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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