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Stochastic Methods in Risk Management Stochastic Methods in Risk Management Dissertation zur Erlangung des akademischen Grades eines Doktors der Wirtschaftswissenschaften (Dr. rer. pol.) durch die Fakult¨atf¨urWirtschaftswissenschaften der Universit¨atDuisburg-Essen, Campus Essen vorgelegt von Jinsong Zheng geboren in Zhejiang, China Essen, 2018 Diese Dissertation wird über DuEPublico, dem Dokumenten- und Publikationsserver der Universität Duisburg-Essen, zur Verfügung gestellt und liegt auch als Print-Version vor. DOI: 10.17185/duepublico/70203 URN: urn:nbn:de:hbz:464-20190624-093546-8 Alle Rechte vorbehalten. Tag der m¨undlichen Pr¨ufung:12. Juni 2019 Erstgutachter: Prof. Dr. R¨udigerKiesel Zweitgutachter: Prof. Dr. Antje Mahayni Zusammenfassung Stochastische Methoden sind in der Finanzbranche weit verbreitet und werden z.B. in der stochastischen Modellierung und Simulation, der risikoneutralen Bewertung, der Derivatebewertung und vielen weiteren Anwendungen eingesetzt. Unter den Rahmenbe- dingungen von Solvency II mussen¨ Versicherungsunternehmen ad¨aquat kapitalisiert sein um jene aus Solvency II erwachsenen Kapitalanforderungen, zum Schutze der Aktion¨are und Versicherungsnehmer, zu erfullen.¨ Daher mussen¨ zwei wesentliche Gr¨oßen betrach- tet werden; das vorhandene Risikokapital (bzw. die Basiseigenmittel) und das ben¨otigte Risikokapital. Im Allgemeinen werden diese Gr¨oßen anhand der Mittelwerte von sto- chastischen Simulationen berechnet und folglich wird ein Economic Scenario Generator (ESG) verwendet, um die potentielle Entwicklung von Risikofaktoren der Okonomie¨ und des Finanzmarktes im Zeitverlauf zu simulieren. Fur¨ die Berechnung des vorhandenen Risikokapitals (definiert als die Differenz zwischen dem Marktwert der Verm¨ogenswerte abzuglich¨ der Verbindlichkeiten) wird ein stochastische Cash-Flow Projektionsmodell verwendet, um eine marktkonsistente Bewertung der Verm¨ogenswerte und der Verbind- lichkeiten, unter Verwendung von risikoneutralen Szenarien, durchzufuhren.¨ Die Berech- nung des ben¨otigten Risikokapitals erfolgt anhand der Wahrscheinlichkeitsverteilung des vorhandenen Risikokapitals uber¨ einen einj¨ahrigen Zeithorizont mithilfe eines Risikoma- ßes. Beispielsweise wird die Solvency Capital Requirement (SCR) anhand des Value-at- Risk zum Konfidenzniveau 99,5% gemessen. Zun¨achst haben wir einen Uberblick¨ uber¨ die bestehende Literatur gegeben. Hierbei haben wir festgestellt, dass die allermeisten Autoren sich bei ihrer Betrachtung hinsicht- lich der Verwendung stochastischer Methoden im Rahmen von Solvency II auf eine der drei Modellkomponenten des interne Partialmodells, also dem Inputmodell, dem Be- wertungsmodell und dem Risikomodell, konzentrieren. In dieser Arbeit wollten wir ein internes Partialmodell mit allen Komponenten aufbauen und Schritt fur¨ Schritt zeigen, wie wir mit stochastischen Methoden eine marktkonsistente Bewertung vornehmen und das ben¨otigte Risikokapital berechnen k¨onnen. Fur¨ das Inputmodel haben anstatt eines akademisch bevorzugten einfachen ESG Mo- dells, mit einem Ein-Faktor Zinsmodells und einem durch eine geometrische Brownsche Bewegung getriebenen Aktienmodell, ein komplexeres Modell entwickelt, welches in der Praxis besser geeignet ist. Fur¨ die Modellierung des Zinssatzes haben wir das erweiterte Drei-Faktoren-Modell von Cox-Ingersoll-Ross, das die drei Hauptkomponenten der Zinss- trukturkurve erfassen kann, verwendet. Wir haben den Preis von Nullkupon-Optionen mithilfe der Fourier-Transformation der charakteristischen Funktion einer Linearkom- bination von Zustandsvariablen sowie einer anschließender Bewertung von Swaptions anhand einer stochastischen Durationsapproximation hergeleitet. Fur¨ die Modellierung von Aktien haben wir ein stochastisches Volatilit¨atsmodell (Heston-Modell) zusammen i mit der oben beschriebenen stochastischen Zinsmodellierung verwendet. In ¨ahnlicher Weise haben wir die geschlossene Form der diskontierten charakteristischen Funktion des logarithmierten Aktienpreises ermittelt, indem wir ein System von gew¨ohnlichen Differentialgleichungen l¨osen, welches von einer affinen partiellen Differentialgleichung stammt. Anschließend haben wir den Preis von europ¨aischen Optionen ebenfalls an- hand Fouriertechniken hergeleitet. Zus¨atzlich haben wir die Methode der Erzeugung ¨okonomischer Szenarien mithilfe einer Monte Carlo Simulation, unter Verwendung eines Euler Diskretisierungsschemas und Varianzreduktionstechniken formuliert. Fur¨ das Bewertungsmodell haben wir ein stochastisches Cashflow-Projektionsmodell entwickelt, um die Entwicklung der Bilanz sowie des aus Kuponanleihen und Aktien bestehenden Verm¨ogensportfolios und des aus uberschussberechtigten¨ Lebensversiche- rungsvertr¨agen bestehenden Passivportfolios zu erfassen. Anschließend haben wir eine marktkonsistente Bewertung der Verm¨ogenswerte und Verbindlichkeiten vorgenommen, die auf den vom stochastischen Modell projizierten Cashflows und den Input risikoneu- traler ¨okonomischen Szenarien basiert. Daruber¨ hinaus haben wir die Managementregeln modelliert. Beispielsweise haben wir eine konstante Asset-Allocation-Strategie entwi- ckelt, um das Asset-Portfolio wieder ins Gleichgewicht zu bringen. Wir haben den nicht realisierten Gewinn und Verlust durch Modellierung des Buchwerts und des Marktwerts von Verm¨ogenswerten berucksichtigt.¨ Daruber¨ hinaus haben wir den MUST-Fall fur¨ die Uberschussbeteiligung¨ zwischen Anteilseigner und Versicherungsnehmer modelliert. Fur¨ das Risikokapitalmodell haben wir zun¨achst die verschachtelte stochastische Simu- lation ( Nested Stochastic Simulation\) implementiert, um das ben¨otigten Risikokapital " zu bestimmen. Da fur¨ die verschachtelte Simulation eine hohe Rechenzeit erforderlich ist, haben wir auch die Proxy-Methoden Least Squared Monte Carlo\ (LSMC), Replica- " " ting Portfolio\ und Curve Fitting\ untersucht. Insbesondere haben wir eine allgemeine " Strategie entwickelt, um ein gutes replizierendes Portfolio zusammenzustellen. Hierbei haben wir zuerst den Aufbau eines Asset-Pools beschrieben. Danach haben wir die Kon- struktion von Sensitivit¨atss¨atzen durch Rekalibrierungs- oder Neugewichtungstechniken veranschaulicht. Als n¨achstes haben wir ein Kalibrierungsverfahren vorgeschlagen, bei dem sowohl die Optimierungsmethode der kleinsten Quadrate als auch die Auswahl von Teilmengen anhand bestimmter Kriterien verwendet werden, um das optimale Replika- tionsportfolio auszuw¨ahlen und das ben¨otigte Risikokapital zu berechnen. Schließlich haben wir anhand einer empirischen Anwendung den gesamten Prozess von der Kalibrierung der ESG-Modelle anhand von realen Marktdaten, der Erzeugung und Validierung der ¨okonomischen Szenarien, der marktkonsistenten Bewertung sowie der Bestimmung des SCR durch die verschachtelte Simulation und des replizierenden Portfolios, illustriert. ii Abstract Stochastic methods, such as stochastic modeling and simulation, risk neutral valuation, derivative pricing, etc., are widely used in the finance industry. Under Solvency II framework, in order to protect the benefit of shareholder and policyholder, the insurance company should be adequately capitalized to fulfill the capital requirement for solvency. Therefore, two main quantities are taken into account, i.e. the available capital (or basic own funds) and the required capital. In general, these two quantities are calculated by means of stochastic simulation and hence an Economic Scenario Generator (ESG) is used to simulate the potential evolution of risk factors of the economies and financial markets over time. For the calculation of available capital (defined as the difference between the market value of assets and liabilities), the stochastic cash flow projection model is used to perform the market consistent valuation of assets and liabilities given the risk neutral scenarios. For the calculation of required capital, the probability distribution of available capital over a one-year time horizon and a risk measure based on such distribution is taken into account. For instance, the Solvency Capital Requirement (SCR) is measured by the Value-at-Risk at confidence level of 99.5%. We began by reviewing the existing literature and found that most authors used stochastic methods in risk management under Solvency II framework on one of the three components of the partial internal model, i.e. the input model, the valuation model or the risk capital model. In this thesis, we aimed to build a partial internal model including all components and show how we can use stochastic methods to do market consistent valuation and calculate the required capital. For the input model, instead of using academic preferred simple ESG models, e.g. one factor short rate interest rate model along with geometric Brownian motion equity model, we developed advanced models that are more suitable in practice. For the modeling of interest rate, we used the extended three-factor Cox-Ingersoll-Ross model, which is able to capture the three main principle components of yield curve. We derived the pricing of zero coupon options by Fourier transformation of the characteristic function of the linear combination of state variables and subsequently the pricing of swaption using stochastic duration approximation. For the modeling of equity, we used the stochastic volatility model (Heston model) along with above-mentioned stochastic interest rate. Similarly, we first showed the closed-form of discounted characteristic function of log equity price by solving a system of Ordinary Differential Equations (ODEs) resulting from an affine Partial Differential Equation
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