Universit¨atUlm Institut f¨urFinanzmathematik Energy-Related Commodity Futures Statistics, Models and Derivatives Dissertation zur Erlangung des Doktorgrades Dr. rer. nat. der Fakult¨at f¨urMathematik und Wirtschaftswissenschaften an der Universit¨at Ulm RSITÄT V E U I L N M U · · S O C I D E N N A D R O U · C · D O O C D E N vorgelegt von Dipl.-Math. oec. Reik H. B¨orger, M. S. Ulm, Juni 2007 ii . iii . Amtierender Dekan: Professor Dr. Frank Stehling 1. Gutachter: Professor Dr. R¨udiger Kiesel, Universit¨at Ulm 2. Gutachter: Professor Dr. Ulrich Rieder, Universit¨atUlm 3. Gutachter: Professor Dr. Ralf Korn, Universit¨at Kaiserslautern Tag der Promotion: 15.10.2007 iv Acknowledgements 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. R¨udiger 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 writing of this thesis and my family Hasso, Ruth and Lars B¨orgerfor their infinite patience. v vi Contents 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 vii viii 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 . 131 CONTENTS 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 162 x CONTENTS Chapter 1 Introduction to Commodity Markets and Summary of the Thesis 1.1 History of Commodity Markets Commodities are raw or primary products and goods which can be traded. The variety of such goods ranges from agricultural products (wheat, coffee, sugar, soybeans and many more) via metals (e. g. iron, copper, zinc) to energy commodities (for example oil, natural gas, coal, power, CO2 certificates). Early commodity markets (mainly agriculturals) date back to Sumerian civilization around 5000 to 3000 BC. Trading basic commodities is a cornerstone of any economy since it allows for division of labor. Since then commodity marketplaces have always existed culminating in today’s highly efficient and standardized markets organized by exchanges. In particular, the trade of the actual commodities (spot trading) has shifted to the trade of contracts promising the exchange of a commodity at a future time (forward trading). One of the earliest known forward contracts regulates the delivery of rice in seventeenth century Japan. The cradle of modern exchange based forward trading is the Chicago Board of Trade, which has been founded in 1848. The importance of Chicago emerged from the central location between Midwestern farmers and east coast consumers. By now, the concept of forward trading has been adapted to other financial markets as well. Traditionally, forward contracts have been signed in order to ensure the sale of production for a predetermined price, thus avoiding the risk of overproduction. On the consumer side, forward contracts can be viewed as an insurance against scarcity of a product at a given future time, for example a bad harvest due to unfavorable weather conditions. Today, these trading strategies are referred to as hedging although forward contracts and hedging have gained a purely financial aspect. In fact, speculators that have no intention in physically buying any commodity have entered the market of forward trading with the goal of participating in price movements. They view such contracts as an alternative asset class. Though they have lost the connection to the physical product, they provide liquidity in commodity markets which is of most importance for efficiency of markets. The range of traded commodities is wide and seems to be ever-growing. The focus of this 1 2 CHAPTER 1. INTRODUCTION AND SUMMARY thesis is on energy-related commodities. Energy-related trading dates back at least to the 1860s with the introduction of crude oil at the Chicago Board of Trade and gains more and more attention by modern societies. On the other hand, one of the most important commodities emerged in the course of the 1990s when many nations in Europe have dereg- ulated their electricity markets so that power became tradable. The physical properties of power, most importantly non-storability, required the setup of tailor-made products and exchanges. The European Energy Exchange in Leipzig (EEX) and Nord Pool in Oslo are prime examples. The youngest energy commodity is formed by CO2 emission allowances – a government issued allowance to produce CO2 which is a major byproduct of power production. While it can be put in question if the right to pollute environment is a com- modity in the original sense of the word, it is a modern attempt to navigate society by means of commodity markets and concepts that have been successful over hundreds of years. Nowadays, actors in energy-related commodity markets such as utility companies and financial institutions cannot have in mind the underlying production process solely, but have to base their trading strategies on financial considerations as well. They have to manage their production process on one hand and financial risks evolving from trades on the other hand as it is standard for actors in other financial markets such as equity and fixed income. This thesis aims at helping to understand the financial risks involved when trading futures and options on energy-related products such as oil, coal, natural gas, power and CO2 allowances. We will describe risks emerging from commodity trading in detail in the corresponding chapters.