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Terminal wealth problems in illiquid markets under a drawdown constraint [Elektronische Ressource] / Marc Wittlinger

114 pages
Terminal wealth problemsin illiquid marketsunder a drawdown constraintDissertationzur Erlangung des Doktorgrades Dr. rer. nat.der Fakult at fur Mathematik und Wirtschaftswissenschaftender Universit at Ulmvorgelegt vonMarc Simon Wittlingeraus NurtingenUlm, im September 2011Amtierender Dekan: Prof. Dr. Paul WentgesErstgutachter: Prof. Dr. Ulrich RiederZweitgutachter: Prof. Dr. Rudiger KieselTag der Promotion: 13. Dezember 2011Fur Friedrich.Contents1. Introduction 11.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2. Overview and contributions of this thesis . . . . . . . . . . . . . . . . . . . 21.3. Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42. Inhomogeneous Levy processes 72.1. De nition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2. Exponential moment and canonical representation . . . . . . . . . . . . . . 112.3. The Markov property of the stochastic exponential . . . . . . . . . . . . . . 142.4. Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.5. A special case: The inhomogeneous Poisson processes . . . . . . . . . . . . 193. The portfolio optimization problem 233.1. Financial Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2. Wealth process and admissible policies . . . . . . . . . . . . . . . . . . . . . 253.3. The terminal wealth problem . . . . . . .
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Terminal wealth problems
in illiquid markets
under a drawdown constraint
Dissertation
zur Erlangung des Doktorgrades Dr. rer. nat.
der Fakult at fur Mathematik und Wirtschaftswissenschaften
der Universit at Ulm
vorgelegt von
Marc Simon Wittlinger
aus Nurtingen
Ulm, im September 2011Amtierender Dekan: Prof. Dr. Paul Wentges
Erstgutachter: Prof. Dr. Ulrich Rieder
Zweitgutachter: Prof. Dr. Rudiger Kiesel
Tag der Promotion: 13. Dezember 2011Fur Friedrich.Contents
1. Introduction 1
1.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2. Overview and contributions of this thesis . . . . . . . . . . . . . . . . . . . 2
1.3. Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2. Inhomogeneous Levy processes 7
2.1. De nition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2. Exponential moment and canonical representation . . . . . . . . . . . . . . 11
2.3. The Markov property of the stochastic exponential . . . . . . . . . . . . . . 14
2.4. Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5. A special case: The inhomogeneous Poisson processes . . . . . . . . . . . . 19
3. The portfolio optimization problem 23
3.1. Financial Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2. Wealth process and admissible policies . . . . . . . . . . . . . . . . . . . . . 25
3.3. The terminal wealth problem . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4. Intensity processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4. A terminal wealth problem with a bounded intensity process 33
4.1. Assumptions and characteristics . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2. Solution via Contracting Markov Decision Process . . . . . . . . . . . . . . 35
4.3. Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.4. Howard’s policy improvement algorithm . . . . . . . . . . . . . . . . . . . . 47
4.5. Separation ansatz for CRRA utility functions . . . . . . . . . . . . . . . . . 48
5. A terminal wealth problem with an unbounded intensity process 51
5.1. Assumptions and properties of the model . . . . . . . . . . . . . . . . . . . 51
5.2. Solution via limsup Markov Decision Process . . . . . . . . . . . . . . . . . 56
5.3. Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.4. Separation ansatz for CRRA utility functions . . . . . . . . . . . . . . . . . 67
6. Convergence Results 69
6.1. Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.2. Convergence of the value function . . . . . . . . . . . . . . . . . . . . . . . . 71
6.3. Conv of the optimal policy . . . . . . . . . . . . . . . . . . . . . . . 72
7. Numerical examples with a Power Utility function 75
7.1. Bounded intensity processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
7.1.1. Constant intensity process . . . . . . . . . . . . . . . . . . . . . . . . 76
viiviii Contents
7.1.2. Time-varying intensity process . . . . . . . . . . . . . . . . . . . . . 78
7.2. Unbounded intensity process . . . . . . . . . . . . . . . . . . . . . . . . . . 81
7.3. Conclusion for practitioners . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
A. Structure Theorem for limsup Markov Decision Processes 85
B. Stochastic processes 89
B.1. Semimartingales and stochastic exponential . . . . . . . . . . . . . . . . . . 89
B.2. Additive processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
C. Fenchel-Legendre transform 95
Bibliography 97
Zusammenfassung 1011. Introduction
1.1. Motivation
In classical portfolio allocation problems an investor may invest his wealth in a nancial
market, which usually consists of a nite number of risky assets and a riskless asset.
Thereby he wants to nd the \best\ investment policy. In order to determine such a policy,
he rst of all needs to choose a performance criteria. A very popular one is, for example,
the maximization of the expected utility of terminal wealth.
Originally, this problem was studied by Robert C. Merton in Merton (1969). In the
framework of a Black-Scholes market and a CRRA utility function, he found out that the
optimal investment policy can be described by a xed proportion of wealth, which should
be put in the risky asset. Furthermore, he showed that this proportion admits an explicit
expression. However, while applying this policy the investor has to adjust his portfolio
continuously, since his wealth is changing at all times.
Moreover, in Merton (1969) it is assumed that the market is perfectly liquid, i.e. the
investor’s transactions do not in uence the asset price and they are executed immediately.
Obviously, most markets are not perfectly liquid, since at least one of these assumptions
fails. Hence, it is more realistic to take the liquidity risk into account.
In the literature there are many ways to model liquidity risk. An overview of the most
common approaches is given in the following:
Transaction costs: In Davis & Norman (1990) the liquidity risk is measured by means
of transaction costs. In principal, the investor can trade whenever he wants, but high
frequency trading is impossible due to large transaction costs.
Price impacts due to transactions: Subramamian & Jarrow (2001) and Cetin et al.
(2004) use an approach to model liquidity risk by price impacts due to transactions.
Such a kind of liquidity risk can for for example be observed when considering large
traders.
Restriction of trading times: In Rogers (2001), Rogers & Zane (2002), Matsumoto
(2003, 2006, 2007, 2009), Pham & Tankov (2008, 2009) and Gassiat et al. (2011)
the liquidity risk is modeled by assuming that trading is only possible at some
exogenous random times, which are given by the jump times of a Poisson process.
Such a situation occurs for example in over-the-counter markets. In those markets
transactions may not be executed immediately, due to the lack of a counter party.
Thus the investor has to wait until his transaction takes place.
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