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Rational Hedging and Valuation with Utility-Based Preferences Dirk

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Rational Hedging and Valuation with Utility-Based Preferences Dirk Rational Hedging and Valuation with Utility-Based Preferences vorgelegt von Diplom-Mathematiker Dirk Becherer aus L¨udenscheid Von der Fakult¨atII - Mathematik und Naturwissenschaften der Technischen Universit¨atBerlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften { Dr. rer. nat. { genehmigte Dissertation Vorsitzender des Promotionsausschusses: Prof. Dr. Fredi Troltzsch¨ { Technische Universit¨atBerlin Berichter: Prof. Dr. Mark H. A. Davis { Imperial College of Science and Technology, London Prof. Dr. Peter Imkeller { Humboldt-Universit¨atzu Berlin Prof. Dr. Martin Schweizer { Ludwig-Maximilians-Universit¨atM¨unchen Tag der wissenschaftlichen Aussprache: 8. Oktober 2001 Berlin 2001 D 83 Abstract In this thesis, we study stochastic optimization problems in which concave functionals are maximized on spaces of stochastic integrals. Such problems arise in mathematical finance for a risk-averse investor who is faced with valuation, hedging, and optimal investment problems in incomplete financial markets. We are mainly concerned with utility-based methods for the valuation and hedging of non-replicable contingent claims which confront the issuer with some inevitable intrinsic risks. We adopt the perspective of a rational investor who aims to maximize his expected exponential utility. Based on these preferences, the issuer's valuation process and hedging strategy are defined via utility indifference arguments. In a general semimartingale model, the solution to this problem is characterized by a stochastic representation problem. Solving the problem amounts to finding a martingale measure whose density process can be written in a particular form. We then specialize our analysis to two stochastic models which satisfy further structural assumptions. In a semi-complete product model, the valuation and hedging methods are shown to be additive when applied to an aggregate of “sufficiently independent" individual claims. We study the impact of diversification and derive a computation scheme. For a second model, we set up a Markovian system of stochastic differential equations which describes the dynamics of an It^oprocess and an additional finite-state process, and permits for various dependencies between both. In the financial market model, the It^oprocess models the price fluctuations of the risky assets while the second process represents some untradable factors of risk. The solution to the pricing and hedging problem is explicitly described by an interacting system of semi-linear partial differential equations { a so-called reaction-diffusion equation. Using Feynman-Kac results and the Picard-iteration method, we establish existence and uniqueness of a classical solution. In a variation of the basic theme, a similar utility indifference approach is applied to quantify the value of additional investment information. On the mathematical side, this involves a mar- tingale preserving measure transformation and martingale representation results for initially enlarged filtrations. Finally, we show that the so-called numeraire portfolio is related to an- other utility-based valuation method which relies on a marginal rate of substitution argument and can be seen as a limiting case of the utility indifference method. i Zusammenfassung Die vorliegende Arbeit behandelt stochastische Optimierungsprobleme, in denen ein konkaves Funktional ¨uber einem Raum von stochastischen Integralen maximiert wird. In der Finanz- mathematik treten derartige Probleme bei der Behandlung von Bewertungs-, Absicherungs-, und Anlageproblemen in unvollst¨andigenFinanzm¨arktenauf. Wir besch¨aftigenuns vornehm- lich mit nutzenbasierten Methoden zur Bewertung und Absicherung von zufallsbehafteten Finanzpositionen, welche unvermeidbare intrinsische Risiken beinhalten. Wir betrachten das Problem aus der Perspektive eines rationalen Investors, dessen Ziel die Maximierung seines erwarteten exponentiellen Nutzens ist. Ausgehend von diesen Pr¨aferenzen, definieren wir mittels Nutzenindifferenz-Argumenten seinen Bewertungsprozess und eine Ab- sicherungsstrategie. In einem Semimartingalmodell kann die L¨osungdurch ein stochastisches Darstellungsproblem charakterisiert werden. Um das Problem zu l¨osen,gilt es ein Martingal- maß zu finden, dessen Dichteprozess eine bestimmte Form hat. Im Weiteren untersuchen wir zwei Modelle, welche gewissen strukturellen Bedingungen gen¨ugen.In einem halbvollst¨andigen Produktmodell wird gezeigt, dass die Nutzenindifferenz-Bewertungs- und Absicherungsme- thode additiv ist, wenn sie auf ein Aggregat von \gen¨ugendunabh¨angigen"Positionen ange- wandt wird. Wir untersuchen Diversifikationseffekte und leiten ein Berechnungsschema her. F¨urdas zweite Modell betrachten wir ein Markovsches System stochastischer Differentialgle- ichungen, welches einen It^o-Prozessund einen weiteren Prozess mit endlichem Zustandsraum beschreibt und verschiedene wechselseitige Abh¨angigkeiten zul¨asst. In unserem Marktmod- ell stellt der It^o-Prozessdie Preise der riskanten Anlagen dar w¨ahrendder zweite Prozess irgendwelche nicht handelbaren Risikofaktoren repr¨asentiert. Die L¨osungdes Bewertungs- und Absicherungsproblems wird durch ein wechselwirkenendes System semilinearer partieller Differentialgleichungen, eine so genannte Reaktions-Diffusions Gleichung, beschrieben. Mit- tels Feynman-Kac Resultaten und der Iterationstechnik von Picard zeigen wir Existenz und Eindeutigkeit einer klassischen L¨osung. Erg¨anzendzu unserem Hauptthema nutzen wir ein ¨ahnliches Indifferenzargument um den Wert von zus¨atzlichen Anlage-Informationen zu quantifizieren. Die wesentlichen Mittel sind eine Martingal erhaltende Maßtransformation und Martingal-Darstellungsresultate f¨uranfangsver- gr¨oßerteFiltrationen. Schließlich zeigen wir, dass das so genannte Numeraire-Portfolio mit einer weiteren nutzenbasierten Bewertungsmethode zusammenh¨angt,welche sich auf ein Gren- znutzenargument st¨utztund als Grenzfall der Indifferenz-Methode angesehen werden kann. iii Contents Introduction 1 I RATIONAL PRICING AND HEDGING OF CONTINGENT CLAIMS 13 1 General results for exponential utility 15 1.1 General semimartingale framework . 16 1.2 Duality results on exponential utility optimization . 17 1.3 Utility indifference pricing . 20 1.4 Utility indifference hedging . 24 2 Results in semi-complete product models 29 2.1 The semi-complete model . 30 2.2 The structure of the equivalent martingale measures . 31 2.3 Additivity and diversification . 33 2.4 A computation scheme and explicit bounds . 36 3 PDE-solutions in incomplete SDE-models 41 3.1 The incomplete SDE-model . 41 3.2 Construction of S-dependent intensities . 44 3.3 Some case studies . 45 3.4 Solutions to the utility maximization problem . 46 3.5 Solutions to the rational pricing and hedging problem . 57 4 Applications 61 4.1 The financial market model . 61 4.2 General solution by reaction-diffusion equations . 62 4.3 Equity-linked life insurance contracts . 64 4.4 Default and credit risk . 67 4.5 Stochastic volatility . 70 v 4.6 Weather derivatives . 72 II INITIAL ENLARGEMENT OF FILTRATIONS AND THE VALUE OF INVESTMENT INFORMATION 77 5 The value of initial investment information 79 5.1 General framework and preliminaries . 80 5.2 Strong predictable representation property . 86 5.3 Utility maximization with non-trivial initial information . 89 5.4 Utility indifference value of initial information . 97 5.5 Examples: Terminal information distorted by noise . 100 III ON THE NUMERAIRE PORTFOLIO 105 6 The numeraire portfolio 107 6.1 General framework and preliminaries . 109 6.2 The growth-optimal numeraire . 112 6.3 The numeraire portfolio . 113 6.4 Examples . 119 IV REACTION DIFFUSION EQUATIONS 125 7 Reaction-diffusion equations 127 7.1 General framework . 127 7.2 Fixed points of the Feynman-Kac representation . 128 7.3 Classical solutions to interacting systems of PDEs . 129 Index of Notation 135 vi Introduction In this thesis, we study stochastic optimization problems in which concave functionals are maximized on spaces of stochastic integrals. Such problems arise in mathematical finance for a risk-averse investor who is faced with valuation, hedging, risk management and optimal investment problems in incomplete financial markets. The emphasis of this thesis is on utility- based methods for the valuation and hedging of non-replicable contingent claims which confront the issuer with some inevitable intrinsic risks. If the payoff of a contingent claim is replicable by dynamic trading, the claim can be perfectly hedged by the replicating strategy, and its price is determined as the replication costs. However, such claims are dynamically redundant and have apparently little reason to exist. In incomplete markets, a contingent claim may be not redundant but incorporate some unavoidable risk. Hence, the issuer's valuation and hedging strategy have to take into account his attitudes and preferences towards risk. We adopt the perspective of a rational investor who aims to maximize his expected utility according to his level of risk aversion. In doing so, the investor balances his two conflicting objectives to maximize return and minimize risk. Based on these preferences, the utility indifference price π of a contingent claim is defined as the amount of money, which makes a potential issuer indifferent { in terms of maximal expected utility { between the opportunity to earn the premium π and take the liability associated with the claim, and the alternative to skip the deal. The hedging strategy for the contingent claim is defined as the adjustment of the investor's optimal
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