Copula-Models in the Electric Power Industry

Transcription

1 University of St.Gallen Graduate School of Business Administration, Economics, Law and Social Sciences (HSG) Copula-Models in the Electric Power Industry Master s Thesis Pascal Fischbach Rislenstrasse Romanshorn Tel.: 071/ pascal_fischbach@hotmail.com Supervisor: Prof. Dr. Karl Frauendorfer Submitted on: 16 August 2010

2 Abstract I ABSTRACT As a consequence of the ongoing deregulation process of electricity markets, prices for electricity are now determined by the forces of supply and demand. The non-storability feature of electricity thereby has a crucial impact on the specific characteristics of spot power prices that differ from most other assets. In order to cope with the correspondingly high degree of uncertainty, providers of trading platforms, such as for instance the EEX, have introduced a variety of spot and derivatives products. Clearly, the companies within the electric power industry are interested in the dependence structure of these contracts. The present thesis therefore applies bivariate Gaussian, t-, Gumbel, Clayton and Frank copula models to various return series of spot and futures contracts traded at the EEX. The majority of the results thereby suggest that the t-copula provides an adequate representation of the empirical dependence structures.

3 Table of Contents II TABLE OF CONTENTS List of Figures... IV List of Tables... V List of Abbreviations and Symbols... VI 1 Introduction Presentation of the Problem and its Relevance Goal Setting and Delimitation Methodology and Structure The Electric Power Industry Trading in Power at the European Energy Exchange (EEX) The European Energy Market The Holding Structure of the EEX and its sub-markets Power Products and Trading Processes at the EEX Trading Spot Power Contracts at the EEX Trading Power Derivatives Contracts at the EEX Trading Volumes and Over the Counter Trading Characteristics of Electricity Prices Spot Price Characteristics Seasonalities Mean Reversion Jumps, Spikes and exceptionally high Volatility Price Characteristics of Futures Contracts Dependence: Linear Correlation, Copulas and Measures of Association Theoretical Background of Copulas and Dependence Pearson s Linear Correlation Copulas Preliminaries Definition of Copula, Sklar s Theorem and Basic Properties Fundamental Copulas Elliptical and Archimedean Copulas Measures of Association Kendall s Tau Spearman s Rho Tail Dependence Fitting Copulas to Data Estimation of Copulas... 40

4 Table of Contents III Full Maximum Likelihood Approach (FML) Inference Method for Margins (IFM) Canonical Maximum Likelihood Approach (CML) Calibration with Kendall s tau and Spearman s rho Nonparametric Method Goodness of Fit Tests for Copulas Applying Copulas to the Electric Power Industry General Remarks about the Estimation Procedure Phelix and Swissix Spot Analysis Data Set and Descriptive Statistics Estimation of the Marginals Phelix vs. Swissix Base vs. Peak Phelix Year Futures Analysis Data Set and Descriptive Statistics Estimation of the Marginals Base vs. Peak vs Further Analysis involving various Phelix Futures Contracts Data Set and Descriptive Statistics Estimation of the Marginals Different Time to Delivery Different Delivery Period Spot vs. Futures Summary of the Results across all Parts of the Analysis Conclusion...79 Appendix A: Additional Figures...81 Appendix B: Additional Tables Appendix C: Codes for Matlab and R Bibliography Declaration of Authorship

5 List of Figures IV LIST OF FIGURES Figure 1 Central West European electricity market as of Figure 2 Submarkets of the EEX... 6 Figure 3 Fréchet-Hoeffding bounds for C(u,v)...27 Figure 4 Gaussian and t-copula densities...31 Figure 5 Gumbel, Clayton and Frank copula densities...33 Figure 6 Relation between Kendall s tau and Spearman s rho...38 Figure 7 Seasonalities during the day, week and year...52 Figure 8 Autocorrelation function for daily log returns of spot products...53 Figure A1 Historical movement of the prices and log returns of the various time series of the Phelix and Swissix spot data set Figure A2 Histogram of the various return series of the Phelix and Swissix spot data set 83 Figure A3 Phelix Day Base vs. Swissix Day Base point clouds...84 Figure A4 Phelix Day Peak vs. Swissix Day Peak point clouds...85 Figure A5 Phelix Hourly vs. Swissix Hourly point clouds...86 Figure A6 Phelix Day Base vs. Phelix Day Peak point clouds...87 Figure A7 Swissix Day Base vs. Swissix Day Peak point clouds...88 Figure A8 Historical movement of the prices and log returns of the various time series of the Phelix Year Futures data set Figure A9 Histogram of the various Phelix Year Futures return series...90 Figure A10 Phelix Jan 2010 Base vs. Phelix Jan 2010 Peak point clouds...91 Figure A11 Phelix Jan 2011 Base vs. Phelix Jan 2011 Peak point clouds...92 Figure A12 Phelix Jan 2010 Base vs. Phelix Jan 2011 Base point clouds...93 Figure A13 Phelix Jan 2010 Peak vs. Phelix Jan 2011 Peak point clouds...94 Figure A14 Historical movement of the synthetic return series...95 Figure A15 Histogram of the various synthetic return series...97 Figure A16 1 Month ahead vs. 2 Months ahead point clouds...98 Figure A17 1 Quarter ahead vs. 2 Quarters ahead point clouds...99 Figure A18 1 Year ahead vs. 2 Years ahead point clouds Figure A19 1 Month ahead vs. 1 Quarter ahead point clouds Figure A20 1 Month ahead vs. 1 Year ahead point clouds Figure A21 1 Quarter ahead vs. 1 Year ahead point clouds Figure A22 Spot vs. 1 Month ahead point clouds Figure A23 Spot vs. 1 Quarter ahead point clouds Figure A24 Spot vs. 1 Year ahead point clouds

6 List of Tables V LIST OF TABLES Table 1 Characteristics of spot and derivatives contracts traded at the EEX...11 Table 2 Trading volume in the spot and derivatives power markets of the EEX...12 Table 3 Copula families, generator functions and permissible parameter ranges...29 Table 4 Relation between the copula parameter and ρ τ, ρ s, λ u and λ l...38 Table 5 Descriptive statistics of the Phelix and Swissix spot data set...55 Table 6 Marginal parameter estimates for the Phelix and Swissix spot data set...58 Table 7 Copula parameter estimates (Phelix vs. Swissix) for the Phelix and Swissix spot return series based on non-central t-distributed marginals...59 Table 8 Copula parameter estimates (Base vs. Peak) for the Phelix and Swissix spot return series based on non-central t-distributed marginals...62 Table 9 Descriptive statistics of the Phelix Year Futures data set...64 Table 10 Marginal parameter estimates for the Phelix Year Futures data set...66 Table 11 Copula parameter estimates (Base vs. Peak) for the Phelix Year Futures return series based on non-central t-distributed marginals...67 Table 12 Copula parameter estimates (2010 vs. 2011) for the Phelix Year Futures return series based on non-central t-distributed marginals...68 Table 13 Descriptive statistics of the synthetic time series...71 Table 14 Marginal parameter estimates for the synthetic return series...72 Table 15 Correlation matrices for the synthetic return series...73 Table 16 Copula parameter estimates (different time to delivery) for the synthetic return series based on non-central t-distributed marginals...74 Table 17 Copula parameter estimates (different delivery period) for the synthetic return series based on non-central t-distributed marginals...75 Table 18 Copula parameter estimates (Spot vs. Futures) for the synthetic return series based on non-central t-distributed marginals...76 Table B3 Copula parameter estimates (Base vs. Peak) for the Phelix Year Futures return series based on empirically distributed marginals Table B4 Copula parameter estimates (2010 vs. 2011) for the Phelix Year Futures return series based on empirically distributed marginals Table B5 Copula parameter estimates (different time to delivery) for the synthetic return series based on empirically distributed marginals Table B6 Copula parameter estimates (different delivery period) for the synthetic return series based on non-central t-distributed marginals Table B7 Copula parameter estimates (Spot vs. Futures) for the synthetic return series based on empirically distributed marginals

7 List of Abbreviations and Symbols VI LIST OF ABBREVIATIONS AND SYMBOLS 10B 10P 11B 11P 1M 1Q 1Y 2M 2Q 2Y AIC APG BIC cdf CML C(u 1,, u d ) C M C n C W C Π Cl Fr Phelix Jan 2010 Base time series Phelix Jan 2010 Peak time series Phelix Jan 2011 Base time series Phelix Jan 2011 Peak time series 1 Month ahead time series 1 Quarter ahead time series 1 Year ahead time series 2 Months ahead time series 2 Quarters ahead time series 2 Years ahead time series Akaike information criterion Austrian Power Grid (Austrian TSO) Bayesian information criterion Cumulative distribution function Canonical maximum likelihood method d-dimensional copula function Comonotonicity copula Empirical copula Countermonotonicity copula Independence copula Clayton copula (with parameter θ) Frank copula (with parameter θ) C Σ Ga Gaussian copula (with parameter Σ) Gu Gumbel copula (with parameter θ) t C Σ,ν t-copula (with parameters Σ and ν) c(u 1,, u d ) D AD D CvM D IAD D KS ECC EEX ENBW ENDEX EU Density of a d-dimensional copula function Anderson-Darling distance measure Cramer-von-Mises distance measure Integrated Anderson-Darling distance measure Kolmogorov-Smirnov distance measure European Commodity Clearing AG European Energy Exchange AG Energie Baden-Württemberg Transportnetze AG (German TSO) European Energy Derivatives Exchange N.V. European Union

8 List of Abbreviations and Symbols VII EUR FML F x 1,, x n F i x i F j x j F j nct x j ; α j nct f x 1,, x n IFM ln L MW MWh OTC pdf PhB PhH PhP RTE RWE r s X i, X j r τ X i, X j SGD Sp SwB SwH SwP TSO T TWh VE κ X1,X 2 λ l X 1, X 2 λ u X 1, X 2 ρ X 1, X 2 ρ s X 1, X 2 ρ τ X 1, X 2 ψ Euro Full maximum likelihood method n-dimensional multivariate (joint) distribution function i-th marginal distribution function (i-th marginal) j-th empirical marginal distribution function j-th non-central t-distributed marginal distribution (with parameter α nct j ) n-dimensional multivariate (joint) density function Inference for margins method Maximum log likelihood function Megawatt Megawatt hour Over the counter Probability density function Phelix Day Base time series Phelix Hourly time series Phelix Day Peak time series Réseau de Transport d Electricité (French TSO) Rheinisch-Westfälisches Elektrizitätswerk AG (German TSO) Estimator of Spearman s rank correlation coefficient (Spearman s rho) Estimator of Kendall s rank correlation coefficient (Kendall s tau) Swissgrid (Swiss TSO) Spot time series Swissix Day Base time series Swissix Hourly time series Swissix Day Peak time series Transmission system operator Generalized inverse Terawatt hour Vattenfall Europe AG (German TSO) Measure of association Lower tail dependence coefficient Upper tail dependence coefficient Pearson s linear correlation coefficient Spearman s rank correlation coefficient (Spearman s rho) Kendall s rank correlation coefficient (Kendall s tau) Generator function of an Archimedean copula

9 Introduction 1 1 INTRODUCTION 1.1 Presentation of the Problem and its Relevance With the initiation of the deregulation process of electricity markets in the last decades, companies within the electric power industry are currently not only subject to volume uncertainties in demand but also to a high degree of uncertainty in electricity prices (He, 2007, p.26). The market participants are hence exposed to an unprecedented amount of financial risk that requires a considerate risk management (Lemming, 2003, p.13). On one side, power producers generate electricity by operating various types of power plants to supply wholesale markets with electricity. On the other side, power providers engage in wholesale trading activities in order to provide their customers with electricity. They sell respectively buy electricity in accordance with projections about their electricity output respectively need. Generally, the loads that are rather certain to occur are sold respectively purchased in advance on the power derivatives market, while the more uncertain components of their portfolios are traded at short notice on spot markets (Lichtblick AG, 2008, p.26). In this sense, the analysis of the dependence structure between the various power products constitutes an integral part of risk management. The application of adequate financial risk management tools is thereby of particular importance, as electricity prices and their respective returns show some characteristics, which crucially differentiate them from the prices and returns of other assets. In fact, the usual simplifying assumption that asset returns follow a normal distribution is not sensible within the context of electricity prices. But this also implies that describing the dependence structures between various random vectors of electricity price returns via a multivariate normal distribution and Pearson s linear correlation coefficient is not appropriate. Consequently, more sophisticated concepts to describe these dependence structures are required. 1.2 Goal Setting and Delimitation Copulas are a tool to capture stochastic dependence of random variables in a far more complex way than with Pearson s linear correlation coefficient. It is important to recognize that the latter only fully characterizes the dependence structure of a set of random variables in the case that they are elliptically distributed. By contrast, the application of copula models is sensible under any kind of distribution. Furthermore, copulas are able to model a broad range of distinct dependence structures, including cases of upper and lower tail dependence, whereas stochastic modelling on the basis of a multivariate normal distribution would imply no tail dependence at all. In this sense, copula models allow to account for joint extreme observations either to the upside, to the downside, or in both directions. Thus, the application of

10 Introduction 2 copula models clearly provides an opportunity to model a specific dependence structure in a way that is closer to the empirically observable reality than with conventional methods. The intention of this thesis is to gain insight into the pair wise dependence structure of the return series of various spot and futures power products. In particular, with respect to the spot products, the dependence between Phelix and Swissix contracts and the dependence between base and peak load contracts shall be analyzed. Moreover, Phelix Year Futures are investigated in order to conclude on the dependence between base and peak load futures contracts as well as between futures products with a different time of delivery. Finally, a further analysis involving synthetic return series for Phelix futures contracts with various delivery periods and times to delivery is employed. There exists a variety of power exchanges throughout Europe. The focus of the present thesis, however, is set on a range of products traded at the European Energy Exchange (EEX). Furthermore, additional products are available for trading at the EEX that are not part of the present analysis. This includes, for instance, various intraday products, contracts with delivery in the French market area, or Phelix options. Delimitations also have to be taken with respect to the specific copula functions under analysis. Whereas this thesis concentrates on the most basic and well-known copula families (i.e. Gaussian, t-, Gumbel, Clayton and Frank), subsequent research could take into account further, more complex copula families. Finally, while this thesis is self-contained in the sense that it attempts to capture all relevant theory and background information with regard to copulas and electricity products as well as prices, some basic concepts within the area of finance and statistics are nevertheless assumed to be prerequisites. 1.3 Methodology and Structure The subsequent Section 2 will commence this thesis by providing a short overview of the electric power industry. In particular, Section 2.1 describes the activities of trading electricity at the EEX. This will include a presentation of the EEX as a trading platform and a description of the various power products traded at the EEX. Section 2.2 continues by elaborating on the characteristics of electricity prices. Section 3 will then shift the focus towards a more statistical context by studying concepts of stochastic dependence. Firstly, Section 3.1 will put forth the basic concepts behind the theory of copulas and various dependence measures. Thereafter, Section 3.2 enters into the discussion of the problem of fitting copulas to a given set of data, covering both the topics of estimating the parameters and testing the goodness of fit of various model specifications. While Section 2 and Section 3 mainly involve a review of the trading possibilities in power at the EEX, the specific characteristics of electricity prices and the relevant body of copula literature, they will together provide the required theoretical

11 Introduction 3 background in order to proceed with the empirical investigation in Section 4. Specifically, Section 4.2, 4.3 and 4.4 will cover a Phelix and Swissix spot analysis, a Phelix Year Futures analysis and a further analysis involving various Phelix futures contracts, respectively. Prior to the presentation of the respective results, the general procedure applied throughout these sub-analyses is presented in Section 4.1. Within each subsection, first the data set under consideration is presented and corresponding descriptive statistics are provided. Following this, the marginal distributions are estimated in order to subsequently present the results of applying copula models onto the various time series of electricity prices (respectively returns) in compliance with the theoretical considerations presented in Section 3. Section 5 finally concludes this paper.

12 The Electric Power Industry 4 2 THE ELECTRIC POWER INDUSTRY With the ongoing deregulation in the electric power industry, electricity markets have experienced substantial changes. Electricity prices are no longer subject to government decisions, but rather result from the trading activities at power exchanges, driven by the forces of supply and demand. Due to the non-existence of decent storage capabilities, electricity as a commodity strongly differs from other financial assets. On one side, this has an impact on the processes of trading electricity at the various power exchanges. For example, trading is possible in hourly contracts as well as block contracts such as peak or base loads. This is clearly a distinguishing feature of electricity markets compared to financial markets, where standard spot products are not subject to the specification of a delivery period. On the other side, electricity prices exhibit certain characteristics, which differentiate them from the prices of common financial assets. According to Blöchlinger (2008), Borchert et al. (2006), Weber (2005) and Weron (2005) this includes seasonal patterns, mean reversion, spikes and jumps, and, as a result, incomparably high volatility. In the following sections both aspects, firstly, regarding electricity trading and secondly, regarding the characteristics of electricity prices, are discussed in more detail. 2.1 Trading in Power at the European Energy Exchange (EEX) The European Energy Market Since 1998, when the liberalization of European energy markets was initiated, a variety of power exchanges has been established in Europe. While competition on the level of power supply is of advantage for end consumers, the establishment of a single European electricity market on the wholesale level has been a major aim of the liberalization process (EEX AG & Powernext SA, 2008a, p.2). Meeus and Belmans (2008, pp.5-10) adhere that the price differences across European countries are still large and that Europe is still far away from an integrated Europe-wide market for electricity. Nevertheless, some promising regional markets have originated that have a potential of being pathbreaking in the process of integration. In 2004, the European Commission recognized that such regional markets may provide an inevitable interim stage in achieving the ambitious goal of a single European electricity market (EEX AG & Powernext SA, 2008a, p.2). As a consequence thereof, the formation of regional markets was actively promoted by the commission. Since then, major attempts to a continuing integration of regional electricity markets have been made in recent years. These include the continuing integration of energy markets in Central Western Europe through the recent cooperation of the German EEX and the French Powernext. Figure 1 illustrates the fact that, with the inclusion of Germany, France, Austria and Switzerland, the cooperation

13 The Electric Power Industry 5 covers a geographical area that constitutes more than one third of the European electricity consumption (EEX AG & Powernext SA, 2008c, p.8). As stated by EU Energy Commissioner Andris Piebalgs, this means great progress for the European electricity market and represents an important step on the path towards a fair and uniform price for Europe (cited in EEX AG & Powernext SA (2008b, p.1,4). According to EEX AG and Powernext SA (2008c, pp.3,6,19) the benefits of this cooperation are an increased security of supply, enhanced competition on the level of supply, price convergence, centralized and increased liquidity, lower transaction costs through harmonization of trading and settlement, integrated clearing over several markets, and a facilitated governance of market splitting and coupling projects. Figure 1 Central West European electricity market as of The Central West European electricity market comprises the products traded on the German EEX AG and the French Powernext SA. In particular, it involves spot and futures markets for Germany, Austria and France and the spot market for Switzerland. Source: EEX AG & Powernext SA, 2008c, p.18. Note, however, that it is neither in the scope of this thesis to extensively describe the various regional power markets within Europe nor to analyze their integration processes. Rather, the aim of this thesis is to analyze the dependence structure between return series of the prices of spot power contracts with delivery within Germany/Austria (Phelix) and Switzerland (Swissix) and various futures contracts on the Phelix. Since all these products are traded on the EEX (respectively on various subsidiaries and joint ventures thereof), the following subsection will focus on examining the EEX and its group structure in a more elaborate way. The intention is to subsequently provide an overview of the trading possibilities in power at the EEX. This will include a detailed specification of the products that are of relevance in the empirical part of this thesis The Holding Structure of the EEX and its sub-markets The Leipzig based European Energy Exchange AG is the leading energy exchange in Continental Europe both in terms of trading participants and turnover. Originating from the 2002

14 The Electric Power Industry 6 merger of the two German power exchanges located in Frankfurt and Leipzig, the EEX has established itself as a leading operator of market platforms for trading in power, natural gas, emission rights and coal (EEX AG, 2010, p.1). The particular corporate structure together with the adoption of an open business model where the spin-offs are able to form partnerships and co-operations with other power exchanges across Europe constitutes a major contribution with regard to the integration of European energy markets. In 2006, as a first step in this process, the EEX outsourced the clearing activities into a subsidiary named European Commodity Clearing AG (ECC). In the same year, ECC started to cooperate with the Dutch European Energy Derivatives Exchange N.V. (ENDEX). Today, ECC provides clearing and settlement services for all products traded on the EEX and its partner exchanges, such as the Powernext SA or the ENDEX. Further spin-offs took place in 2007 and 2008, resulting in the transfer of the spot and derivatives trading activities into separate entities. The EEX now offers a market place on several distinct sub-markets: EPEX Spot Market, EEX Spot Market and EEX Derivatives Market. Figure 2 provides an overview of these sub-markets, the products traded on each of them and the respective subsidiaries of the EEX. Figure 2 Submarkets of the EEX. The submarkets of the EEX comprise the EPEX Spot Market, the EEX Spot Market, the EEX Power Derivatives Market and the EEX Derivatives Market. Source: The EPEX Spot SE, established in late 2008, is a joint venture of the EEX and the French Powernext SA, each of them holding a 50% stake in the Paris based company (EEX AG & Powernext SA, 2008b, p.5). Both companies transferred their entire spot power trading activities into the newly founded company that is now offering products on a day-ahead basis for France, Germany/Austria (Phelix) and Switzerland (Swissix) and intraday markets for France and Germany. In the cooperation with the Powernext SA, the EEX also agreed on the creation of the EEX Power Derivatives GmbH, which offers trading in German and French power derivatives. Powernext which contributed with the injection of its trading platform for French

15 The Electric Power Industry 7 power futures holds 20% in this joint venture, while the EEX holds the remaining 80% (EEX AG & Powernext SA, 2008b, p.5). As illustrated in Figure 2, the product range of the EEX further includes day ahead spot trading in natural gas and emission rights and derivative contracts in natural gas, emission rights and coal (EEX AG, 2010, p.3) Power Products and Trading Processes at the EEX The fact that electricity is a commodity which must be consumed immediately after being produced together with the fact that electricity is a non-storable good has a decisive influence on how power products are specified for trading purposes. In particular, an important feature of power products, compared to most other commodities, is the necessity to determine a delivery period during which electricity is delivered at a constant rate. In its most basic form, a contract may involve the constant delivery of a certain amount of power (e.g. 1 MW) over a certain period of time (e.g. a single hour on a specific day in the future), leading to the total contract volume measured in energy units (e.g. 1 MWh in our case). Apparently, it is possible to acquire a certain amount of power during a delivery period which extends one hour by purchasing a portfolio of these hourly contracts. For instance, suppose that we buy 24 hourly contracts, one for each hour of the day, to get a constant delivery of power during an entire day. In general, however, it is not necessary to purchase these hourly contracts individually, as power exchanges offer the possibility to buy electricity on a block basis. A block order thereby covers a constant delivery over several consecutive hours where the individual hours depend on each other with respect to their execution, in the sense that all or none of the hours are executed (EPEX Spot SE, 2010, p.8). A base load contract, for instance, covers a constant delivery from hour 1 to hour 24 on any day of the week. The counterpart to the base load is the peak load that covers 12 hours of constant delivery from 8 a.m. to 8 p.m. (i.e. hours 9 to 20). Furthermore, for a delivery period of more than one week, a peak load contract typically covers only the days from Monday to Friday. The price of such a contract, measured in MWh, is straightforward, as it is the arithmetic average over the prices of the underlying hourly contracts. Thus, as no separate pricing takes place, block contracts cannot, in a strict sense, be seen as distinct products (EPEX Spot SE, 2009, p.5). But clearly, any other price would result in arbitrage opportunities. According to the European Federation of Energy Traders (2008, p.38), base and peak load contracts have become prevalent as standard contracts in power trading mainly due to the fact that a low number of contracts allows for high liquidity while still providing an acceptable mapping of the typical demand load pattern Trading Spot Power Contracts at the EEX Spot power trading encompasses the physical delivery of power on a short-term basis. According to Wenzel (2007, p.14), spot trading mainly serves for balancing short-term devia-

16 The Electric Power Industry 8 tions in the purchase and selling portfolios of the market participants. Wenzel (2007, p.14) further addresses the fact that risk and price expectations have an influence on whether larger or smaller parts of one s strategy are transacted on the spot segment. The EEX offers contracts on two distinct trading platforms within the spot market segment, namely continuous trading on the intraday market and closed auction trading on a day-ahead basis 1. Both are offered via the EEX and the Powernext SA s joint venture EPEX Spot SE. An in-depth representation of the concrete trading processes at the EPEX Spot SE can be found in EEX AG (2008a), EEX AG (2010), EPEX Spot SE (2009) and EPEX Spot SE (2010), on which also the following remarks are based on. Through its continuous intraday trading, the EEX offers its market participants a platform to buy and sell power at very short notice, with delivery taking place on the same (or the next) trading day. In general, the order book is open within intraday trading, so that price and volume information is visible to the market participants. Each offer is thus immediately checked for executability with a matching offer. With regard to the place of delivery, intraday trading is divided into the market areas France and Germany. It is important to note that the trading specifications correspondingly vary up to a certain extent. German and French intraday trading comprises the individual hourly contracts of the current day, which can be traded until 75 respectively 60 minutes before the start of the corresponding delivery period. Trading in the hourly (and block) contracts of the following day is possible from 3 p.m., respectively a.m. onwards. Besides hourly contracts, intraday trading involves base and peak load contracts. French intraday trading further allows for a number of other standardized block contracts and any user-defined block contracts, while German intraday trading only provides the latter. Trading on the German intraday market takes place continuously 24/7, whereas the trading hours of intraday trading for France are confined to between 7.30 a.m. and 11 p.m. Day-ahead auction trading is offered for the market areas France, Germany/Austria (Phelix) and Switzerland (Swissix) and encompasses the trading of power contracts for individual hours of the next day as well as various blocks thereof. Phelix hereby stands for Physical Electricity Index which, according to the EEX AG (2010, p.6), provides market participants both on and off the exchange with a reference price for power traded on the wholesale spot market in Continental Europe. The Phelix is calculated daily as the arithmetic average of the auction prices of the hours 1 to 24 (Phelix Day Base), respectively of the hours 9 to 20 (Phelix Day Peak) for the market area Germany/Austria without taking into account any power transmission bottlenecks. The Swiss Electricity Index, abbreviated as Swissix, represents the 1 Note that the EPEX Spot SE actually offers a third trading platform for spot power contracts, which consists of continuous trading of electricity block contracts with delivery on the following day within the French transmission system (cf. EEX AG, 2010, p.9). As it combines day-ahead with continuous trading, it basically represents a combination of the two previously mentioned platforms.

17 The Electric Power Industry 9 corresponding index for the Swiss market area (EEX AG, 2009b, p.5). Like the Phelix, it is calculated and published daily on a base and peak load basis under the notion Swissix Day Base and Swissix Day Peak, respectively. Both indexes are also calculated on a monthly basis. As we will see later, the monthly Phelix represents the settlement price of the Phelix power futures. Similar to intraday trading, market participants can make bids for the hourly contracts, base and peak load block bids and other standardized and user-specific block contracts with delivery on the following day. However, contrary to intraday trading, the price is determined in an auction procedure. Orders for all contracts can be entered into the system starting fourteen days before their respective delivery period begins (Phelix and Swissix), respectively on the Wednesday preceding the week when delivery takes place (French day-ahead). According to the EPEX Spot SE (2010, p.6), the order book then remains open 24 hours per day until the pricing process is initiated on the exchange trading day before delivery, timed at 11 a.m., 12 p.m. (noon) and a.m., for the French, German/Austrian and Swiss market area, respectively. The bids are hereby aggregated to demand and supply curves, with the intersection providing the market clearing price and volume Trading Power Derivatives Contracts at the EEX Due to the incomparably high uncertainty related to spot power prices (cf. Section 2.2.1), market participants typically try to cover only a minor part of their portfolio with spot products (Wenzel, 2007, p.14). In particular, those parts of their portfolio which are not subject to uncertainties in demand can be purchased or sold in advance on the long-term derivatives market (Lichtblick AG, 2008, p.26). A potential use of futures contracts further lies in the support of risk management by providing hedging possibilities against future price risk (EEX AG, 2010, p.13). Clearly, a long position in a derivatives contract can serve as a hedge against increasing prices, while a short position provides an insurance against declining prices. Besides, market participants who primarily act as speculators or arbitrageurs may likewise heavily rely on power derivatives (EEX AG, 2008b, p.5). Finally, Wenzel (2007, p.14) also points out that while the spot power market provides market participants with information about the current price, forwards and futures contracts disclose the market s expectations about future spot price movements. The EEX Power Derivatives GmbH offers derivatives contracts both with an unconditional and a conditional nature. Futures (as well as forwards) are unconditional contracts, in the sense that they comprise the obligation to buy (i.e. long position) or sell (i.e. short position) a predefined underlying at a certain point of time in the future at a price specified today. Note that a futures contract usually refers to a standardized contract, usually traded on an exchange, while a forward contract represents a non-standardized contract, for instance traded over the counter. Options on the other side are conditional contracts, as they convey the right

18 The Electric Power Industry 10 but not the obligation to buy (i.e. call option) or sell (i.e. put option) a predefined underlying at maturity (i.e. European option) or until maturity (i.e. American option) of the contract at a price specified today. Regarding the subsequent remarks, we shall refer to EEX AG (2008b), EEX AG (2010) and EEX Power Derivatives GmbH (2009). The Power futures that are tradable at the EEX Power Derivatives GmbH can be subdivided into French and German futures as well as Phelix futures contracts. French and German power futures involve the delivery of electricity on a base or peak load basis into the respective market area during the entire delivery period in the future, where the delivery period can be chosen to be a month, a quarter or a year 2. Moreover, contracts are traded for the current and the next six months, the next seven quarters and the next six years. French/German power futures imply a physical fulfillment as mandatory. Year and quarter futures are traded until three days before the delivery period begins, where they cascade into the respective quarter or month futures. Thus, no direct physical delivery takes place at this stage, as the contracts are replaced by contracts of the next lower delivery period. For the month futures, however, partial physical deliveries will be taking place on each day during the entire delivery month. The corresponding final settlement price, which is determined on the last trading day, stays constant over the whole settlement period and represents the basis for the payments. The contract volume of the respective month futures contract is thereby reduced with the execution of every partial delivery. As a result, the month futures contract can still be traded during the delivery month until the final partial delivery has taken place. Besides a different underlying, Phelix futures differ from the above mentioned futures contracts mainly with respect to their fulfillment, which is of a financial nature. Furthermore, contracts are traded for the current and the next nine months (Phelix Month Futures), the next eleven quarters (Phelix Quarter Futures) and the next six years (Phelix Year Futures). Year and quarter futures cascade in the same way as the French/German power futures and the settlement price is established in a similar way too, based on the value of the corresponding monthly Phelix index. During the delivery month, however, the futures contract will be settled financially by accounting for the daily variation margin. Alternatively, the market participants can also opt for a physical delivery at the EPEX Spot SE on the day-ahead auction segment for the market area Germany/Austria (Phelix). 2 For a monthly base respectively peak load contract this corresponds to between 672 (= 28 x 24) and 744 (= 31 x 24), respectively 240 (= (30 5 x 2) x 12) to 276 (= (31 4 x 2) x 12) hours. A quarterly contract covers 2160 (= ( ) x 24) to 2208 (= ( ) x 24), respectively 768 (= ( x 2) x 12) to 792 (= ( x 2) x 12) hours. Lastly, for a yearly base respectively peak load contract, the number of hours of delivery is between 8760 (= 365 x 24) and 8784 (= 366 x 24), respectively between 3120 (= ( x 2) x 12) and 3132 (= ( x 2) x 12).

19 Table 1 Characteristics of spot and derivatives contracts traded at the EEX. The table compares the market areas Germany and France for the intraday segment and Germany/Austria, France and Switzerland for the day-ahead segment. Furthermore, French and German power futures, Phelix futures and Phelix options are included in the comparison. Based on: EEX AG (2008a), EEX AG (2008b), EEX AG (2010), EPEX Spot SE (2009), EPEX Spot SE (2010), EEX Power Derivatives GmbH (2009). Trading procedure Size (min. volume increment) Tick (min. price increment) Underlying Contracts (load profile, delivery period and time to delivery) Trading of individual contracts until Trading hours Settlement Place of delivery Clearing of OTC transactions Spot intraday continuous trading France Continuous 1 MW EUR 0.01/MWh Electricity traded for delivery on the same day and, from am, all hours of the next day Individual hours, base and peak load contracts, user-defined blocks 60 minutes before delivery 7.30 am 11 pm, year-round Physical RTE No Germany Continuous 0.1 MW EUR 0.01/MWh (negative prices authorized) Electricity traded for delivery on the same day and, from 3 pm, all hours of the next day Individual hours, base and peak load contracts, other standard blocks, userdefined blocks 75 minutes before delivery Continuous (24/7), year-round Physical RWE, EON, VE, ENBW Yes Spot day-ahead auction trading France Daily auction 1 MW EUR 0.01/MWh Electricity traded for delivery on the next day Individual hours, base and peak load, other standard blocks, userdefined blocks Auction at 11 am Order book open 24h, year-round Physical RTE No Germany/ Austria (Phelix) Daily auction 0.1 MW EUR 0.1/MWh (negative prices authorized) Electricity traded for delivery on the next day Individual hours, base and peak load, other standard blocks, userdefined blocks Auction at 12 pm Order book open 24h, year-round Physical RWE, EON, VE, ENBW, APG No Switzerland (Swissix) Daily auction 0.1 MW EUR 0.1/MWh Electricity traded for delivery on the next day Individual hours, base and peak load, other standard blocks, userdefined blocks Auction at am Order book open 24h, year-round Physical SGD No Futures French/ German Power Futures Continuous 1 MW EUR 0.01/MWh Electricity with delivery over the respective delivery period into the area of the French/ German TSOs Base and peak load contracts, current and next 6 months, 7 quarters and 6 years 3 days before delivery (quarter and year), month futures can still be traded during delivery month 8.25 am 4 pm, year-round (except week-ends & statutory holidays) Physical RTE/Amprion Yes Phelix Futures Continuous 1 MW EUR 0.01/MWh Phelix base load or Phelix peak load over the respective delivery period Base and peak load contracts, current and next 9 months, 11 quarters and 6 years 3 days before delivery (quarter and year), month futures can still be traded during delivery month 8.25 am 4 pm, year-round (except week-ends & statutory holidays) Financial (Phys.) - Yes Options Phelix Options Continuous 1 MW EUR 0.001/MWh Corresponding Phelix base load or peak load futures contract Base and peak load contracts, next 5 months, 6 quarters and 3 years Exercisement on the last trading day 8.25 am 4 pm, year-round (except week-ends & statutory holidays) Financial (Phys.) - Yes The Electric Power Industry 11

20 The Electric Power Industry 12 In addition to French/German power futures and Phelix futures, the EEX also offers trading in Phelix options. The underlying security is a Phelix futures contract, either on a base or peak load basis. With respect to the delivery period of the underlying futures contract, monthly, quarterly and yearly Phelix options exist. Moreover, Phelix options are of the European type and can thereby only be exercised by the buyer of the option on the last trading day. After being exercised, Phelix options are fulfilled by opening a position in corresponding Phelix futures at the given exercise price. The main specifications of the various spot and forward contracts described above are summarized in Table 1 to provide a direct comparison of the various product categories across contract type and market area. Finally, note that the EEX has recently introduced two product innovations, namely Phelix Week Futures and Phelix futures contracts with a delivery during the off-peak hours of the respective delivery period. Hence, these products extend the product range of the EEX with respect to the delivery period as well as with respect to the load profile of the contracts Trading Volumes and Over the Counter Trading Table 2 lists the main product categories traded at the EEX together with their respective trading volumes in 2008 and The figures indicate towards the relative importance of derivatives contracts in comparison with spot contracts, where the prior constitute more than 80% of the total trading volume. Within the spot market segment, the volume of day ahead trading clearly exceeds that of intraday trading with the latter presenting merely 5% or even less of the total spot volume traded. Table 2 Trading volume in the spot and derivatives power markets of the EEX (all numbers in TWh). Based on: EEX AG (2009a, pp.80-81), EPEX Spot SE and EEX Power Derivatives GmbH (2010) (incl. Powernext) 2008 (excl. Powernext) Spot Volume Day ahead Auction Germany/Austria France Switzerland Intraday Market Germany France Power Derivatives OTC Clearing Exchange Trading Total Trading Volume

21 The Electric Power Industry 13 Besides the categorization into spot and derivatives trading, the wholesale market for trading in power can also be classified into exchange based and over the counter (OTC) trading. Basically, trading over the counter allows for custom-tailored contracts (i.e. forwards), whereas on an exchange only certain standardized products (i.e. futures) are offered. However, Wenzel (2007, p.13) points out that the exchange based and OTC market have assimilated more and more so that comparable products are now being traded on both platforms. In order to prevent the emergence of price differences and thus of arbitrage possibilities, market participants operating in the OTC market closely follow the prices given through the trading activities on the exchange (cf. Lichtblick AG, 2008, p.24). In the following analysis, we will hence focus on the prices of futures contracts, assuming that the prices of futures and forward contracts are approximately equal. According to RWE (2009, pp.2-3) one of the greatest differences is that forward contracts traded over the counter generally allow for physical or financial fulfillment while exchange traded futures (i.e. Phelix futures) mainly involve financial fulfillment. It is important to recognize that the EEX does not only provide clearing and settlement for exchange traded products but also offers clearing services for certain OTC transactions (compare last row in Table 1). The corresponding trading volume for power derivatives is presented in Table 2, amounting to TWh in 2008 and TWh in Hence, the share of OTC trading amounts to above 70%, leaving the volume of exchange traded contracts at roughly 30%. According to RWE (2009, pp.1-2), in 2008 further 2705 TWh were traded in Germany apart from the EEX on electronic trading facilities. The importance of the EEX is nevertheless undisputable as the prices implied by the exchange act as a reference prices also off the exchange. 2.2 Characteristics of Electricity Prices Commodity markets generally allow for the build-up of an inventory so that shortages and surpluses in a certain commodity can be compensated by corresponding adjustments in the inventory. According to Borchert et al. (2006, p.51) this has the favorable effect that sudden changes in supply or demand have only a limited effect on the price of the respective commodity. Electricity, however, has physical attributes that crucially differentiate it from other commodities. Most importantly, electricity must be consumed immediately, i.e. it is a nonstorable good. The absence of efficient storing possibilities has the consequence that supply and demand imbalances directly push through onto the market prices of electricity, leading to the enormous price fluctuations generally observed in spot power markets. At this place, it is worth mentioning, though, that some restricted possibilities for storing electricity do exist. For instance, managers of hydro storage power plants are faced with the task of managing the water level of the reservoirs by deciding on the timing of pumping and turbining activities. Since the water in the reservoir represents a storable commodity that can be used to instan-

22 The Electric Power Industry 14 taneously produce electricity by activating the turbines, the storability property inherent to water is to some extent transferred to electricity (cf. He, 2007, p.38 and Weber, 2005, p.15). Indeed, managers of hydro storage power plants may decide to pump water into the reservoirs using electricity at one point in time (typically when electricity prices are low) in order to make use of it to operate the turbines and produce electricity at a later point in time (when prices have risen above a certain level). It is hence not surprising that in hydro-dominated systems, as for instance in the Nordic power market, electricity prices behave much more like other commodity prices, i.e. they are considerably less exposed to price fluctuations in the short-term (Weber, 2005, p.15). However, considering that in most countries hydro storage power plants are rather scarce and that pumping generally leads to an energy loss of approximately 30%, it is reasonable to say that electricity is not storable, at least not in an adequately efficient and conventional way and at sufficiently large volumes (Burger et al., 2003, p.2). A further implication of the non-storability feature of electricity is the requirement that power supply and demand must exactly equal each other at any location and at each point in time (Borchert et al., 2006, p.51). Any shortfalls or surpluses that occur through imbalances in supply and demand have the potential to destabilize the entire electricity grid, with the resulting frequency and voltage fluctuations being capable of inflicting serious damage onto generation and transmission equipment (Lemming, 2003, p.3). The situation is aggravated by the fact that the interregional exchange of excess units is limited due to the grid dependence of electricity transmission and various bottlenecks (Borchert et al., 2006, p.51). Additionally, electricity is an important input factor in many domestic and industrial processes so that electricity demand is highly inelastic in the short-term, preventing any balancing attempts from the demand side (He, 2007, p.56). It is hence in the duty of the grid operators to ensure that the amount of power being produced exactly equals the amount being consumed. Transmission system operators such as Swissgrid (cf. therefore have to balance out unforeseen fluctuations in production and consumption in order to ensure a secure supply of electricity at a constant frequency. For this to be achieved, power plants and other suppliers of electricity may be required to increase or decrease the volume of energy that they inject into or withdraw from the system. Briefly worded, the characteristics of electricity prices are to a large extent driven by this interrelation between non-storability of electricity on one side and the requirement of instantaneous equilibrium of power supply and demand on the other side. Various studies have examined electricity prices and their characteristics. For instance, Borchert et al. (2006) analyze EEX (i.e. Phelix) electricity spot prices on a hourly and daily base load basis for the period from 2000 to 2004 as well as the prices of various futures contracts for the period from 2003 to Blöchlinger (2008) extends this period by analyzing historical Phelix spot price

23 The Electric Power Industry 15 data from 2001 to April 2007 and Phelix futures prices between 2003 and With regard to Swissix based products, Giger (2008) analyzes spot prices for a period ranging from December 2006 to March For the Nordic power market, one of the pioneering markets with regard to the world-wide deregulation process, Weron (2005) studies the hourly spot prices of the Nordic Power exchange Nord Pool for a period from 1992 to In a crosssection analysis, Meyer-Brandis and Tankov (2007) further compare electricity spot prices of several power exchanges across Europe and the USA with historical data up to In the following, the main results of these studies shall be discussed in a general setting. The most fundamental characteristics of electricity spot prices can thereby be summarized under the notions strong seasonalities, mean reverting behaviour and jumps, spikes as well as extreme volatility Spot Price Characteristics Seasonalities Blöchlinger (2008, p.8) states that, compared to other commodities, the seasonal patterns of electricity spot prices are among the most complicated. In order to see why electricity prices exhibit seasonal fluctuations, we must first recognize that the demand for electricity itself underlies fluctuations. Electricity demand (and thus electricity prices, as we will see later) thereby typically exhibits three different types of seasonalities: within a single day, during the week, and during the year. Simonsen et al. (2004, p.6) and Weron (2005, p.4) remark that the amount of power demanded at various points in time depends to a large extent on the level of human activity as well as weather and climate conditions. During the day, we can generally observe a drastic increase in electricity demand between 5 a.m. and 8 a.m. when people get up and business activities are initiated (cf. e.g. Borchert et al., 2006, p or Blöchlinger, 2008, p.8). On the other side, around 8 p.m. electricity demand declines rapidly, as most human activities come to a halt. As shown by Blöchlinger (2008, p.8-10), the specific hourly pattern differs largely with respect to the yearly seasons. While we can generally observe two peaks on a typical winter day, a less pronounced at noon and a more pronounced around 7 p.m., we typically only see one peak at noon during a summer day. Clearly, shortterm electricity demand is strongly affected by temperature, with significant amounts of electricity being used for heating and air conditioning purposes (cf. Lemming, 2003, p.4). Furthermore, electricity consumption is lower on Saturdays and Sundays, which is due to the fact that many businesses do not operate during the weekend. Lastly, electricity demand during the year seems to depend on the respective power market under analysis. For instance, Simonsen et al. (2004, p.6) find that in some Nordic countries, especially when electricity is used country-wide for heating, electricity consumption is significantly higher during the winter compared to the summer. Simonsen (2004, p.6) further notes that the opposite

24 The Electric Power Industry 16 may be true for other countries or regions. For instance, in California, the permanent use of air conditioning systems during the hot summer months results in higher electricity consumption than during mild winter months. For the spot power market of the EEX, Blöchlinger (2008, p.8) does not find any clear evidence for either of these extreme cases. Once we have accepted that electricity demand shows seasonal behavior, we must demonstrate that the demand fluctuations translate into corresponding seasonalities in electricity prices. This relation is for instance shown in an empirical context by Borchert et al. (2006, p.52-53). Due to the highly inelastic demand, combined with the lack of sufficient and efficient storing possibilities, market prices for power are to a large extent determined by the power supply, which is based on the merit order of power generation technologies (Burger et al., 2003, p.2). The merit order is established by stacking the various power plants of a power system according to their marginal costs, creating a strictly increasing curve that returns the marginal costs for any possible accumulated power load. To the left, the curve includes all power plants with high fixed and low variable costs, such as for instance nuclear, coal and run-of-river hydro power plants (He, 2007, p.57). Since these power plants are characterized by very low marginal costs once they are running, they are usually called in first and consequently form the base load. More to the right, the merit order includes power plants that have high variable costs, such as gas or oil power plants. Hence, they are only operated in peak hours, i.e. when demand is too high to just rely on the other power plants. Interrelating this with the demand curve, we can note that a low demand leads to an intersection of supply and demand curves at a low level, so that during these hours only the power plants with the lowest marginal generation costs are in operation, resulting in a relatively low market clearing price. The higher the load the more power plants with steadily increasing marginal costs are employed in the generation process, leading to correspondingly higher electricity prices. Besides demand, also power supply may be subject to climate conditions and seasonal variations, leading to further fluctuations in power prices. For instance, hydro storage power plants highly depend on rainfall and snow melting, with precipitation following annual cycles (Weron, 2005, p.4; Lemming, 2003, p.4) Mean Reversion According to He (2007, p.59), there exists a strong consensus among researchers that electricity prices show a mean reverting behavior, as do most commodities. This means that electricity prices show significant jumps and spikes in the short-term, but in the longer-term they are always pulled back towards a long-term mean (cf. Borchert et al., 2006, p.45-55). Geman and Roncoroni (2006, p.1227) further specify that commodities typically exhibit a mean reverting behavior towards a price level that is characterized by the marginal costs of production and that may be constant, periodic or periodic with a trend. Clearly, electricity

25 The Electric Power Industry 17 prices mean revert around a periodical trend driven by the above mentioned seasonalities, outages of power plants as well as fluctuations in demand and supply due to changing weather condition. According to Pilipovic (2007, p.24), the rate of mean reversion depends on how fast the responsible events dissipate and on the ability of the supply side to react to these events by rebalancing supply and demand. Burger et al. (2003, p.3) further state that mean reversion is rather fast and usually takes place within days or weeks, at most. In the long-run, however, the mean reverting level itself can be subject to permanent shifts, for instance, initiated by changes in the availability and prices of energy sources or by sustainably changing demand patterns (Borchert et al., 2006, p.55) Jumps, Spikes and exceptionally high Volatility Besides seasonal patterns and mean reverting behavior, electricity prices typically also exhibit jumps. Note, however, that since prices do not persist on the level they initially jump up to, it makes sense to rather use the notion spike (Blöchlinger, 2008, p.10). According to Geman and Roncoroni (2006, p.1228), a spike can thereby be regarded as one or several subsequent upward jumps, which are directly followed by a significant down movement in the price. The initial price jumps can be initiated by unexpected outages of power plants (Weber, 2005, p.15) or sudden increases in demand due to extreme weather conditions (Geman and Roncoroni, 2006, p.1228). As a result, the demand curve will intersect the merit order further to the right, indicating that power plants with higher marginal costs are being operated. However, as these shocks in demand or supply are typically of a short-term nature, prices shortly return to the normal level. Again, efficient storing possibilities could mitigate this process and lead to smoother price movements. As this is not the case, the large price movements, which are responsible for a significant part of the total variation in electricity prices, explain the extremely high volatilities in spot power markets (Weron, 2005, p.3). In fact, various studies, for instance Pilipovic (1998, cited in Weber, 2005, p.15) have shown that electricity prices show the highest volatilities among all traded commodities. Corresponding to He (2007, p.58), annualized volatilities of 1000% are not uncommon to observe in hourly spot price data. He (2007, p.59) further states that volatility is heteroskedastic as it tends to be high when prices are at a high level. This corresponds to the observations of Weron (2005, p.3) and Simonsen et al. (2004, p.11) that spikes are more likely to occur during peak hours and in months of high demand for electricity Price Characteristics of Futures Contracts On most power exchanges trading platforms for futures contracts were offered shortly after trading in spot contracts was established (Weber, 2005, p.17). The intention of this was to provide market participants with the ability to cope with the uncertainties in electricity spot prices (cf. Blöchlinger, 2008). This is particularly important when considering the extreme

26 The Electric Power Industry 18 volatility of electricity spot prices, as shown in the precedent section. Weber (2005) reasons that the prices of power futures contracts for electricity exhibit characteristics that are very distinct to those of spot power contracts but similar to that of other financial contracts. Hence, futures prices do not show strong seasonal patterns, have no mean reversion in general and feature a much lower volatility than spot prices. According to Pilipovic (2007, p.26), the volatilities of futures prices decrease with increasing expiration of the respective contracts. This is a result of the fact that it is reasonable to expect that supply and demand are balanced in the long run. Consequently, futures prices reflect the corresponding equilibrium price level, which is relatively stable over time. Due to these reasons, and given the fact that they are storable, futures and forward contracts are often regarded as the basic tradable assets within power markets (Blöchlinger, 2008, p.1).

27 Dependence: Linear Correlation, Copulas and Measures of Association 19 3 DEPENDENCE: LINEAR CORRELATION, COPULAS AND MEAS- URES OF ASSOCIATION In risk management, it is of utmost importance to get an idea about how certain random variables move together. Thus, whenever we are dealing with the issue of modeling dependencies among random variables, methods such as correlation measures or copula models come into play. However, unlike the concept of Pearson s linear correlation, copulas allow for a far more complex and more flexible way to describe and model such dependence structures. This section starts with the presentation of the theoretical background behind copula models. Compared to other theoretical contributions discussing the issue of modeling dependence structures, the first subsection may follow a somewhat unconventional but yet logic structure. We commence with the definition of Pearson s linear correlation as a dependence measure and immediately proceed to discuss the limitations which are inherent to this concept. Having recognized the fact that Pearson s linear correlation can only be applied in a sensible way in certain cases, we move on to cover the more general copula models, which help us to conceive dependence at a deeper level. Finally, we will derive Spearman s rho and Kendall s tau as alternative dependence measures, i.e. measures of association. Since they are consistent with the afore mentioned concept of copulas, they represent improved and adequate alternatives to the initially discussed linear correlation. Following the theoretical considerations about dependence structures, the transition to the empirical part will be initiated by discussing two topics that are central when fitting copulas to data. Firstly, this includes the presentation of the main estimation methods and secondly, the evaluation of different copula model specifications based on a variety of goodness of fit measures. 3.1 Theoretical Background of Copulas and Dependence Pearson s Linear Correlation Measures of dependence are commonly used to represent the dependence structure of a pair of random variables by a scalar (cf. Schmidt, 2007, p.21). Out of these, Pearson s linear correlation is undeniably the most popular and most frequently applied measure in practice. Hence, we will begin our analysis of stochastic dependence herewith. Definition 3.1 (Pearson s linear correlation) Let X 1 and X 2 be two vectors of random variables with finite and nonzero variances, then Pearson s linear correlation coefficient is

28 Dependence: Linear Correlation, Copulas and Measures of Association 20 ρ X 1, X 2 = Cov X 1, X 2 Var X 1 Var X 2 (3.1) with Cov X 1, X 2 and Var X i being the covariance of X 1 and X 2, respectively the variance of X i (cf. Embrechts et al., 2001, p.9). Pearson s linear correlation is a measure of a specific kind of dependence, namely a linear one. Thus, X 1 and X 2 being perfectly linearly dependent in the sense of X 2 = α + βx 1 with α R and β R\{0} is equivalent to ρ X 1, X 2 = 1. To be more precise, β > 0 implies a perfectly positive and β < 0 implies a perfectly negative linear dependence (cf. Embrechts et al., 2001, p.10; McNeil et al., 2005, p.202). It is exactly these values that form the bounds of the possible range of values that can be taken by Pearson s linear correlation coefficient. In any other case we can hence state that 1 < ρ X 1, X 2 < 1. Particularly, in the case that β = 0, i.e. X 1 and X 2 are independent, it applies that ρ X 1, X 2 = 0. It is, however, important to recognize that the inverse of this statement does not hold in general, i.e. a correlation of zero does not per se imply independence. Furthermore, it holds for β 1, β 2 > 0 that ρ α 1 + β 1 X 1, α 2 + β 2 X 2 = ρ X 1, X 2. This directly implies that Pearson s linear correlation is invariant under strictly increasing linear transformations. On the other side, it is not invariant under nonlinear strictly increasing transformations of the form T: R R, i.e. ρ T(X 1 ), T(X 2 ) ρ X 1, X 2. As we will see later, this is an important point when distinguishing Pearson s linear correlation from other measures of dependence. According to Embrechts et al. (2001, p.10), the reasons for making Peason s linear correlation the first choice in many application lies in the fact that it is simple to calculate and that it is a natural measure of dependence when working within the family of elliptical distributions, such as for instance the multivariate normal or the multivariate t-distribution. Their name comes from the fact that elliptical distributions are distributions which have a density that is constant on ellipsoids, e.g. in the bivariate case the contour lines of the density surface form ellipses (cf. Embrechts et al., 1999, p.2). Elliptical distributions are fully characterized by a vector of means, a variance-covariance matrix and a characteristic generator function 3. Later on, it will become evident that means and variances are determined by the marginal distributions, so that in the case of elliptical distributions, copulas only depend on the correlation matrix and the characteristic generator function. In this sense, the correlation matrix has a natural parametric role in this class of distributions (McNeil et al., 2005, p.201). Thus, as long as we have a multivariate normal distribution (or any other elliptical distribution), it is completely unproblematic to use a correlation matrix based on Pearson s linear correlation to get an idea about the dependence structure of the underlying variables (Embrechts et al., 1999, 3 For a more elaborate presentation of elliptical distributions and their characteristics compare for instance Asche (2004, p.29), Lindskog (2003, p.3) and Schmidt (2007, p.21).

29 Dependence: Linear Correlation, Copulas and Measures of Association 21 p.2). However, as remarked by Lindskog (2000, p.1), empirical studies suggest that most financial data is not adequately represented by a multivariate normal distribution, mainly due to heavy tails and extreme events occurring more frequently than implied by a multivariate normal distribution. As indicated by the specific spot price characteristics presented in Section 2.2, the same may hold for electricity prices. Considering this, the elliptical world must be left behind and Pearson s linear correlation can no longer be seen as a suitable measure of dependence. Following the argumentation of Schmidt (2007, pp.21-22), Embrechts et al. (1999, pp.1-7) and McNeil et al. (2005, pp )), further limitations and pitfalls regarding the use of Pearson s linear correlation as a dependence measure can be summarized as follows. To start with, Pearson s linear correlation is only a scalar measure of dependence and as such it is unable to capture the whole dependence structure of the underlying variables. Secondly, a correlation coefficient of zero does not in general imply independence. Although there is an equivalence between zero correlation and independence for normal distributions, this does not even apply to t-distributions, although they belong to the same family of elliptical distributions. Thirdly, Pearson s linear correlation is invariant under strictly increasing linear transformations, but not under more general nonlinear strictly increasing transformations. The consequence of this is that for instance two logarithmically transformed random variables may have a different correlation than the untransformed random variables. Fourthly, Pearson s linear correlation is only defined when the variances of the underlying variables are finite. In this sense, Pearson s linear correlation is not an adequate measure of dependence for very heavy-tailed variables. Fifthly, depending on the marginal distributions of the underlying variables, it is not necessarily the case that all values in the interval [-1, 1] are attainable by the linear correlation coefficient. Although it holds true if the variables follow an elliptical distribution, we cannot claim that the same applies for other distributions. Hence we must adhere to the statement that perfectly negatively dependent variables do not per se have a correlation of minus one and perfectly positively dependent variables do not necessarily exhibit a correlation of plus one. Furthermore, the interpretation that small correlations imply weak dependence may be misleading. Lastly, it does not in general hold that a pair of random variables with some marginal distributions and a specific linear correlation uniquely determines the multivariate distribution. In other words, unless we restrict ourselves to elliptical distributions, two bivariate distributions may have a completely different dependence structure despite having identical marginal distributions and an identical linear correlation. To summarize the key result of the last two pitfalls, we may state that it can be quite dangerous to conclude on the dependence structure solely based on information about the underlying marginal distributions and linear correlations.

30 Dependence: Linear Correlation, Copulas and Measures of Association 22 The previous remarks have shown us that the imprudent use of Pearson s linear correlation can lead to severe misinterpretations, especially in the case of distributions outside of the elliptical world. Or to say it in the words of Embrechts et al. (1999, p.1) [Pearson s linear] correlation is a minefield for the unwary. This renders it necessary to introduce other techniques to model dependence structures which avoid at least some of these problems. For this reason, we will now turn the focus towards copulas and subsequently to some copula related measures of association. They seem to be promising concepts in a broader range of applications Copulas According to Durante and Sempi (2009, pp.1-2), Trivedi and Zimmer (2007, p.3) and Quesada-Molina et al. (2003, p.499), the history of copulas dates back as far as the 1940s and 1950s when, among others, Hoeffding (1940), Fréchet (1951) and Gumbel (1958) released papers on copula related subjects. Nevertheless, it was not until 1959 when Sklar (1959) first made use of the term copula. Beyond that, by proving the theorem that is now carrying his name, Sklar achieved the deepest result in the context of copulas (Durante and Sempi, 2009, pp.1-2). At that time, however, the use of copulas merely encompassed the construction of the theory of probabilistic metric spaces (Quesada-Molina, 2003, p.499). It is thus not surprising, that Nelson (2006, p.1) refers to copulas as being a rather modern phenomenon, stating that they have only recently been rediscovered. For one part, this is due to several conferences devoted to copulas that were being held in the 1990s and early 2000s which have strongly contributed to the further development of the field. The publications by Joe (1997) and Nelson (1999) emerged as standard references of copula theory, further increasing the popularity of copulas. Moreover, copula models started to attract the interest by researchers of various applied sciences, most notably finance. Charpentier et al. (2006, p.1) go as far as saying that copulas have become a standard tool in finance, with applications comprising for instance credit derivatives, option pricing and risk management (cf. also Cherubini et al., 2004). Due to these developments, the copula related body of literature has seen a tremendous increase over the past twenty years Preliminaries Before we can enter into the deeper mathematics of copulas, we need to introduce some basic notions and concepts which turn out to be central in the context of copulas. Firstly, and although it is assumed that the reader is familiar with basic statistics, we will shortly recapitulate the definition of the distribution function of a continuous random variable, with the intention to clarify the notation used in the subsequent remarks (cf. Asche, 2004, pp.6-10).

31 Dependence: Linear Correlation, Copulas and Measures of Association 23 Definition 3.2 (Cumulative distribution function) The cumulative distribution function (cdf), often just referred to as distribution function, denotes the probability that the random variable X takes at most a value x, i.e. F x = P X x. Whenever we are working with a set of random variables X = X 1,, X n, as it is inevitably the case when we intend to analyze dependence structures, we have to further distinguish between multivariate and marginal distribution functions. A multivariate or joint cdf indicates the probability that each of the random variables takes at most a certain value, i.e. F x 1,, x n = P X 1 x 1,, X n x n. By contrast, the i-th marginal cdf defines the distribution of a single component X i of X independent of the distribution of the other components, i.e. F i x i = P X i x i. The relation between joint and marginal cdf (in the bivariate case) is given by the following limit (for the case of the first component): F 1 x 1 = lim x 2 F x 1, x 2 (3.2) Basically, probabilities are defined as an integral of the probability density function (pdf), i.e. b P a X b = f x a dx. As for the cdfs, it is possible to express multivariate and marginal densities. The relation between the cdf and the pdf is given by x 1 x n F x 1,, x n = P X 1 x 1,, X n x n = f x 1,, x n dx 1 dx n (3.3) and, in the case of continuously differentiable cdfs, by df(x) dx = f(x) (3.4) Next, let us have a look at the notion of the so called generalized inverse, as it is stated by McNeil et al. (2005, pp ). Definition 3.3 (Generalized inverse) Let T be an increasing function such that y > x T(y) T x for all pairs and T y > T x for some pair, then the (left-continuous) generalized inverse of T is defined as T y = inf {x T(x y)} Applying the idea of the generalized inverse to distribution functions leads to the notion of the quantile function and the following proposition (cf. McNeil et al., 2005, pp ): Proposition 3.4 (Quantile and probability transformation) If G denotes a distribution function and G is its generalized inverse, then the quantile transformation states that P G (U) x = G(x) given that U ~ U(0, 1). Conversely, the probability transformation states that if Y has a distribution function G, then G Y ~ U(0, 1). In fact, the quantile and the probability transformation represent two sides of the same coin. The probability transformation represents the usual way of reading a cdf, i.e. applying a

32 Dependence: Linear Correlation, Copulas and Measures of Association 24 probability transformation on a random variable that follows a certain distribution provides us with the corresponding probability information. By contrast, the quantile transformation is of central importance in stochastic simulation. It basically states that by generating uniformly distributed random variables U and applying the inverse of a distribution function G on them, we can generate random variables of the desired cdf G Definition of Copula, Sklar s Theorem and Basic Properties The following remark will provide us with the standard operational definition of a copula based on McNeil et al. (2005, p.185) and Trivedi & Zimmer (2007, pp.9-10) 4. Definition 3.5 (Copula) A copula C u = C(u 1,, u d ) in d dimensions is a d-dimensional distribution function on [0,1] d whose marginals are uniformly distributed. Thus, a copula is a mapping of the unit hypercube into the unit interval, i.e. C: [0,1] d [0,1]. Since copulas represent multivariate distributions, copulas have properties analogous to those of any other joint cdf. Following McNeil (2005, p.185) three properties must hold for a function in order to be qualified as a copula: Firstly, C(u 1,, u d ) is an increasing function in each component u i. Secondly, by setting all the components u j = 1 with j i we obtain the marginal component u i, i.e. C 1,,1, u i, 1,,1 = u i. Or, to put it in the words of Trivedi and Zimmer (2007, p.10), given that we know d 1 of the random variables with marginal probability one, then the joint probability of the d outcomes corresponds to the probability of the remaining outcome. Clearly, this property can be seen as the requirement of marginal distributions that are uniform. Thirdly, for a random vector (U 1,, U d ) having a distribution function C and for values a 1,, a d and b 1,, b d [0,1] d with a i b i the probability P a 1 U 1 b 1,, a d U d b d has to be non-negative. This last property is often referred to as rectangle inequality. Together, the three properties characterize a copula and hence provide an alternative way to define a copula. As a result, we can state that any function which fulfills these properties represents a copula. As a concluding remark, it is important to note that other papers use slightly distinct expressions for these properties. For instance, according to Asche (2004, p.12) and Embrechts (2001, p.3), C must be grounded and d-increasing. Trivedi and Zimmer (2007, p.10) further elaborate the term grounded as C u 1,, u d = 0 if u i = 0 for any i d. Hence, if the marginal probability of one outcome is zero, the joint probability of all outcomes is zero, too. Moreover, the above mentioned rectangle inequality is equivalent to the expression that C is d -increasing. In the bivariate case, this property is often represented by C u 1,2, u 2,2 C u 1,2, u 2,1 C u 1,1, u 2,2 + C(u 1,1, u 2,1 ) 0 for two marginals u 1 and u 2 with two values each and u 1,1 u 1,2 and u 2,1 u 2,2. 4 For a more abstract definition that interprets copulas as a subset of general multivariate distributions see for instance Embrechts et al. (2001), Asche (2004) or Nelson (2006).

33 Dependence: Linear Correlation, Copulas and Measures of Association 25 But let us now have a closer look at what copulas actually are. According to Nelson (2006, p.2), the latin word copula can be translated as a link, tie or bond. And this is exactly what a copula is. A copula couples a multivariate distribution to its univariate marginal distributions. This relation between marginal and multivariate cdfs is covered in Sklar s theorem. It is a key result in the context of the application of copulas and shall now be discussed. Theorem 3.6 (Sklar, 1959) Given a multivariate distribution function F with marginal distributions F 1,, F d, then there exists a copula C such that for all x 1,, x d [, ] F x 1,, x d = C F 1 x 1,, F d x d (3.5) As stated by Embrechts et al. (2001, p.4), the elegance of Sklar s theorem lies in the way it shows us how the univariate marginal distributions and the multivariate dependence structure, represented by the copula function, can be separated. Basically, Sklar s theorem can be interpreted in two ways. On one side, if C is a copula and F 1,, F d represent univariate distribution functions, then the multivariate distribution function F is defined as in the formula above. A joint distribution function can thus be generated by coupling the marginals with a copula. On the other side, a copula can be extracted from a joint distribution function and the corresponding marginal distributions. Sklar s theorem hence also shows that all multivariate distribution functions contain a copula. In particular, the copula function C is unique if the marginals are continuous. To see this second interpretation more clearly, we can rewrite Sklar s theorem by applying the concept of the generalized inverse x i = F i u i on the lefthand side and F x i = u i on the right-hand side (cf. McNeil et al., 2005, p.187): C u 1,, u d = F F 1 u 1,, F d u d (3.6) The expressions (3.5) and (3.6) are essential in the application of copulas (cf. Schmidt, 2007, p.7 and McNeil et al., 2005, p.187). The importance of the second approach is rather theoretical and lies in the extraction of a copula from a multivariate distribution function. Later on, we will see that the derivation of the Gaussian and the t-copula are examples of this approach. By contrast, the first way of interpreting Sklar s theorem is the starting point of many empirical applications (cf. Trivedi and Zimmer, 2007, pp.10-12). This approach allows us to conclude on the joint distribution function by separately specifying the marginal distribution for each random variable and the copula function. This makes the estimation of the joint cdf very flexible, for instance letting the marginal distributions stem from different families. In particular, we can express (3.6) as F x 1,, x d ; θ = C F 1 x 1,, F d x d ; θ (3.7) where θ represents the parameter vector of the copula, characterizing the dependence between the marginal distributions. Note, in the case of independence the copula is simply the

34 Dependence: Linear Correlation, Copulas and Measures of Association 26 product of the marginals, reducing the problem to the rather trivial task of estimating the individual marginal distributions. The definition of copulas implies that they are cumulative distribution functions. For certain applications, however, it may be more convenient to dispose of copula densities. On one side, this may involve the illustration of the dependence structure through plotting the pdf, which is in many cases more intuitive than plotting the cdf (Schmidt, 2007, p.8). More importantly, copula densities are required whenever we intend to fit copulas to a data set using a maximum likelihood approach. Although not all copula functions have densities, all the parametric copulas discussed throughout this section do so. Following McNeil et al. (2005, p.197), we may characterize the copula density c of a copula C (given differentiability) as c u 1,, u d = d C u 1,, u d u 1 u d (3.8) When discussing Pearson s linear correlation in the precedent subsection, we realized that it is only invariant under strictly increasing linear transformations. Copulas, on the contrary, possess the superior property that they are invariant under any strictly increasing, i.e. monotonic transformation of the marginal distributions. As a consequence, the dependence structure of the respective random variables will remain unchanged after the transformation. For instance, logarithmically transformed random variables will still have the same dependence structure, expressed by the copula, as the untransformed variables. The following proposition, whose proof can be found in McNeil et al. (2005, p.188), formalizes this property. Proposition 3.7 (Invariance of copulas) Let X 1,, X d be random variables with continuous marginal distributions and copula C. By referring to T 1,, T d as strictly increasing functions, the transformed random variables T 1 X 1,, T d X d will have the same copula C. The Fréchet-Hoeffding bounds, which will be discussed next based on McNeil et al. (2005, pp ), Schmidt (2007, pp.10-12) and Trivedi and Zimmer (2007, pp.9-14), constitute another important result in the context of copulas. Theorem 3.8 (Fréchet-Hoeffding bounds) The Fréchet-Hoeffding bounds represent universal bounds in that sense that any cumulative distribution function, and hence every copula, is bounded by the lower and upper bounds max d i=1 u i + 1 d, 0 C(u) min u 1,, u d (3.9) According to Trivedi and Zimmer (2007, pp.12-13), the practical relevance of the Fréchet- Hoeffding bounds becomes evident when we intend to select a reasonable copula. It may thereby be sensible to choose a copula that covers the whole space between the lower and

35 Dependence: Linear Correlation, Copulas and Measures of Association 27 the upper bound. Furthermore, if the copula parameter θ reaches its upper (lower) limit within the permissible range, the copula should converge to the upper (lower) Fréchet-Hoeffding bound. However, depending on the parametric form of a certain copula, not the full range of dependence structures is attainable. This makes the application of certain copulas more or less reasonable depending on what data set is analyzed. Figure 3 Fréchet-Hoeffding bounds for C(u,v). The upper Fréchet-Hoeffding bound is represented by the front surface of the pyramid-shaped body, while the surface spanned by the bottom and rear side corresponds to the lower bound. Source: Schmidt, 2007, p.11. It is important to note that the Fréchet-Hoeffding bounds allow for specific interpretations with regard to dependence. Particularly, in the bivariate case, the Fréchet-Hoeffding bounds are copula functions themselves 5. These copulas, known as comonotonicity and countermonotonicity copula, together with the independence copula constitute the class of the fundamental copulas. As their name suggests, these copulas represent some fundamental dependence structures. Before elaborating on these copulas, the illustration of the Fréchet-Hoeffding bounds for the two dimensional case (cf. Figure 3) shall be commented. The surface spanned by the bottom and rear side of the pyramid represents the lower bound, while the surface given by the front side equals the upper bound. In accordance with Theorem 3.8, every copula must lie within the interior of this pyramid. 5 It should be particularized that the lower Fréchet-Hoeffding bound is not a copula for d 3, while the upper bound actually satisfies the definition of a d-dimensional copula function for all d 2.

36 Dependence: Linear Correlation, Copulas and Measures of Association Fundamental Copulas The independence copula C Π u 1,, u d = d i=1 u i (3.10) refers to a dependence structure where there is no dependence between the random variables. Sklar s theorem directly implies that random variables are independent if and only if the independence copula describes their dependence structure (McNeil et al., 2005, p.189). According to Trivedi and Zimmer (2007, p.15), the importance of the independence copula lies in its function as a benchmark for independence. The comonotonicity copula C M u 1,, u d = min u 1,, u d (3.11) is the upper Fréchet-Hoeffding bound and relates to the case of perfect positive dependence. Following McNeil et al. (2005, p.190), a number of random variables is referred to as perfectly positively dependent if they are almost surely strictly increasing functions of each other, i.e. X i = T i X 1 for i = 2,, d. The countermonotonicity copula C W (u 1, u 2 ) = max u 1 + u 2 1, 0 (3.12) is the two-dimensional lower Fréchet-Hoeffding bound and describes the other extreme, namely perfect negative dependence. Two random variables are perfectly negatively dependent if one random variable is almost surely a strictly decreasing function of the other. Formally, it holds X 2 = T X 1 with T being a strictly decreasing function Elliptical and Archimedean Copulas Obviously, there exist many functions that fulfill the definition of a copula and the body of copula literature is characterized by a correspondingly vast number of different copulas. In this subsection, some important parametric copula families will be presented in more detail. Together, they represent a broad spectrum of dependence structures, allowing for the reconstruction of many characteristics of empirical data. Consequently, these copulas are not only popular in the literature but also frequently applied in empirical studies (cf. Trivedi and Zimmer, 2007, p.15). Moreover, as expressed in equation (3.7), each copula family is determined by a single parameter or a vector thereof. More precisely, all copulas taken into consideration are, in the bivariate case, characterized by a single parameter, except for the t-

37 Dependence: Linear Correlation, Copulas and Measures of Association 29 copula, which is subject to an additional second parameter 6. Durante and Sempi (2009, p.14) list a number of requirements that should preferably be fulfilled by a copula family. In particular, they should allow for a probabilistic interpretation, represent a flexible and wide range of dependence and be easy to handle. With respect to the range of dependence Trivedi and Zimmer (2007, p.13) further state that a copula family should comprise the independence, comonotonicity and countermonotonicity copula in order to be qualified as comprehensive. Table 3 thereby reveals for each copula family under consideration firstly, what permissible parameter ranges are and secondly, for what parameter values the three fundamental copulas are obtained, provided that they are attainable. Since we restrict ourselves to some basic copula families (i.e. Gaussian, t-, Gumbel, Clayton and Frank copula), the third of the above mentioned requirements holds as well. Table 3 Copula families, generator functions and permissible parameter ranges. The first column provides the generator functions for the Archimedean copula families The second column exhibits the permissible parameter range for the Gaussian (C Ga t ρ ), t- (C ρ,ν ), Gumbel (C Gu θ ), Clayton (C Cl θ ) and Frank (C Fr θ ) copula families. Furthermore, the table indicates for which parameter values the countermonotonicity (C W ), the independence (C Π ) and the comonotonicity (C M ) copulas are obtained. Note that the entries for the Gaussian and t-copula as well as for the countermonotonicity copula uniquely refer to the bivariate case. Based on: McNeil et al., 2005, p.220; Cherubini et al., 2004, pp Copula family Generator function ψ t Permissible parameter range C W C Π C M C ρ Ga N/A ρ [ 1,1] ρ = 1 ρ = 0 ρ = 1 Ct ρ,ν N/A ρ [ 1,1] ρ = 1 ρ 0 ρ = 1 Gu ln t θ θ [1, ) Not attainable θ = 1 θ Cl 1 θ t θ 1 θ [ 1, ) \ {0} θ = 1 θ 0 θ Fr ln e θt 1 e θ 1 θ R \ {0} θ θ 0 θ In the section about Sklar s theorem, it was mentioned that it is possible to extract copulas from multivariate distributions. The copulas within the elliptical class originate exactly from that approach, representing copulas inherent to multivariate elliptical distributions. In particular, the Gaussian and the t-copula are derived from the multivariate normal respectively t- distribution by applying expression (3.6). 6 In the following, the copula functions are represented in their d-variate form. In the subsequent analysis, however, we will investigate the dependence structure of return series of electricity prices in a pair wise manner. Consequently, we will then restrict ourselves to the bivariate form of the individual copula families.

38 Dependence: Linear Correlation, Copulas and Measures of Association 30 The Gaussian copula is given by Ga C Σ u 1,, u d = Φ Σ Φ 1 u 1,, Φ 1 u d (3.13) where Φ Σ denotes the cdf of a d-variate standard normal distribution with correlation matrix Σ and Φ is the univariate standard normal distribution (cf. McNeil et al., 2005, p.191). The Gaussian copula covers dependence structures between perfect positive dependence, independence and perfect negative dependence, with the strength of dependence being determined by Σ. In the bivariate case, it is basically sufficient to know the linear correlation coefficient ρ between the two random variables, which constitutes the only parameter, as Σ = 1 ρ ρ 1 (3.14) This reinforces the previous statement that Pearson s linear correlation fully describes the dependence structure of normal distributions respectively elliptical distributions in general (cf. Schmidt, 2007, p.14). The parametric role of Σ can be ascribed to the fact that through the standardization procedure of the marginal distributions the random variables Y~N μ, Ω are transformed to X~N 0, Σ by strictly increasing transformations. As we know from Proposition 3.7, such transformations leave copulas unaffected. Cherubini et al. (2004, p.114) point out that the Gaussian copula results in a multivariate normal distribution only if it is combined with normal marginal distributions. This does not imply, however, that, once it has been extracted, the Gaussian copula cannot be applied to some arbitrary marginal distributions. Rather, the resulting joint distribution would be non-normal, further enlarging the range of multivariate distributions that can be modeled. Following McNeil et al. (2005, p.193), these multivariate (non-normal) distributions are referred to as meta-gaussian. For instance, prominent credit risk models use exponential marginal cdfs together with the Gaussian copula to model companies default times (McNeil et al., 2005, p.193). In the same way as the meta-gaussian distribution refers to the Gaussian copula, the notation of a meta-distribution can be extended to other copulas. The fact, that we can construct a meta-gaussian, meta-t-, meta- Gumbel etc. distribution with the same marginals and the same correlation again clarifies the limitations of Pearson s linear correlation as a single measure of dependence whenever we do not combine the marginals of an elliptical distribution with the corresponding copula. The t-copula is represented as t 1 1 C Σ,ν u 1,, u d = t Σ,ν t ν u 1,, t ν u d (3.15) with t Σ,ν and t ν describing the cdf of the d-variate respectively univariate t-distribution with ν degrees of freedom (cf. McNeil et al., 2005, p.191). Analogous to the Gaussian copula, the t- copula is extracted from a multivariate t-distribution, provided that the marginals are also t- distributed. According to Trivedi and Zimmer (2007, p.17), the t-copula converges to the

39 Dependence: Linear Correlation, Copulas and Measures of Association 31 Gaussian copula as ν approaches infinity. Σ again corresponds to the correlation matrix and, analogously to the Gaussian copula, the t-copula is parameterized by the linear correlation coefficient in the bivariate case. As for all elliptical distributions, with the exception of the normal distribution, zero correlation in the components does not imply independence. Hence, while comonotonicity and countermonotonicity can be achieved in the same way as in the case of the Gaussian copula, it is not possible to obtain the independence copula with ρ = 0 as long as ν <. Figure 4 compares the copula densities of a Gaussian copula and a t-copula for ρ = 0.3 and ν = 2. Firstly, we can observe that both copulas are symmetric, with the lower left quadrant being equally pronounced as the upper right quadrant. Secondly, despite being quite similar in the center, the behavior at the four corner points differs substantially. In particular, the t- copula features lower and upper tail dependence, while the Gaussian copula does not show any tail dependence for ρ ± 1 (cf. McNeil et al., 2005, pp ). Tail dependence hereby refers to the occurrence of joint extremal events in the sense that there is a tendency for X 2 to take extreme values when X 1 takes extreme values and vice versa. Data exhibiting joint extremal events can hence be more accurately modeled via a t-copula than via a Gaussian copula. Subsection will further elaborate on the topic of tail dependence. Figure 4 Gaussian and t-copula densities. (a) illustrates the Gaussian copula density, which is characterized by its symmetry and the absence of tail dependence. (b) shows the t-copula density, which is also symmetric but shows both upper and lower tail dependence. Source: Schmidt, 2007, p.14. Elliptical copulas are easily parameterized by the linear correlation matrix, but they are not without drawbacks. To begin with, empirical data often does not follow a joint elliptical distribution. This can be partially compensated by combining arbitrary marginals with e.g. a Gaussian copula, creating a meta-gaussian distribution. Embrechts et al. (2001, p.24) further state that we are in no way restricted to stay within a single distributional family with regard to the marginals. By using different marginal distributions for the individual components further flexibility in the modelling of multivariate distributions is gained. It is hence not surprising, that, as pointed out by Asche (2004, p.20), many applications in finance achieve a good representation of the empirical dependence structure by simply using a Gaussian or t-copula.

40 Dependence: Linear Correlation, Copulas and Measures of Association 32 What remains unsolved, however, is that the elliptical copulas have radial symmetry, so that asymmetric dependence structures cannot be modeled with these copulas (cf. Embrechts et al., 2001, p.15). Yet, we may have empirical data which exhibits stronger tail dependence either to the up- or downside. The Archimedean copulas, a second class of copulas beside the elliptical ones, thereby have fundamentally different features. They describe specific dependence structures often found in empirical data, such as the cases of upper or lower tail dependence. Furthermore, Archimedean copulas have simple closed forms (cf. Gartner, 2007, p.39). Contrary to the elliptical copulas, Archimedean copulas are not extracted from multivariate distributions but rather originate from mathematical construction. This is also why McNeil et al. (2005, p.190) refer to them as explicit copulas in contrast to the elliptical, implicit copulas. Before having a closer look at some popular Archimedean copula families, we will first examine the general definition of an Archimedean copula. Definition 3.9 (Archimedean copulas) Given a continuous, strictly decreasing, convex function ψ from [0,1] to [0, ] with ψ 1 = 0 and its pseudo-inverse ψ 1 : 0, [0,1], a copula which fulfils C u 1,, u d = ψ ψ 1 u 1 + ψ 1 u ψ 1 u d (3.16) is called an Archimedean copula (cf. Embrechts et al., 2001, p.31; Durante and Sempi, 2009, p.15 and Cherubini et al., 2004, p.121). The function ψ is denoted as the generator function. Note that the respective Archimedean copula is only strict if ψ 0 =, in which case ψ 1 corresponds to the usual inverse ψ 1. The generator functions that lead to the following copula families are provided in Table 3. The Gumbel copula Gu u 1,, u d = exp lnu 1 θ + + lnu d θ 1 θ (3.17) covers dependence structures between independence and perfect positive dependence, hence it is not comprehensive. As it becomes evident from Figure 5 (a), the Gumbel copula exhibits strong upper but only weak lower tail dependence. As such, it may propose an appropriate model for the joint distribution of random variables where the outcomes are strongly correlated at high values and to a less extent at low values (Trivedi and Zimmer, 2007, p.19).

41 Dependence: Linear Correlation, Copulas and Measures of Association 33 The Clayton copula Cl u 1,, u d = u 1 θ + + u d θ d θ (3.18) is only strict if θ (0, ). In this case, the Clayton copula is not comprehensive, as it does not cover the case of countermonotonicity (in the bivariate case). By letting θ 1, \ {0} this can be solved, however, at the cost of the Clayton copula losing its property of being strict. Furthermore, this implies that the Clayton copula is not given by the expression shown above, but by the maximum of this term with zero (cf. McNeil et al., 2005, p.220). The Clayton copula can be used to model strong lower tail dependence while holding upper tail dependence relatively low (cf. Figure 5 (b)). This may represent an adequate model for applications in finance, where we can observe a strong correlation across the components in down markets. The Frank copula Fr u 1,, u d = 1 θ ln 1 + exp θu 1 1 exp θu d 1 exp θ 1 d 1 (3.19) is comprehensive as it interpolates between perfect negative and perfect positive dependence, at least in the bivariate case where the countermonotonicity copula is attainable. According to Embrechts et al. (2001, p.32), the Frank copula is the only Archimedean family showing radial symmetry, comparable to the Gaussian or t-copula. As visible from Figure 5 (c), the Frank copula exhibits strongest dependence in the center, while having no tail dependence. According to Trivedi and Zimmer (2007, p.19), the Frank copula is often used in empirical studies. Figure 5 Gumbel, Clayton and Frank copula densities. (a) illustrates the density of the Gumbel copula with its upper tail dependence. (b) plots the density of the counterpart to the Gumbel copula, namely the Clayton copula. It is characterized by lower tail dependence. (c) visualizes the Frank copula density which, similar to the Gaussian copula density, is symmetric and does not exhibit any tail dependence. Source: Schmidt, 2007, p.18.