Detailansicht

Applications of non-convex optimization in portfolio selection

David Osswald Wozabal

Art der Arbeit

Dissertation

Universität

Universität Wien

Fakultät

Fakultät für Wirtschaftswissenschaften

Betreuer*in

Georg Pflug

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DOI

10.25365/thesis.1625

URN

urn:nbn:at:at-ubw:1-29929.36326.285869-4

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(Print-Exemplar eventuell in Bibliothek verfügbar)

Abstracts

Abstract

(Deutsch)

Die vorgelegte Arbeit befasst sich mit nicht-konvexer Optimierung in dem Gebiet der Portfolio Selection. Thematisch lässt sich die Arbeit in zwei Teilgebiete strukturieren: (1) Das Lösen von Mean-Risk Problemen mit Value-at-Risk als Risikomaß: Es werden Methoden zum Auffinden von effizienten Portfolios für den Fall von diskret verteilten Asset Returns vorgestellt. Die behandelten Probleme sind (wegen der Nicht-Konvexität des Value-at-Risk) nicht konvex und lassen sich als Differenz von konvexen Funktionen darstellen. Es werden sowohl Branch-and-Bound als auch approximative Lösungsverfahren angewandt. Die globalen Lösungen des Branch-and-Bound werden mit den Lösungen der approximativen Verfahren verglichen. (2) Robustifizierung von Portfolio-Selection Problemen: In den letzten Jahren gibt es in der Literatur verstärkt Bemühungen Optimierungsprobleme bezüglich Unsicherheiten in den Parametern zu robustifizieren. Robustifizierte Lösungen haben die Eigenschaft, dass moderate Variationen von Parametern nicht zu dramatischen Verschlechterungen der Lösungen führen. Im Rahmen der robusten Portfolio Optimierung geht es hauptsächlich darum, Lösungen in Bezug auf Abweichungen in den Verteilungen der Gewinne der verwendeten Finanzinstrumente zu kontrollieren. In der gegenständlichen Arbeit werden mit Hilfe von Wahrscheinlichkeitsmetriken sogenannte Ambiguity Mengen definiert, welche alle Verteilungen enthalten, die aufgrund der Datenlage als mögliche Verteilungen in Frage kommen. Die verwendete Metrik, die sogenannte Kantorovich (Wasserstein) Metrik, ermöglicht es mittels Ergebnissen der nichtparametrischen Statistik, die Ambiguity Mengen als Konfidenzmengen um die empirischen Verteilungschätzer zu interpretieren. Mittels der beschriebenen Methoden werden Mean-Risk Probleme robustifiziert. Diese Probleme sind zunächst infinit und werden in einem weiteren Schritt zu nicht konvexen semi-definiten Problemen umformuliert. Die Lösung dieser Probleme basiert einerseits auf einem Algortihmus zum Lösen von semi-definiten Problemen mit unendlich vielen Nebenbedingungen und andererseits auf Methoden zum approximativen Lösen von nicht konvexen Problemen (dem sogenannten Difference of Convex Algorithm).

Abstract

(Englisch)

The thesis is concerned with application of non-convex programming to problems of portfolio optimization in a single stage stochastic optimization framework. In particular two different classes of portfolio selection problems are investigated. In both the problems a scenario based approach to modeling uncertainty is pursued, i.e. the randomness in the models is always described by finitely many joint realizations of the asset returns. The thesis is structured into three chapters briefly outlined below: (1) A D.C. Formulation of Value-at-Risk constrained Optimization: In this Chapter the aim is to solve mean risk models with the Value-at-Risk as a risk measure. In the case of finitely supported return distributions, it is shown that the Value-at-Risk can be written as a D.C. function and the mentioned mean risk problem therefore corresponds to a D.C. problem. The non-convex problem of optimizing the Value at Risk is rather extensively treated in the literature and there are various approximative solution techniques as well as some approaches to solve the problem globally. The reformulation as D.C. problem provides an insight into the structure of the problem, which can be exploited to devise a Branch-and-Bound algorithm for finding global solutions for small to medium sized instances. The possibility of refining epsilon-optimal solutions obtained from the Branch-and-Bound framework via local search heuristics is also discussed in this Chapter. (2) Value-at-Risk constrained optimization using the DCA: In this part of the thesis the Value-at-Risk problem is once again investigated with the aim of solving problems of realistic sizes in relatively short time. Since the Value at Risk optimization can be shown to be a NP hard problem, this can only be achieved by sacrificing on the guaranteed globality of the solutions. Therefore a local solution technique for unconstrained D.C. problems called Difference of Convex Algorithm (DCA) is employed. To solve the problem a new variant of the DCA the so called 'hybrid DCA' is proposed, which preserves the favorable convergence properties of the computationally hard 'complete DCA' as well as the computational tractability of the so called 'simple DCA'. The results are tested for small problems and the solutions are shown to actually coincide with the global optima obtained with the Branch-and-Bound algorithm in most of the cases. For realistic problem sizes the proposed method is shown to consistently outperform known heuristic approximations implemented in commercial software. (3) A Framework for Optimization under Ambiguity: The last part of the thesis is devoted to a different topic which received much attention in the recent stochastic programming literature: the topic of robust optimization. More specifically the aim is to robustify single stage stochastic optimization models with respect to uncertainty about the distributions of the random variables involved in the formulation of the stochastic program. The aim is to explore ways of explicitly taking into account ambiguity about the distributions when finding a decision while imposing only very weak restrictions on possible probability models that are taken into consideration. Ambiguity is defined as possible deviation from a discrete reference measure Q (in this work the empirical measure). To this end a so called ambiguity set B, that contains all the measures that can reasonably be assumed to be the real measure P given the available data, is defined. Since the idea is to devise a general approach not restricted by assuming P to be an element of any specific parametric family, we define our ambiguity sets by the use of general probability metrics. Relative to these measures a worst case approach is adopted to robustify the problem with respect to B. The resulting optimization problems turn out to be infinite and are reduced to non-convex semi-definite problems. In the last part of the paper we show how to solve these problems numerically for the example of a mean risk portfolio selection problem with Expected Shortfall under a Threshold as the risk measure. The DCA in combination with an iterative algorithm to approximate the infinite set of constraints by finitely many ones is used to obtain numerical solutions to the problem.

Autor*innen

David Osswald Wozabal

Haupttitel (Englisch)

Applications of non-convex optimization in portfolio selection

Paralleltitel (Deutsch)

Anwendungen nicht konvexer Optimierung in Portfolio-Optimierungsproblemen

Publikationsjahr

2008

Umfangsangabe

93 S. : graph. Darst.

Sprache

Englisch

Beurteiler*innen

Georg Pflug ,

Jitka Dupacova

Klassifikationen

31 Mathematik > 31.48 Variationsrechnung ,

31 Mathematik > 31.80 Angewandte Mathematik

AC Nummer

AC05038826

Utheses ID

1304

Studienkennzahl

UA | 084 | 136 | |

Detailansicht

Abstracts

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