Unit-linked life insurance in Lévy-process financial markets [Elektronische Ressource] : modeling, hedging and statistics / vorgelegt von Martin Riesner

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• Social work

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• Publié par: universitat_ulm
• Publié le: 01 janvier 2006
• Nombre de lectures: 32
• Langue: English
• Poids de l'ouvrage: 1 Mo

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Universitat Ulm¨
Abteilung Zahlentheorie und Wahrscheinlichkeitstheorie
Unit-linked life insurance in
L´evy-process financial markets
Modeling, Hedging and Statistics
Dissertation
zur Erlangung des Doktorgrades
Dr. rer. nat.
der Fakultat fur Mathematik und Wirtschaftswissenschaften¨ ¨
der Universita¨t Ulm
vorgelegt von
Dipl.-Math. oec. Martin Riesner, M.S.
aus
Schorndorf
Ulm, im Juli 2006
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Amtierender Dekan: Professor Dr. Ulrich Stadtmu¨ller
1. Gutachter: Professor Dr. Ulrich Stadtmu¨ller, Universit¨at Ulm
2. Gutachter: Professor Dr. Ru¨diger Kiesel, Universit¨at Ulm
3. Gutachter: Professor Dr. Ralf Korn, Universitat Kaiserslautern¨
Tag der Promotion: 13.10.2006Preface
Bringing together the L´evy-process financial world and the theory of unit-
linked life insurance contracts, this thesis shall contribute to the ongoing
research on the interface between financial and actuarial mathematics. It
concludes my doctorate research which was carried out at the Department
of Number Theory and Probability Theory at the University of Ulm in
the period from October 2003 to July 2006. My work was supervised by
Professors Ulrich Stadtmu¨ller and Ru¨diger Kiesel, both University of Ulm.
Acknowledgements
First of all, I would like to express my deepest gratitude to Professor Ulrich
Stadtmu¨ller for being an excellent supervisor and teacher. During the last
three years we had many extremely fruitful discussions, I received valuable
advice and I always got the greatest possible support. Moreover, I very
much enjoyed being a member of the Department of Number Theory and
Probability Theory and to contribute to its teaching duties.
IalsowouldliketoexpressmysincerethankstoProfessorRu¨digerKiesel
for his brilliant supervision and numerous valuable suggestions. Especially,
I am very grateful for always having been invited to the great social events
of his Department of Financial Mathematics.
Special thanks go to Professor Ralf Korn, University of Kaiserslautern,
for being an external referee of this thesis and to Professor Nick Bingham,
University of Sheffield, for his very helpful and encouraging comments on
some of my preprints.
Furthermore, I would like to thankthe Head of theDepartment of Num-
ber Theory and Probability Theory at the University of Ulm, Professor
Helmut Maier, for offering me a pleasant research and work environment.
I am also very grateful to Professor Ulrich Rieder, University of Ulm, by
whom I enjoyed having my fistcourses on Stochastic Analysis and Financial
Mathematics.
I extend my thanks to Anja Blatter for all her support and to my
colleagues and friends Reik Bo¨rger, Dr. Stefan Kassberger, Dr. Hartmut
Lanzinger, Gregor Mummenhoff, Clemens Prestele, Matthias Scherer and
Monika Thalmaier for many interesting discussions.iv
Finally, I very much appreciate the scholarship by the federal state
Baden-Wu¨rttemberg (LGFG Baden-Wu¨rttemberg) for the entire phase of
my doctorate and the financial supportgranted by the Department of Num-
ber Theory and Probability Theory and the Department of Financial Math-
ematics at the University of Ulm.
Ulm, July 2006
Martin RiesnerContents
Preface iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
1 Introduction and summary 5
1.1 Historical overview and existing literature . . . . . . . . . . . 5
1.2 Summary, contributions and outline of this thesis . . . . . . . 8
2 Basic concepts 15
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 L´evy processes . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.1 Definition and basic properties . . . . . . . . . . . . . 17
2.2.2 Common examples . . . . . . . . . . . . . . . . . . . . 26
2.2.3 Finite activity L´evy-processes: jump-diffusions . . . . 26
2.2.4 Infinite activity L´evy-processes . . . . . . . . . . . . . 28
2.2.5 Variance gamma L´evy-process . . . . . . . . . . . . . 28
2.2.6 Normal inverse Gaussian L´evy-process . . . . . . . . . 30
2.2.7 Generalized hyperbolicdistributions and L´evy processes 32
2.3 Financial market . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.1 L´evy-process model . . . . . . . . . . . . . . . . . . . 35
2.3.2 Fo¨llmer-Schweizer measure . . . . . . . . . . . . . . . 37
2.3.3 Other equivalent changes of measure . . . . . . . . . . 39
2.4 Life insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4.1 Multi-state Markov model . . . . . . . . . . . . . . . . 40
2.4.2 Typical portfolio of insured lives . . . . . . . . . . . . 42
2.4.3 Analytical laws of mortality . . . . . . . . . . . . . . . 43
2.4.4 Conventional life-insurance risk diversification . . . . . 44
2.4.5 Risk-neutrality . . . . . . . . . . . . . . . . . . . . . . 45
2.5 Combined financial and life insurance model . . . . . . . . . . 45
2.5.1 Combined model . . . . . . . . . . . . . . . . . . . . . 45
2.5.2 Hedging theory of F¨ollmer and Sondermann . . . . . . 46
2.5.3 Local risk-minimization of Schweizer . . . . . . . . . . 49
12 CONTENTS
3 Arbitrage-free price process 51
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 A Feynman-Kac type formula . . . . . . . . . . . . . . . . . . 51
3.3 Galtchouk-Kunita-Watanabe decomposition . . . . . . . . . . 57
4 Hedgingofunit-linkedpureendowmentandterminsurance 63
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Unit-linked life insurance contracts . . . . . . . . . . . . . . . 64
4.2.1 Unit-linked pure endowment . . . . . . . . . . . . . . 65
4.2.2 Unit-linked term insurance . . . . . . . . . . . . . . . 65
4.3 Risk-minimizing hedging strategy . . . . . . . . . . . . . . . . 66
4.3.1 Unit-linked pure endowment . . . . . . . . . . . . . . 66
4.3.2 Unit-linked term insurance . . . . . . . . . . . . . . . 73
4.4 Why use local risk-minimization? . . . . . . . . . . . . . . . . 79
5 Payment stream hedging for semimartingales 81
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.3 Local risk-minimization . . . . . . . . . . . . . . . . . . . . . 88
5.4 Finding a locally risk-minimizing strategy . . . . . . . . . . . 94
6 Hedging of general unit-linked life insurance contracts 97
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.2 General unit-linked benefit and premium payments . . . . . . 98
6.3 Locally risk-minimizing hedging strategies . . . . . . . . . . . 100
6.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.4.1 Single unit-linked life annuity insurance . . . . . . . . 109
6.4.2 Single unit-linked term insurance . . . . . . . . . . . . 112
6.4.3 Portfolio of n unit-linked life annuity contracts . . . . 114
6.4.4 Portfolio of n unit-linked term insurance contracts . . 117
7 Jump-diffusion stock model 121
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.2 The asymmetric double exponential distribution . . . . . . . 123
7.3 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
7.3.1 The asymmetric double exponential jump-diffusion . . 128
7.3.2 Financial market . . . . . . . . . . . . . . . . . . . . . 130
7.4 Risk-neutral measure . . . . . . . . . . . . . . . . . . . . . . . 133
7.4.1 F¨ollmer-Schweizer measure . . . . . . . . . . . . . . . 133
7.4.2 Kou’s risk-neutral measure . . . . . . . . . . . . . . . 139
7.5 Arbitrage-free price process . . . . . . . . . . . . . . . . . . . 140
7.5.1 Valuation of an European Call . . . . . . . . . . . . . 140
7.5.2 Valuation of unit-linkedpureendowmentandtermin-
surance with guarantee . . . . . . . . . . . . . . . . . 144CONTENTS 3
7.5.3 Spatial derivative . . . . . . . . . . . . . . . . . . . . . 144
8 Statistics for jump-diffusions 147
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
8.2 Model of stock price and returns . . . . . . . . . . . . . . . . 148
8.3 Estimation method . . . . . . . . . . . . . . . . . . . . . . . . 151
8.4 Mathematics behind the estimation . . . . . . . . . . . . . . . 157
8.5 Future research . . . . . . . . . . . . . . . . . . . . . . . . . . 163
A Supplementary Material 165
Bibliography 169
List of Tables 175
List of Figures 177
Zusammenfassung 1794 CONTENTSChapter 1
Introduction and summary
1.1 Historical overview and existing literature
Unlike traditional life insurance the benefits and possibly the premiums of
an unit-linked life insurance are random and linked to the development of
some specified reference portfolio, a mutual fund or simply a stock index.
Insurance companies offering these kind of products often collaborate with
mutual funds and offer the policy-holder a huge variety of investment op-
portunities including funds that concentrate on specific countries, areas or
industry sectors or even mixtures of