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Energy related commodity futures [Elektronische Ressource] : statistics, models and derivatives / vorgelegt von Reik H. Börger

177 pages
ECOD·ODNEICS·MLUTÄTUniversit¨at UlmInstitut fur¨ FinanzmathematikEnergy-Related Commodity FuturesStatistics, Models and DerivativesDissertation zur Erlangung des DoktorgradesDr. rer. nat.der Fakult¨at fur¨ Mathematik und Wirtschaftswissenschaftenan der Universit¨at Ulmvorgelegt vonDipl.-Math. oec. Reik H. B¨orger, M. S.Ulm, Juni 2007ISREVINU·ODNARUC·ODNii.iii.Amtierender Dekan: Professor Dr. Frank Stehling1. Gutachter: Professor Dr. Rudig¨ er Kiesel, Universit¨at Ulm2. Gutachter: Professor Dr. Ulrich Rieder, Universit¨at Ulm3. Gutachter: Professor Dr. Ralf Korn, Universit¨at KaiserslauternTag der Promotion: 15.10.2007ivAcknowledgementsThis thesis would not have been possible without the financial and scientific support byEnBW Trading GmbH.In particular, I received instructive input from Dr. Gero Schindlmayr. He suggestedmany of the problems that have been covered in this work. In numerous discussions hegave insight into physical and financial details of commodities and commodity markets.I also benefited from his suggestions on aspects of the mathematical models and theirapplicability to practical questions.I take the opportunity to thank my academic advisor Professor Dr. Ru¨diger Kiesel whoinitiated the collaboration with EnBW from the university’s side and who supported mystudies in every possible respect.
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Universit¨at Ulm
Institut fur¨ Finanzmathematik
Energy-Related Commodity Futures
Statistics, Models and Derivatives
Dissertation zur Erlangung des Doktorgrades
Dr. rer. nat.
der Fakult¨at fur¨ Mathematik und Wirtschaftswissenschaften
an der Universit¨at Ulm
vorgelegt von
Dipl.-Math. oec. Reik H. B¨orger, M. S.
Ulm, Juni 2007
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Amtierender Dekan: Professor Dr. Frank Stehling
1. Gutachter: Professor Dr. Rudig¨ er Kiesel, Universit¨at Ulm
2. Gutachter: Professor Dr. Ulrich Rieder, Universit¨at Ulm
3. Gutachter: Professor Dr. Ralf Korn, Universit¨at Kaiserslautern
Tag der Promotion: 15.10.2007ivAcknowledgements
This thesis would not have been possible without the financial and scientific support by
EnBW Trading GmbH.
In particular, I received instructive input from Dr. Gero Schindlmayr. He suggested
many of the problems that have been covered in this work. In numerous discussions he
gave insight into physical and financial details of commodities and commodity markets.
I also benefited from his suggestions on aspects of the mathematical models and their
applicability to practical questions.
I take the opportunity to thank my academic advisor Professor Dr. Ru¨diger Kiesel who
initiated the collaboration with EnBW from the university’s side and who supported my
studies in every possible respect. I highly appreciate his confidence in my work and his
encouragement which resulted in an enjoyable working environment that goes far beyond
the usual conditions.
I thank the members of the Institute of Financial Mathematics at Ulm University, in
particular Gregor Mummenhoff, Clemens Prestele and Matthias Scherer, for the many
mathematical and non-mathematical activities that enriched my time in Ulm.
I am also indebted to Professor Fred Espen Benth (University of Oslo) and Professor
´Alvaro Cartea (Birkbeck College) for their perpetual willingness to answer my questions
and for giving helpful comments on my work.
Further, I want to express my gratitude to my friends Oliver Horn and Markus Kunze for
all the lively discussions during the many years we know each other and Christin Sautter
and Berthold Wespel who have always been available when I needed them, never asking
for a return.
Last but not least I want to thank Stefanie Piechulla for her caring support during the
writingofthisthesisandmyfamilyHasso,RuthandLarsB¨orgerfortheirinfinitepatience.
vviContents
Acknowledgements v
1 Introduction to Commodity Markets and Summary of the Thesis 1
1.1 History of Commodity Markets . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Statistical Properties of Commodity Forwards . . . . . . . . . . . . . . . . . 2
1.3 Approaches to Stochastic Commodity Forward Modeling . . . . . . . . . . . 3
1.4 Current Issues in Forward Pricing & Risk Management. . . . . . . . . . . . 5
1.5 Objective of the Thesis and Contribution . . . . . . . . . . . . . . . . . . . 5
1.6 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Multivariate Generalized Hyperbolic Distributions, L´evy Processes and
Option Pricing 11
2.1 Multivariate Generalized Hyperbolic Distributions . . . . . . . . . . . . . . 11
2.1.1 Definition & Properties of Generalized Hyperbolic Distributions . . 11
2.1.2 Estimation of Generalized Hyperbolic Distributions . . . . . . . . . 18
2.2 Financial Model Building with L´evy Processes . . . . . . . . . . . . . . . . 19
2.2.1 Definition & Properties of L´evy Processes . . . . . . . . . . . . . . . 19
2.2.2 Financial Model Building with Exponential L´evy Processes . . . . . 21
2.3 Option Pricing and Black’s Formula . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Previous Applications to Finance . . . . . . . . . . . . . . . . . . . . . . . . 26
3 A Multivariate Commodity Analysis and Applications to Risk Manage-
ment 29
3.1 Literature Overview & Contribution . . . . . . . . . . . . . . . . . . . . . . 32
3.2 Statistical Tools. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3 Data Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3.1 Data Preparation and Construction of Data Sets . . . . . . . . . . . 39
3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.5 Application to Risk Management . . . . . . . . . . . . . . . . . . . . . . . . 46
3.5.1 Computation of Risk Measures . . . . . . . . . . . . . . . . . . . . . 46
viiviii CONTENTS
3.5.2 Numerical Examples Analyzing Power Plants . . . . . . . . . . . . . 50
3.6 Summary & Related Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4 Modeling Futures with Delivery Over Periods 63
4.1 Literature Overview & Contribution . . . . . . . . . . . . . . . . . . . . . . 64
4.2 The EEX Futures and Options market . . . . . . . . . . . . . . . . . . . . . 65
4.3 No-Arbitrage Considerations Implied by Delivery Periods . . . . . . . . . . 67
4.4 Description of the Model and Option Pricing . . . . . . . . . . . . . . . . . 72
4.4.1 General Model Formulation . . . . . . . . . . . . . . . . . . . . . . . 72
4.4.2 Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.5 The Special Case of a Two-Factor Model . . . . . . . . . . . . . . . . . . . . 75
4.5.1 Model Formulation and Option Pricing . . . . . . . . . . . . . . . . 75
4.6 Estimating the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.6.1 Calibration Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.6.2 Calibration to Option Prices . . . . . . . . . . . . . . . . . . . . . . 78
4.7 Summary & Related Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5 A L´evy-driven Futures Model 83
5.1 Literature Overview & Contribution . . . . . . . . . . . . . . . . . . . . . . 86
5.2 Presentation of a L´evy-driven Two-factor Model . . . . . . . . . . . . . . . 88
5.2.1 Risk-neutral Drift Condition . . . . . . . . . . . . . . . . . . . . . . 89
5.2.2 Options on Futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2.3 Options on Sums of Futures . . . . . . . . . . . . . . . . . . . . . . . 94
5.2.4 Modeling Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.3 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.4 Choice of an Objective Function . . . . . . . . . . . . . . . . . . . . . . . . 106
5.4.1 Prices vs. Volatilities . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.4.2 Modifications of the Objective Function . . . . . . . . . . . . . . . . 108
5.5 Data Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.6 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.7 Summary & Related Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6 Distributions of Arithmetic Averages – A Simulation Study 119
6.1 Literature Overview & Contribution . . . . . . . . . . . . . . . . . . . . . . 120
6.2 Model Presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.3 Underlying, Simulating & Approximating Distributions . . . . . . . . . . . 123
6.4 Parameter Inference by Moment-Matching . . . . . . . . . . . . . . . . . . . 126
6.5 Measures of Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131CONTENTS ix
6.6 Statistical Proceeding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.7 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.7.1 LIBOR Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.7.2 Energy Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.8 Summary & Related Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
A Option Pricing Using Fast Fourier Transforms 145
A.1 Option Pricing Using Fourier Transforms . . . . . . . . . . . . . . . . . . . 145
A.2 Application of Fast Fourier Transforms . . . . . . . . . . . . . . . . . . . . . 146
List of Tables 149
List of Figures 150
Bibliography 152
Curriculum Vitae 159
Zusammenfassung 162x CONTENTS

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