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Advances in the Analysis of Energy Commodities
and of Multivariate Dependence Structures
Inaugural-Dissertation
zur Erlangung des akademischen Grades eines Doktors
der Wirtschafts- und Sozialwissenschaften
(Dr. rer. pol.)
der Friedrich-Alexander-Universitat Erlangen-Nurn berg
vorgelegt von: Diplom-Wirtschaftsmathematiker Stephan Schluter
aus: Munc henx
Kumulative Promotion
Erstreferent: Professor Dr. Ingo Klein
Zweitreferent: Professor Dr. Wolfgang Stummer
Letzte Prufung: 27.01.1011Wer mit den wenigsten und einfachsten Symbolen
das Meiste und Bedeutendste ausspricht,
der ist der gr o te Kunstler.
Heinrich HeineAcknowledgements
First and foremost I would like to express my deep and sincere gratitude to my
doctoral advisor Professor Dr. Ingo Klein for his invaluable support and advice. He
encouraged me to write this thesis and gave me every support necessary to conduct
this project. Besides that I would like to thank Dr. Matthias Fischer for the fertile
cooperation. I would also like to thank Klaus Herrmann for his advice and the great
company during the many years we spent together at the department. Furthermore,
I am indebted to our wonderful secretary Mrs. Enhuber, my great colleagues and all
those people and friends who helped me on my way. Eventually I would like to show
my deepest gratitude to my family for their unconditional and invaluable support.Contents
1 Introduction 3
2 A New Model for Daily Electricity Prices 17
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 A Discussion of Models for Power Prices . . . . . . . . . . . . . . . . 20
2.3 Goodness of Fit Measures . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Data Analysis and Preprocessing . . . . . . . . . . . . . . . . . . . . 24
2.5 Estimation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3 Wavelet Based Forecasting 41
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Some Basics of Time Series Analysis . . . . . . . . . . . . . . . . . . 44
3.3 Wavelet Based Forecasting . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4 An Empirical Comparison of Di erent Forecasting Methods . . . . . . 54
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4 A Student T Tail Quantile Approximation 73
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2 Notation and De nitions . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3 Approximating High Gaussian Quantiles . . . . . . . . . . . . . . . . 78
4.4 Approximating High Student t Quantiles . . . . . . . . . . . . . . . . 80
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
IContents
5 Valuation of a European Gas Storage Facility 85
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2 The TTF Day Ahead Gas Price . . . . . . . . . . . . . . . . . . . . . 89
5.3 Valuation of a Gas Storage Facility . . . . . . . . . . . . . . . . . . . 91
5.4 Analyzing a Realistic Scenario . . . . . . . . . . . . . . . . . . . . . . 98
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6 An Empirical Analysis of Multivariate Copula Models 111
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2 Constructing Multivariate Non-Elliptical Copulas . . . . . . . . . . . 114
6.3 Goodness of Fit Measures . . . . . . . . . . . . . . . . . . . . . . . . 124
6.4 The Data Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.5 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
7 Tail Dependence in the Elliptical Generalized Hyperbolic Family 141
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.2 Quantifying Extremal Probabilities Using Tail Dependence Coe cients144
7.3 The Elliptical Generalized Hyperbolic Distribution . . . . . . . . . . 146
7.4 The Weak Tail Dependence Coe cient of the Elliptical Generalized
Hyperbolic Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.5 Technical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8 Conclusion 159
IIChapter 1
Introduction
With the Directive 96/92/EC in 1996 the European Parliament enacted the liber-
alization of the electricity markets in the member states of the European Union. In
Germany, this led to the launch of two power exchanges which merged into the Eu-
ropean Energy Exchange (EEX) located in Leipzig in 2002. Since then the liquidity
on the EEX has been steadily increasing, so that we can speak from about 2005 on
of a liquid though not complete market.
In the course of these events the German natural gas market was liberalized as
well. As a result several smaller gas trading hubs were launched. However, in con-
trast to the electricity market, the German gas market remained fragmented. At
the beginning of 2007 there were still 19 di erent market areas which made it hard
to buy and ship gas across Germany (cf. IEA, 2009). Recent mergers increased the
trade signi cantly, though. Still, the market is too small to be independent, and the
short-term gas price is heavily linked to the developments of continental Europe’s
major gas hub, the Dutch Title Transfer Facility (TTF).
Signs of an advancing market liberalization can be seen when looking at power and
gas prices in Germany. The TTF day ahead gas price history shows distinct jumps
around 2005, which diminish from 2006 on. This indicates an increase in market
liquidity. Historical power prices, in turn, show a structural break between 2005 and
2006.
3Chapter 1. Introduction
As the structure of both prices is changing, we cannot use gas or power price
models from past research. Which is the most appropriate price model for the latest
data needs to be investigated. Answering this question is important as especially
1the short-term power prices show a very high volatility compared to other energy
commodities. This induces a substantial nancial risk on a market where the ma-
jority of participants is active not for speculation purposes but because of a physical
need for these commodities.
The major reason for the high price volatility is that once produced electricity is
not economically storable (let alone some small capacities of pumped hydro power).
On the contrary, gas can be stored relatively easily in underground facilities like
salt caverns. Historically, these have been mainly used to bene t from seasonal
price oscillations: Gas is cheap in the summer and if it is stored it can be sold at
a higher price in the winter. With the recently developing gas spot markets new
possibilities arise. Storage facilities with high injection and withdrawal rates can be
used as hedging positions in option businesses. This strategy o ers more pro t but
is also more risky. Moreover, it becomes more complex to determine the optimal
operational strategy for a storage facility as well as its value.
Above we described the modeling of gas and power as isolated problems. However,
in most scenarios contracts on gas and electricity are integrated in a larger portfolio
comprising further energy commodities like oil or coal. To some extent these fuels
are used to generate power, so rising fuel prices have a positive e ect on power
prices. High power prices, in turn, imply that expensive but exible power plants
like gas turbines are more pro table, i.e. they are likely to be switched on more
often. The demand and therefore the price of gas is rising as well. Hence, there is a
complex multidimensional dependence structure. Moreover, the individual portfolio
components are quite di erent: Power prices show a high daily volatility, whereas oil
or coal prices are dominated by a long-term pattern. What is required is a concept
that models dependence appropriately but allows a exible behavior of each of the
1We understand volatility here as standard deviation, i.e. as a measure for dispersion.
4Chapter 1. Introduction
portfolio’s components.
In total there is a broad range of questions which are addressed in the six chapters
of this work. Initially, power prices are studied. There is already a considerable
amount of literature about modeling German power price indices and we can use
it as a basis for future research. In contrast, the local gas markets have attracted
less interest so far. There are some models for the North American market, but not
for the European ones. However, with the advancing liberalization in Europe the
academic interest for these markets is growing, which is also due to the improving
data situation. The European markets now have an almost su cient liquidity and
the time series are of a length that allows a credible econometrical analysis. The rst
years of trading at the EEX, for example, can only be analyzed with reservation. The
major reason is insu cient liquidity which prevents us from using some standard
arguments of time series analysis, e.g. that the action of a single trader does not
in uence the price.
There are a few other features which make power a special commodity. Its
demand shows a characteristical intra-day pattern and further di erent seasonal os-
cillations, for instance. Prices are also rather local due to transportation constraints.
In the EEX forward market most of these e ects are not observed, mainly because
market participants can plan long term. The prices’ oscillation there is relatively
small and their structure resembles those of other commodities (e.g. coal or oil).
More interesting are short-term power prices, e.g. hourly or (average) day ahead
prices. There the special characteristics of power have signi cant impact and we
observe di erent kinds of seasonality and a comparatively high volatility.
Modeling Power Prices
The rst step towards understanding power prices is the analysis of their seasonal
structure. There are two major reasons for that: First, when using a parametric
price model we need to adjust for seasonal e ects in order to prevent the parameter
estimates from being biased. Secondly, we can apply the extracted seasonality to
5Chapter 1. Introduction
compute rough forecasts. To identify and adjust for seasonality, methods like the
Kalman lter, the Hodrick-Prescott lter or the Fourier transform may be applied.
However, these concepts do not capture dynamics in a patterns’ period and intensity.
This is of relevance for our analysis. The German short-term power prices, for
example, are in uenced among others by the daily temperature, which has a seasonal
pattern that is not completely constant over the years. A solution to this problem is
to apply the wavelet transform, which is a generalization of the Fourier transform,
as it captures these e ects. Wavelets are localized wave functions, i.e. they are
centered around the origin and diminish in plus and minus in nity. Each wavelet
has a xed period which can be changed by a scaling operation. Moreover, the
wavelet can be shifted by translation towards an arbitrary point on the real axis.
Convolving a time series with a wavelet corresponds to its orthogonal projection
on the wavelet in the point of time where it has been translated to. Thus, we can
identify for each point of time the part of the data which is explained by a pattern
with the wavelet’s (scaled) period.
In Chapter 2 we apply the wavelet transform to lter seasonal e ects in power
prices from four di erent European countries, namely Germany, the Netherlands, the
UK and the Nordic market (comprising Denmark, Sweden, Norway and Finland).
We

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