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Risk, Uncertainty and Optimisation on Infinite Dimensional Ordered Spaces

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Risk, Uncertainty and Optimisation on Infinite Dimensional Ordered Spaces Risk, Uncertainty and Optimisation on Infinite Dimensional Ordered Spaces Dissertation an der Fakult¨atf¨urMathematik, Informatik und Statistik der Ludwig-Maximilians-Universit¨atM¨unchen vorgelegt von Felix-Benedikt Liebrich 7. August 2019 1. Gutachter: Priv.-Doz. Dr. Gregor Svindland 2. Gutachter: Prof. Dr. Michael Kupper 3. Gutachterin: Prof. Dr. Francesca Biagini Tag der m¨undlichen Pr¨ufung:19. Februar 2020 Eidesstattliche Versicherung (Siehe Promotionsordnung vom 12.07.11, § 8, Abs. 2 Pkt. 5.) Ich versichere hiermit an Eidesstatt, dass die vorgelegte Dissertation von mir selbst¨andig,ohne unerlaubte Beihilfe angefertigt ist. M¨unchen, den 7. August 2019 Felix-Benedikt Liebrich Zusammenfassung Die vorliegende Arbeit beleuchtet mathematische Aspekte der Theorie ¨okonomischer und fi- nanzieller Risiken. In den drei versammelten Beitr¨agenwerden das Fortsetzungsproblem f¨ur Risikomaße unter Knight’scher Unsicherheit sowie die Optimierung von Risiko- und Nutzenauf- teilungen studiert. Der erste Beitrag, Model Spaces for Risk Measures, widmet sich dem Fortsetzungsproblem f¨ur Risikomaße. Letztere sind als Kapitalanforderungen definiert, welche durch das Zusammenspiel einer konvexen Menge akzeptabler Positionen und eines mehrdimensionalen Security-Markts zur Absicherung nicht akzeptabler Positionen induziert werden. Die Nettoverluste, deren Risiko gemessen wird, werden durch Zufallsvariablen auf der Menge m¨oglicher k¨unftigerZust¨andeder Okonomie¨ modelliert. Wir betrachten allerdings Risikomaße, welche auch Knight’sche Unsicher- heit abbilden: Die Realisierung k¨unftigerZust¨andel¨asstsich nur partiell oder ¨uberhaupt nicht durch einen Zufallsmechanismus mit bekannten Parametern beschreiben. In der Fortsetzungs- frage gehen wir deshalb von einer simplen Situation aus, in der das Risiko von beschr¨ankten Nettoverlusten durch szenarioweise Uberlegungen¨ und ohne ein zugrundeliegendes Wahrschein- lichkeitsmodell bestimmt wird. Wir zeigen, dass unter milden Annahmen ein solches Risikomaß eine intrinsische probabilistische Sichtweise auf die Realisierung k¨unftigerZust¨andeerlaubt. Diese kann als Ausgangspunkt einer Fortsetzung dienen, da sie im Vergleich zur modellfreien Betrachtung keinen Informationsverlust verursacht. Um das Risiko komplexerer Verluststprofile bestimmen zu k¨onnen,muss ein geeigneter gr¨oßererDefinitionsbereich gefunden werden. Wir konstruieren einen Banachverband von Zufallsvariablen, dessen analytische Struktur durch die urspr¨ungliche Risikomessung diktiert wird. Er ist invariant unter allen relevanten probabilis- tischen Modellen und fungiert als maximaler nat¨urlicher Definitionsbereich des urspr¨unglichen Risikomaßes, da er verschiedene Fortsetzungen desselben auf unbeschr¨ankteZufallsvariablen er- laubt. Diese beleuchten und vergleichen wir detailliert. Die Diskussion von Bedingungen, unter welchen regul¨are Subgradienten, d.h. sinnvolle Preisregeln, existieren, schließt sich an. Der zweite Beitrag, Risk Sharing for Capital Requirements with Multidimensional Security Markets, studiert die Existenz optimaler Risikoaufteilungen. Betrachtet wird ein System von n Agenten, welche in einer Handelsperiode individuelle Nettoverluste erleiden. Deren Risiko quan- tifiziert jeder Agent mit einem individuellen Risikomaß, welches wie im ersten Beitrag eine aus Akzeptanzmenge und endlichdimensionalem Security-Markt resultierende Kapitalanforderung ist. Die Frage ist nun, ob die aggregierten (d.h. addierten) individuellen Risiken durch restlose Umverteilung der individuellen Verlustpositionen minimiert werden k¨onnen.Nettoverluste wer- den als Elemente eines zugrundeliegenden Riesz-Raums modelliert. Individuell relevante und individuell akzeptable Verlustpositionen sowie individuelle Security-M¨arktek¨onnensehr hetero- gen sein. Wir treffen einzig eine schwache Kooperationsannahme, welche den Agenten ¨uberhaupt erst ein gemeinsames Pooling der Risiken erm¨oglicht. Es wird gezeigt, dass nur Pareto-optimale Allokationen (Aufteilungen) eines gegebenen Gesamtverlusts die Aggregation der Einzelrisiken minimieren. Das minimale aggregierte Risiko eines aufgeteilten Gesamtverlusts stellt eine Kapi- talanforderung f¨urden Markt dar, die sich durch eine Markt-Akzeptanzmenge und einen globalen Security-Markt charakterisieren l¨asst.Das Problem l¨asstsich also mittels eines repr¨asentativen Agenten formulieren. F¨urAgentensysteme mit polyhedralen und verteilungsinvarianten Akzep- i tanzmengen wird die Existenz von Pareto-optimalen Allokationen und Equilibria gezeigt und ihre Stabilit¨atunter Perturbation des aggregierten Verlusts analysiert. Als Anwendung wer- den optimale Portfolio-Aufteilungen unter Tochterfirmen in M¨arktenmit Transaktionskosten untersucht. Der dritte Beitrag, Efficient Allocations under Law-Invariance: A Unifying Approach, wech- selt die Perspektive vom Risiko von Verlusten auf den Nutzen, den die Aufteilung eines durch stochastische Faktoren unsicheren k¨unftigenGuts unter n Agenten mit heterogenen Pr¨aferenzen erzeugt. G¨uter werden als Zufallsvariablen ¨uber einem atomlosen Wahrscheinlichkeitsraum (Ω, F, P) modelliert. Die Pr¨aferenzen der einzelnen Agenten sind durch quasikonkave vertei- lungsinvariante Nutzenfunktionale kodiert. Alle Agenten sind also insofern kompatibel, als dass sie indifferent zwischen zwei G¨uternmit derselben Verteilungsfunktion unter dem Referenz- maß P sind. Wir nehmen an, dass die individuellen Nutzen mit einer allgemein gehaltenen Funktion zu einem Gesamtnutzen aggregiert werden, beispielsweise durch (gewichtete) Sum- mation. Allokationen sind effizient, wenn sie bei Aggregation zum optimalen, d.h. maximalen aggregierten Nutzen unter allen m¨oglichen Allokationen f¨uhren. F¨urgroße Klassen individu- eller Nutzenfunktionen und Aggregationsfunktionen zeigen wir die Existenz komonotoner ef- fizienter Allokationen. Hiermit verallgemeinern wir viele der bereits bekannten Resultate zu optimaler Nutzenaufteilung mit verteilungsinvarianten Nutzenkriterien und f¨uhrensie zu einem Meta-Theorem zusammen. Als Anwendungen der allgemeinen Existenztheorie zeigen wir die Ex- istenz von Allokationen mit verschiedenen ¨okonomisch motivierten Optimalit¨atseigenschaften: (schwache, verzerrte und individuell rationale) Pareto-Optimalit¨at,systemisch faire Allokatio- nen und Kern-Allokationen. ii Abstract This dissertation examines mathematical aspects of the theory of economic and financial risks. The three collected contributions study the extension problem for risk measures under Knightian uncertainty as well as the optimisation of risk and utility allocations. The first contribution, Model Spaces for Risk Measures, addresses the extension problem for risk measures. The latter are defined as capital requirements which are induced by the interplay of a convex set of acceptable positions and a multidimensional security market for securing non-acceptable positions. The net losses whose risk is to be measured are modelled by random variables on the set of future states of the economy. However, we consider risk measures which can also account for Knightian uncertainty: The realisation of future states of the economy can only partially or not at all be described with known parameters. In the extension question, we thus depart from a simple situation in which the risk of bounded net losses is determined by scenariowise considerations without an underlying probabilistic model. We show that under mild assumptions such a risk measure admits an intrinsic probabilistic perspective on the realisation of future states of the economy. It can serve as starting point for an extension as it does not lead to any loss of information compared to the model-free risk measure. In order to determine the risk of more complex loss profiles, a suitable larger domain of definition must be found. We construct a Banach lattice of random variables whose analytic structure is dictated by the initial risk measure. It is invariant under all relevant probabilistic models and plays the role of a maximal natural domain of definition of the initial risk measure, as it admits several extensions thereof to unbounded random variables. We examine and compare them in detail. A discussion of conditions under which regular subgradients exist, i.e. sensible pricing rules, concludes. The second contribution, Risk Sharing for Capital Requirements with Multidimensional Se- curity Markets, studies the existence of optimal risk allocations. We consider a system of n agents who incur individual net losses in a one-period market. Each agent quantifies their risk with an individual risk measure which is a capital requirement resulting from an acceptance set and a finite dimensional security market as in the first contribution. This raises the question whether the aggregated (i.e. added) individual risks can be minimised by complete redistribu- tion of the individual losses. Net losses are modelled as elements of an ambient Riesz space. Individually relevant and individually acceptable losses as well as individual security markets may be very heterogeneous. We only make a weak cooperation assumption which allows the agents to pool their risks in the first place. It is shown that only Pareto optimal allocations of a given aggregated loss minimise the aggregation of the individual risks. The minimal aggregated risk of an allocated aggregated loss defines a capital requirement for the market which may be characterised by a market acceptance set and a global security market. The problem can hence be formulated with a representative agent. For agent systems with polyhedral
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