neumann and Morgenstern representation for measurable stochastic kernels and extend then to a conditional version of the variational preferences. We finally clarify the interplay between model risk and distributional risk on the axiomatic level.
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On some aspects of coherent risk measures and their applicationsAssa, Hirbod 07 1900 (has links)
Le sujet principal de cette thèse porte sur les mesures de risque. L'objectif général est d'investiguer certains aspects des mesures de risque dans les applications financières. Le cadre théorique de ce
travail est celui des mesures cohérentes de risque telle que définie dans Artzner et al (1999). Mais ce n'est pas la seule classe de mesure du risque que nous étudions. Par exemple, nous étudions aussi quelques aspects des "statistiques naturelles
de risque" (en anglais natural risk statistics) Kou et al (2006) et des mesures convexes du risque Follmer and Schied(2002). Les contributions principales de cette
thèse peuvent être regroupées selon trois axes: allocation de capital, évaluation des risques et capital requis et solvabilité.
Dans le chapitre 2 nous caractérisons les mesures de risque avec la propriété de Lebesgue sur l'ensemble des processus bornés càdlàg (continu à droite, limité à gauche). Cette caractérisation nous permet de présenter deux applications dans l'évaluation des risques et l'allocation de
capital. Dans le chapitre 3, nous étendons la notion de statistiques naturelles de risque à l'espace des suites infinies.
Cette généralisation nous permet de construire de façon cohérente des mesures de risque pour des bases de données de n'importe quelle taille. Dans le chapitre 4, nous discutons le concept de "bonnes affaires" (en anglais Good Deals), pour notamment caractériser les situations du marché où ces positions pathologiques
sont présentes. Finalement, dans le chapitre 5, nous essayons de relier les trois chapitres en étendant la définition de "bonnes affaires" dans un cadre plus
large qui comprendrait les mesures de risque analysées dans les chapitres 2 et 3. / The aim of this thesis is to study several aspects of risk measures particularly in the context of financial applications. The primary framework that we use is that of coherent risk measures as defined in Artzner et al (1999). But this is not the only class of risk measures that we study here. We also investigate the concepts of natural risk statistics Kou et al (2006) and convex risk measure Follmer/ and Schied (2002). The main contributions of this Thesis can be classified in three main axes: Capital allocation, risk measurement and capital requirement and solvency. In chapter 2, we characterize risk measures with the Lebesgue property on bounded càdlàg processes. This allows to present two applications in risk assessment and capital allocation. In chapter 3, we extend the concept of natural risk statistics to the space of infinite sequences. This has been done in order to introduce a consistent way of constructing risk measures for data bases of any size. In chapter 4, we discuss the concept of Good Deals and how to deal with a situation where these pathological positions are present in the market. Finally, in chapter 5, we try to relate all three chapters by extending the definition of Good Deals to a larger set of risk measures that somehow includes the discussions in chapters 2 and 3.
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[en] TACTICAL ASSET ALLOCATION FOR OPEN PENSION FUNDS USING MULTI-STAGE STOCHASTIC PROGRAMMING / [pt] ALOCAÇÃO TÁTICA DE ATIVOS PARA EMPRESAS DE PREVIDÊNCIA COMPLEMENTAR VIA PROGRAMAÇÃO ESTOCÁSTICA MULTIESTÁGIOTHIAGO BARATA DUARTE 11 July 2016 (has links)
[pt] Uma importante questão que se coloca para entidades abertas de
previdência complementar e sociedades seguradoras que operam previdência
complementar é a definição de uma gestão dos ativos e passivos (do inglês ALM
– Asset and Liability Management). Tal questão se torna mais relevante em um
cenário de alta competitividade, margens operacionais decrescentes, garantias
mínimas de rentabilidade para um passivo estocástico de longo prazo e um
período de queda da rentabilidade dos instrumentos financeiros, sendo estes
muitas vezes de difícil precificação e pouco previsíveis num mercado volátil
como o brasileiro. Somada a estas dificuldades, as companhias deste mercado
estão sujeitas a uma regulação baseada em riscos, oriunda de práticas
internacionais, adotada pelo órgão superior, Susep, que impõe restrições
regulamentares para a manutenção da solvência das companhias, o que eleva a
dificuldade da definição de um modelo. Diante deste cenário, esta dissertação
apresenta uma proposta de ALM baseada em um modelo de programação
estocástica multiestágio que tem como objetivo definir dinamicamente a alocação
ótima dos ativos, incluindo títulos com pagamentos de cupons, e mensurar o risco
de insolvência da companhia para o horizonte de planejamento. / [en] An important issue of open pension funds and insurance companies that operate supplementary pension is the definition of an asset and liability management (ALM) framework. Such a question becomes more relevant in a scenario of high competition, declining operating margins, minimum guaranteed returns to a stochastic long-term liability and a period of falling returns on financial instruments, these being often difficult to pricing and predictable in a volatile market such as Brazil. Added to these issues, those companies are subject to a risk-based regulation, derived from international practices adopted by the national insurance regulator, Susep, which imposes constraints to maintain solvency of companies and therefore increases the complexity of an ALM framework. Due this condition, this dissertation presents a proposition of ALM based on a multistage stochastic programming model, which aims to define a dynamic optimal asset allocation, including bonds with coupons payment, and measure the company s insolvency risk for the planning horizon.
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On the design of customized risk measures in insurance, the problem of capital allocation and the theory of fluctuations for Lévy processesOmidi Firouzi, Hassan 12 1900 (has links)
No description available.
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