Energy markets [Elektronische Ressource] : risk management, optimal liquidation, and derivatives / vorgelegt von Kevin Metka

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• Mathematics

Informations

• Publié par: universitat_ulm
• Publié le: 01 janvier 2011
• Nombre de lectures: 4
• Langue: English
• Poids de l'ouvrage: 5 Mo

Extrait

Universität Ulm
Institut für Finanzmathematik
Energy Markets
Risk Management, Optimal Liquidation, and Derivatives
Dissertation zur Erlangung des Doktorgrades
Dr. rer. nat.
der Fakultät für Mathematik und Wirtschaftswissenschaften
an der Universität Ulm
Vorgelegt von
Dipl.-Math. oec., MSc., Kevin Metka
im Januar 2011Amtierender Dekan: Prof. Dr. Werner Kratz
1. Gutachter: Prof. Dr. Rüdiger Kiesel
2.hter: Dr. Ulrich Rieder
Tag der Promotion: 24.03.2011Acknowledgements
Apart from the many contributing individuals, this thesis would not have been possi-
ble without the financial support through a scholarship, granted by the EnBW Trading
GmbH. In particular, I would like to thank Dr. Sven-Olaf Stoll and Klaus Wiebauer for
their scientific support and trust in my work. In addition, they suggested many of the
practical problems addressed in this thesis. By answering uncounted questions about
energy markets, they contributed as a helpful and valuable source.
I am especially indebted to my Doktorvater, Professor Dr. Rüdiger Kiesel, who has
been conducive to this thesis in multiple ways. (1) He initiated the joint work with
the EnBW Trading GmbH, establishing the financial support. (2) Technical, as well as
conceptional issues were always discussed and resolved. This held the scientific work on
track. (3) His confidence in my work kept me motivated and made me adhere to the
main objectives, which constitute the contributions of this thesis.
Professor Dr. Ulrich Rieder contributed by sharpening my senses for technical details.
This enhanced the thesis considerably.
Further thanks go to the members of the Institute of Mathematical Finance at the
University of Ulm, for the many fruitful discussions about and beyond the scope of
mathematical finance. In particular, I would like to thank my mate Andreas Rupp for
the many refreshing and insightful comments.
A special thank goes to Dr. Reik Börger, who motivated the practical question about
retail contracts in Chapter 6 considerably, and to Stefan Ehrenfried, who gave valuable
comments on Chapter 5 and 6.
Last but not least, my special thanks go to my mother, Beate Heine, her Husband
Jürgen Heine, my brother Danny Metka, and my Sister Linn Bartscht for their infinite
patience over the years.
iiiContents
Acknowledgements i
1 Introduction to Energy Markets 1
1.1 Energy? Markets? Energy Markets? . . . . . . . . . . . . . . . . . . . . 1
1.2 Current Issues in Modeling Energy Markets . . . . . . . . . . . . . . . . 3
1.3 Relevant Markets and Portfolio Setup . . . . . . . . . . . . . . . . . . . 3
1.4 Structure of the Thesis and Main Results . . . . . . . . . . . . . . . . . 10
2 A Multivariate Commodity Analysis 15
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.1 Related Literature & Contribution . . . . . . . . . . . . . . . . . 16
2.1.2 General Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Multivariate Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.1 Volatility Patterns in Energy Markets? . . . . . . . . . . . . . . . 19
2.2.2 Vy Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.3 Marginal Distributions . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2.4 Dependence Structure . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Application to Risk Management . . . . . . . . . . . . . . . . . . . . . . 36
2.3.1 Portfolio Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3.2 Backtesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4 Conclusions & Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3 Optimal Liquidation Strategies 47
3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 Market Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3 Solution Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3.1 Static Liquidation . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3.2 Dynamic . . . . . . . . . . . . . . . . . . . . . . . . . 56
4 Optimal Static Liquidation Strategies 57
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1.1 Related Literature & Contribution . . . . . . . . . . . . . . . . . 57
4.2 Optimal Liquidation as a Convex Optimization Problem . . . . . . . . . 58
4.2.1 General Structural Properties . . . . . . . . . . . . . . . . . . . . 58
iiiiv CONTENTS
0
4.2.2 Conditions for Optimal Solutions of (P ) . . . . . . . . . . . . . . 63
4.2.3 Conditions for of (P ) . . . . . . . . . . . . . . 66
4.3 Solution Methods and Case Study . . . . . . . . . . . . . . . . . . . . . 67
4.3.1 A Cutting Plane Algorithm . . . . . . . . . . . . . . . . . . . . . 68

4.3.2 Further Properties of the Management Rule PR,α . . . . . . . 68
4.3.3 Negative Market Drift . . . . . . . . . . . . . . . . . . . . . . . . 70
4.3.4 Positive Market Drift . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3.5 No Market Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3.6 Comparison of Different Models for a Fixed Management Rule . 75
4.3.7 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.4 Conclusions & Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5 Optimal Dynamic Liquidation Strategies 79
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.1.1 Related Literature & Contribution . . . . . . . . . . . . . . . . . 79
5.1.2 General Formulation as a Stochastic Dynamic Program . . . . . 81
0
5.2 Solving (P ): Explicit Optimal Liquidation Strategies . . . . . . . . . . 83
5.2.1 The Bellman Equation and Existence of Optimal Solutions . . . 83
5.2.2 Frictionless Markets . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.2.3 Markets with Liquidity Risk . . . . . . . . . . . . . . . . . . . . . 93
5.3 Solving (P ): Extended Settings and Numerical Schemes . . . . . . . . . 110
5.3.1 The Bellman Equation and Existence of Optimal Solutions under
a Target Wealth . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.3.2 An Extended Grid Algorithm . . . . . . . . . . . . . . . . . . . . 115
5.3.3 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.4 Conclusions & Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.5 Extensions: Margin Modeling . . . . . . . . . . . . . . . . . . . . . . . . 126
6 Pricing of Structured Retail Contracts 129
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.1.1 Related Literature & Contribution . . . . . . . . . . . . . . . . . 130
6.1.2 The Retail Electricity Price . . . . . . . . . . . . . . . . . . . . . 131
6.1.3 Retail Contract Structure . . . . . . . . . . . . . . . . . . . . . . 132
6.1.4 Customer Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.2 Valuation Framework for Structured Contracts . . . . . . . . . . . . . . 134
6.2.1 Analysis of Payment Flows . . . . . . . . . . . . . . . . . . . . . 134
6.2.2 Market Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.2.3 Endowing Payment Flows with Call Rights . . . . . . . . . . . . 138
6.2.4 Further Properties of PV . . . . . . . . . . . . . . . . . . . . . 144t0
6.2.5 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.3 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.3.1 Example Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.3.2 Calibration Results . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.3.3 Analysis of Contracts without Call Rights . . . . . . . . . . . . . 154CONTENTS v
6.3.4 Analysis of Contracts with Call Rights . . . . . . . . . . . . . . . 155
6.3.5 Value-Neutral Specifications . . . . . . . . . . . . . . . . . . . . . 160
6.4 Conclusions & Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.5 Extensions: Incorporating Volume Risk . . . . . . . . . . . . . . . . . . 161
A Technical Details & Supplementary Material 169
A.1 The Arithmetic Ornstein-Uhlenbeck Process . . . . . . . . . . . . . . . . 169
A.2 Supplementary Material for Chapter 2 . . . . . . . . . . . . . . . . . . . 172
A.3tary for Chapter 5 . . . . . . . . . . . . . . . . . . . 183
∗
A.3.1 Distribution of W under π . . . . . . . . . . . . . . . . . . . 183T +1
A.3.2 Distribution of W under πˆ . . . . . . . . . . . . . . . . . . . . 183T +1
A.4 Supplementary Material for Chapter 6 . . . . . . . . . . . . . . . . . . . 185
List of Tables 185
List of Figures 188
Bibliography 191
Zusammenfassung 200vi CONTENTSChapter 1
Introduction to Energy Markets
‘There is not the slightest indication that
energy will ever be obtainable from the atom.’
1
Albert Einstein
1.1 Energy? Markets? Energy Markets?
Energy is a physical quantity that describes the amount of work that can be performed
by a force. This definition is commonly attached to the term ‘energy’. From an economic
point of view this term is more difficult to specify. Many people understand energy as
a class of energy sources, which are comprised of coal, natural gas, crude oil, and many
more. These commodities have a