The Information Premium on Electricity Markets A New Spot-Forward Relationship for non-storable Underlyings

Transcription

1 Universiä Duisburg-Essen Fakulä für Mahemaik The Informaion Premium on Elecriciy Markes A New Spo-Forward Relaionship for non-sorable Underlyings Disseraion zur Erlangung des Dokorgrades Dr. rer. na. der Fakulä für Mahemaik der Universiä Duisburg-Essen vorgeleg von Richard Biegler-König, M.Sc. Essen, im Februar 23. Guacher: Prof. Dr. Rüdiger Kiesel 2. Guacher: Prof. Dr. Fred Espen Benh Tag der Dispuaion:

2

3 Universiä Duisburg-Essen Fakulä für Mahemaik The Informaion Premium on Elecriciy Markes A New Spo-Forward Relaionship for non-sorable Underlyings Disseraion zur Erlangung des Dokorgrades Dr. rer. na. der Fakulä für Mahemaik der Universiä Duisburg-Essen vorgeleg von Richard Biegler-König, M.Sc. Essen, im Februar 23

4

5 v Amierender Dekan: Prof. Dr. Ulrich Görz. Guacher: Prof. Dr. Rüdiger Kiesel 2. Guacher: Prof. Dr. Fred Espen Benh

6

7 Acknowledgmens I wish o express my greaes graiude o my firs supervisor Prof. Rüdiger Kiesel. He has been an inspiring scienific menor and always assised wih good advice. His chair has been an enormously friendly and producive working environmen. I was a pleasure geing o know he academic world under his guidance, in paricular when joining him abroad for conferences. My second supervisor, Prof. Fred Espen Benh, no only iniiaed he subjec of my hesis bu also invariably provided valuable assisance and new ideas. I was always a grea pleasure being his gues in Oslo. I am very graeful for his suppor. Las bu no leas, I wish o hank my parens, Cornelia and Friedrich, no only for proofreading his manuscrip bu also, apar from everyhing else, for enabling a myriad of possibiliies. Richard Biegler-König Essen, April 23

8

9 Absrac Wih he deregulaion of elecriciy markes and he esablishmen of elecriciy exchanges Quaniaive Financial Modelling in his area has become increasingly imporan. There are wo main objecs raded on such markes: he spo (usually a day-ahead price) and fuures and forwards wih various delivery periods. These are seled agains he spo. I is hus essenial o relae spo and forward prices. In he lieraure his is usually done by making use of he classical spo-forward relaionship which gives he forward price as he risk-neural expecaion of he spo price under he hisorical filraion. Sill, elecriciy is special in he way ha i is non-sorable and hus addiional informaion influences fuure prices while leaving spo prices unaffeced. Consequenly, he radiional relaionship fails heoreically as well as pracically. Therefore, his hesis proposes a new exended spo-forward relaionship for elecriciy ha does no only ake risk-adjusmen ino consideraion bu also he so-called marke filraion. This is consruced by enlarging he hisorical filraion wih relevan fuure informaion. Furhermore, building on he ideas of he reference paper Benh and Meyer-Brandis (29) we quanify he impac of his approach by means of he informaion premium, i.e. he difference beween he forward price under he marke filraion and he radiional forward price. In he firs chapers of his hesis we lay he foundaions for he consequen analysis. In he firs chaper we moivae he new relaionship discussing wo EEX marke scenarios: he inroducion of he second rading phase of he European emission cerificaes in 28 and he German Aom Moraorium of 2. Boh exhibi large informaion premia. The second chaper hen discusses he (laer o be used) mahemaical heory of he enlargemen of filraion while he hird chaper inroduces a popular sochasic spo model. This model will be our work horse for he analyical calculaions of his hesis. The main par of he hesis conribues o he academic lieraure by analysing various aspecs of he impac he new spo-forward relaionship has on Financial Modelling. Analyically, we show how o use he mahemaical heory o calculae closed-form expressions for he informaion premium for differen ypes of fuure informaion abou boh he spo and correlaed processes. Furhermore, we explore pricing of derivaives on forward conracs in he presence of fuure informaion. We also calculae indifference prices and marke power of differen marke raders. Empirically, we provide he firs horough invesigaion of he inerplay of a nonsorable commodiy and addiional fuure informaion. We propose a saisical es for he exisence of he informaion premium. We exemplify by analysing he wo marke siuaions menioned above. Our es is based on Hilber-basis represenaion and can be more generally applied o assess measurabiliy of wo ime series. Summarising, we advocae he necessiy and show he feasibiliy of using a new spo-forward relaionship in he conex of Quaniaive Financial Modelling of elecriciy markes.

10

11 Zusammenfassung Durch die in den lezen Jahren sagefundene Deregulierung des Elekriziäsmarkes und die Gründung von Srombörsen seig die Relevanz des finanzmahemaischen Modellierens in diesem Bereich. Der Handel an solchen Börsen finde zumeis am Spo- und Terminmark sa. Während der Spo ein Day-ahead Preis is, sind die wichigsen Produke am Terminmark Forwards und Fuures mi verschiedenen Lieferperioden. Diese werden gegen den Spopreis abgerechne, so dass das Aufsellen eines möglichs präzisen Verhälnisses zwischen beiden Produken von großer Bedeuung is. In der akademischen Fachlieraur wurde bislang allerdings ausschliesslich das klassische Spo-Forward Verhälnis übernommen. Dieses gib den Preis des Forwards als risikoneuralen Erwarungswer bezüglich der naürlichen (hisorischen) Filraion an. Nun is jedoch elekrischer Srom insofern ein besonderes Underlying, als dass er nich speicherbar is. Zur Verfügung sehende Informaionen über die zukünfige Enwicklung werden selbsversändlich zukünfige Preise beeinflussen, nich aber momenane Preise. Deshalb is das klassische Spo- Forward Verhälnis weder heoreisch noch prakisch zur Modellierung geeigne. In dieser Arbei wird deshalb ein neues, erweieres Spo-Forward Verhälnis präsenier, das nich nur die Risikoanpassung der Markeilnehmer berücksichig, sondern auch die sogenanne Markfilraion. Diese enseh durch Vergrößerung der naürlichen Filraion um relevane zusäzliche Zukunfsinformaionen. Bezugnehmend auf das Referenzpaper Benh and Meyer-Brandis (29) werden hier die Auswirkungen dieses neuen Verhälnisses mi Hilfe der Informaionsprämie quanifizier. Diese is als Differenz zwischen dem Forwardpreis bezüglich der vergrößeren Filraion und demjenigen bezüglich der naürlichen Filraion definier. Im ersen Kapiel dieser Arbei werden zwei EEX Markszenarien diskuier, die zur empirischen Moivaion für das erweiere Spo-Forward Verhälnis dienen. Auf der einen Seie is dies die Einführung der zweien Phase der europäischen Emissionszerifikae Anfang 28, auf der anderen Seie das deusche Aom Moraorium des Jahres 2. Für beide Szenarien finden wir signifikane Informaionsprämien. Das zweie Kapiel beschreib die späer verwendeen mahemaischen Grundlagen, insbesondere die Theorie der Filraionsvergrößerung. Im drien Kapiel sellen wir ein populäres Spomodell vor, das im weieren Verlauf dieser Arbei Verwendung finden wird. Im Haupeil der vorliegenden Arbei werden die zenralen Resulae vorgesell. Die Arbei räg zur akademischen Lieraur durch die Diskussion verschiedener finanzmahemaischer Aspeke des erweieren Spo-Forward Verhälnisses bei. Wir zeigen analyisch, wie die mahemaische Theorie angewende werden kann, um geschlossene Ausdrücke für die Informaionsprämie zu finden und zwar sowohl mi Informaionen über den zukünfigen Spopreis, als auch mi Informaionen, die korreliere Prozesse bereffen. Des Weieren widmen wir uns der Frage, wie zusäzliche Informaionen über die zukünfige Enwicklung die finanzmahemaische Preisfin-

12 xii dung von Derivaen auf Forwards beeinflussen. Außerdem berechnen wir Indifferenzpreise von Forwards für verschiedene Typen von Händlern und ziehen Rückschlüsse auf ihre jeweilige Markmach. Empirisch wird in dieser Arbei die erse grundlegende Unersuchung zum Zusammenspiel eines nich speicherbaren Underlyings und dem Mark zur Verfügung sehender Informaionen über die zukünfige Enwicklung durchgeführ. Um die Exisenz der Informaionsprämie zu zeigen, sellen wir einen empirischen Tes vor, den wir auf die beiden obigen Markszenarien anwenden. Dieser Tes basier auf Hilberraumrepräsenaionen und is generell auch zur Fessellung der Messbarkei zweier Zeireihen anwendbar. Zusammenfassend propagieren wir in dieser Arbei die Nowendigkei, im Konex der finanzmahemaischen Modellierung von Elekriziäsmärken ein neues, erweieres Spo-Forward Verhälnis zu verwenden und verdeulichen die analyischen und empirischen Möglichkeien, die sich hieraus ergeben.

13 Conens. Inroducion and Summary of he Thesis.. Elecriciy Markes Hisory of Elecriciy Markes Marke Design and Producs Modelling Elecriciy Markes - Lieraure Classificaion Spo-Forward Relaionships and he Informaion Approach The Beginning of he Second Phase of he EUETS The German Aom Moraorium Objecives and Conribuion of he Thesis Srucure of he Thesis Daa, Programming and Compuer Algebra Publicaions Mahemaical Background Lieraure Overview and Summary Enlargemen of Filraion Enlargemen of Filraion and Maringale Decomposiion Enlargemen of Filraion and Change of Measure Malliavin Calculus and Imkeller s Mehod Enlargemen of Filraion and Linear Differenial Equaions Condiional Expecaion and Hilber Space Represenaions The Arihmeic Spo Price Model Lieraure Overview and Summary Descripion of he Model Two-facor Model wih Double-exponenially Disribued Jumps Forward Pricing wih Delivery Period Forward Pricing under he Real-world Measure Forward Pricing under a Risk-neural Measure Esimaing he Model Parameers Filering Spikes The Seasonal Componen The Gaussian Ornsein-Uhlenbeck Process The Lévy Ornsein-Uhlenbeck Process Simulaing he Model Empirical Sudy EEX Spo from 22 unil EEX Spo from /22 unil 5/

14 xiv Conens 4. Risk Premium and Informaion Approach Lieraure Overview and Summary Spo-Forward Relaionships and Premia The Model of Benh/Meyer-Brandis Fuure Lévy Informaion Fuure Brownian Informaion Furher Calculaions The Informaion Premium wih Delivery Period Lieraure Overview and Summary Calculaing he Informaion Yield Addiional Informaion abou he Lévy Componen Addiional Informaion abou he Brownian Moion Addiional Informaion abou he Base Componen Base Componen Threshold Informaion The Informaion Premium for Differen Seups of he Time Axis Informaion Premium wih Spike Informaion Informaion Premium wih Base Componen Informaion Sylised Examples and Discussion Muliple Pieces of Fuure Informaion Two Pieces of Fuure Informaion The General Resul Sylised Examples and Discussion The Informaion Premium for Correlaed Processes The Temperaure Model Calculaing he Informaion Premium Sylised Examples and Discussion Conribuion and Discussion Opion Pricing and Informaion Approach Lieraure Overview and Summary Insider Trading and Opion Pricing Asses, Risk-neuraliy Prices of Vanilla Opions on Forward Conracs The Tradiional Price of a Vanilla Call The Price of a Call wih Addiional Marke Informaion Mixed Cases Sylised Examples and Discussion Conribuion and Discussion Informaion Premium and Marke Power Lieraure Overview and Summary The Model of Benh/Carea/Kiesel and Framework Indifference Forward Price wih Fuure Informaion The Disribuion of he Exponenial Brownian Inegral The Expecaion of he Exponenial Brownian Inegral

15 Conens xv The Addiional Variance Term Conribuion and Discussion Exisence of he Informaion Premium: A Saisical Tes Lieraure Overview and Summary A Tes for he Informaion Premium Excursus: The Leas Squares Mone Carlo Mehod Furher Remarks on he Tes Empirical Measure Change and Srucure of Forward Prices Fuures Daa and Measure Change The Srucure of Differen Forward Prices Empirical Sudies The Second Phase of he EUETS The German Aom Moraorium Furher Regressors and Robusness of Tes Furher Regressors Robusness Conribuion and Discussion Appendices 6 A. Miscellaneous B. Calibraion Resuls for Chaper B.. The Daase from /23 unil / B.2. The EUETS Daa Se B.3. The Moraorium Daa Se C. Regression Resuls for Chaper C.. The EUETS Daa Se C.2. The Moraorium Daa Se Bibliography 67 Lis of Figures 77 Lis of Tables 79 Index 8

16

17 Chaper. Inroducion and Summary of he Thesis Alhough usually characerised as a commodiy, elecriciy is special, in paricular when considered as an underlying for financial markes. I possesses a number of properies ha influence is use on he one hand and marke design on he oher hand. The mos sriking and influenial one of hese feaures is is non-sorabiliy (a leas in relevan quaniies). Once elecriciy has been produced, i needs o be consumed. I is his unique propery and is inerconnecion wih he principles of Financial Mahemaics ha will moivae he heoreical analysis as well as he empirical sudies in he disseraion a hand. This inroducory chaper will lay he foundaion and provide general informaion for he disseraion. In Secion. we will briefly inroduce elecriciy markes. This will include an overview of he hisorical developmen as well as a descripion of relevan issues of marke design. Furhermore, we will provide a brief summary of he academic lieraure in he fields of Financial Modelling and Financial Mahemaics. Secion.2 will hen moivae he main subjec of his disseraion, he impac of a non-sorable underlying when i comes o he relaionship beween he basic objecs on financial markes. This moivaion will also be illusraed by making use of wo marke scenarios ha will be discussed in much deail over he course of his disseraion. In Secion.3 we will sae and formulae he objecs of he disseraion and briefly summarise is conribuion o he academic lieraure. An overview of he srucure of his disseraion as well as imporan informaion on daa sources, compuer programming and he publicaions ha form he basis of some chapers and he resuls herein will finally be brough forward in Secion.4... Elecriciy Markes... Hisory of Elecriciy Markes For more han years elecriciy has been indispensable for he welfare of modern socieies. This is he reason why is generaion, disribuion and reailing used o be organised by inegraed monopoliss and in close collaboraion wih sae holdings or local auhoriies (cf. Sröbele e al. (2, Secion.2)). In he 99s, deregulaion laws were passed in a number of developed counries and elecriciy exchanges were esablished. This was o allow for more compeiion and a higher degree of efficiency. In Norway, he Nord Pool was founded in 993 and laer joined by Sweden as well as Denmark and Finland (cf. Burger e al. (27, page 33 ff.)). Afer a European Union direcive (96/92/EC, cf. Sröbele e al. (2, page 28)) various oher European exchanges were sared, for example, for Cenral Europe, he EEX in Leipzig in

18 2. Inroducion and Summary of he Thesis 22 (by he merger beween he Leipzig Power Exchange and he European Energy Exchange). Oher European exchanges include he APX (Neherlands) or he APX UK (Grea Briain). In he USA, he PJM (he Pennsylvania-New Jersey-Maryland marke) has been influenial and rend-seing. In his hesis he EEX will be our reference marke and we will consider EEX ime series as well as EEX producs. The EEX is he larges power exchange in Europe and ogeher wih is subsidiaries (in paricular EPEX forspomarkes)icovers energy and elecriciy rading in France, Germany, Ausria and Swizerland. In 2 more han TWh were raded on is derivaives plaform and around 3 TWh on he spo marke (mos of i in Germany, we refer o he annual repor EEX (22b, page 88)). For a comparison, he overall amoun of generaed elecriciy in Germany was around 6 TWh (cf. Bundeswirschafsminiserium (22, Secion Energieräger)). Amongs oher European exchanges, he EEX is also a marke place for European emission cerificaes (EUAs). Following he Kyoo proocol of 997, he European Union Emissions Trading Scheme (EUETS) was inroduced as a marke-based means o reduce he emission of greenhouse gases (see Benh e al. (28b, page 6)). The firs (rial) period of he EUETS sared in 25 and ended in 27. The second period commenced in 28 and came o an end in 22. For hese periods, cerificaes were iniially allocaed o paricipans and were valid for one ime period. Generally, hey allow he holder o emi one onne of CO Marke Design and Producs In his secion we will briefly discuss he objecs raded on ypical elecriciy exchanges in general and he EEX in paricular. Furhermore, we will only consider hose producs and hose properies ha will be relevan over he course of his hesis. For he EEX area, as menioned above, spo rading is organised by he EPEX. Alhough inra-day rading exiss, spo prices under consideraion here are dayahead base prices. Prices for individual hours of he following day are calculaed by maching bid- and offer-curves (we will commen on his process laer). The socalled Phelix day base price is hen he simple average of he prices of he 24 hours (i.e. base as well as peak load hours, we refer o bookle EPEX Spo (22) for deails). I is his day-ahead base price ha will be called spo price in his hesis. The imporance of he spo hus consruced is ha i consiues he reference price for he derivaives marke. The mos imporan producs in erms of liquidiy and quaniy on elecriciy exchanges are fuures conracs. For Germany, hese are called Phelix fuures on he EEX. These are conracs ha are seled eiher physically or in cash. As elecriciy is a flow commodiy, fuures conracs have a delivery period raher han a delivery dae. On he EEX delivery periods available are he curren and subsequen four weeks, he curren and he subsequen nine monhs as well as various quarers and years (cf. EEX (22c) for deails). Hence, fuures conracs on elecriciy markes have a swap-like srucure. Oher derivaives raded on he EEX are various forms of opions. The underlying of hese opions mosly are fuures conracs as described

19 .. Elecriciy Markes 3 above, hus hey are no wrien direcly on he spo. In he empirical pars of his hesis, we will mainly examine he curren and he nex six monh-fuures conracs (hese being mos liquidly raded). The number of marke players on he EEX is relaively low when compared o radiional markes. The annual repor EEX (22b) gives heir number a around 2 paricipans regisered a boh he EPEX and he EEX. Clearly, for more specialised producs his already hins owards liquidiy issues and, in fac, derivaives oher han fuures are highly illiquid. The difference beween fuures and he more basic forward conracs is ha he laer is an agreemen beween wo paries whereas he former is organised by an exchange and feaures a margining process. I is also well-known ha one can consider hese wo relaed producs o be equivalen in case ineres raes are deerminisic. In his hesis, we will follow he sandard approach of he lieraure and assume his equivalence. This will ease hings from a modelling perspecive. Furhermore, concerning he daa used in he empirical pars of he disseraion, he assumpion of deerminisic ineres raes is no unrealisic, firsly because ime horizons under consideraion will usually be less han half a year and secondly because ineres raes have generally been very low over he las years. Concerning he maching of demand and capaciy on elecriciy exchanges such as he EEX we sill need o menion he meri order andwereferovonroonand Huck (2) for more deails. The meri order is he lis of available power plans, sored in ascending order by heir individual marginal coss. The mos expensive power plan needed o saisfy demand hen ses he price. For he German marke his plan is usually eiher fired wih hard coal or gas...3. Modelling Elecriciy Markes - Lieraure Classificaion In his secion we will provide a brief overview of he differen branches of he scienific lieraure on modelling elecriciy markes. We will also classify he approach chosen in his hesis. The firs basic decision ha has o be made is wheher one models he spo price or he price of forwards. The laer approach is closely conneced o he Heah-Jarrow- Moron framework known in Financial Mahemaics from he field of ineres rae modelling. These models have he advanage ha one can commence pricing of derivaives direcly, raher han having o calculae forward prices firs. Imporan examples of his branch of he lieraure are Clewlow and Srickland (999), Börger e al. (29) or Benh and Koekebakker (28). The oher possible modelling approach is o consruc a model for he spo price and hen derive forward prices using some relaionship. There are wo very differen branches of spo models. The firs one is ha of srucural or fundamenal models. Here, one akes ino consideraion driving facors of he elecriciy spo such as demand, capaciy or fuel prices (gas, coal). One hen deduces a price for elecriciy from hese facors. Some influenial examples of his approach are Aïd e al. (29), Coulon and Howison (29) and Burger e al. (24). The second approach is ha of reduced-form models. Here, one chooses some sochasic model for he spo price direcly. This is he ypical approach in Financial

20 4. Inroducion and Summary of he Thesis Mahemaics and well-known from he famous Black-Scholes world. One model of his ype will be he workhorse of his hesis and we will provide a deailed lieraure overview for his class in Chaper 3. The reason why we choose his approach is ha in his hesis we propose a cerain fundamenal relaionship beween spo and forward prices..2. Spo-Forward Relaionships and he Informaion Approach Afer having presened he necessary background informaion we are now going o moivae he subjec of his hesis. We have seen in he previous secion ha elecriciy forward conracs are seled agains he spo price. Hence, i is naural o relae spo and forward prices. Secion 4.2 will feaure a deailed discussion abou his relaionship on elecriciy markes as well as he corresponding lieraure. Here, we will discuss in a moivaing manner only. We will use he following noaion: Noaion.2.. Spo price and forward conrac. LeS denoe he spo price a ime. Furhermore le he forward price a wih mauriy a ime T be denoed by F (, T ). In probabilisic erms and by he principle of risk-neuraliy, he well-known (classical) spo-forward relaionship is given by F (, T )=E Q [S T F ] (.) Here, Q is a risk-neural probabiliy and F is he hisorical filraion of he spo process (i.e. he filraion generaed by pas and presen values of he spo). As he elecriciy marke is no complee (we will sae reasons laer), we conclude by he second fundamenal heorem of asse pricing (cf. Bingham and Kiesel (24, Theorem 4.3.) or Shreve (24, page 232)) ha he measure need no be unique. One can choose he specific measure Q and usually a parameric measure is chosen ha minimises he disance beween observed forward prices and prices calculaed. Furhermore, he expecaion in Equaion. is aken under he hisorical filraion, i.e. given he informaion of he spo price evoluion hus far. This is usually jusified by assuming he validiy of he (weak) efficien marke hypohesis (cf. Hull (28, page 78)) which says ha curren (spo) marke prices reflec all publicly available informaion. As saed by Benh and Meyer-Brandis (29, page 2), he relaionship requires he oal available informaion o be equal o he flow of informaion generaed by he spo price process, an assumpion accepable for classical, raded asses. The relaionship is hen proved by seing up a replicaing porfolio for a shor posiion in a forward. Obviously, his porfolio consiss of borrowing money and going long in he underlying. A mauriy one can hen hand over he underlying and repay he loan. This ype of hedging sraegy is called buy-and-hold sraegy. Equaion. is he relaionship used in he quaniaive lieraure on energy markes and no much hough has been invesed o wheher i consiues he righ approach o relae spo and forward prices of elecriciy. Even more so, i is used despie a very serious shorcoming: he mos imporan inrinsic (and quie unique)

21 .2. Spo-Forward Relaionships and he Informaion Approach 5 propery of elecriciy as an underlying is ha i is no sorable (in relevan quaniies). Thus, Equaion. fails heoreically as well as pracically. Pracically, i is no possible o se up he buy-and-hold sraegy o hedge an elecriciy forward. We canno sore spo elecriciy unil mauriy. Theoreically, considering he spo marke isolaed from forward and derivaives markes, he efficien marke hypohesis is no valid eiher. Curren prices migh no reflec available informaion abou he fuure. To illusrae his, consider a power generaing company announcing he mainenance of one of heir power plans in wo monhs ime. If he hypohesis held, hen he decrease in producion capaciy should lead o an increase in he spo price, no only in wo monhs bu also immediaely. Bu no arbirage sraegy is available. The spo price will no reac o his informaion. Sill, he use of he hisorical filraion in Equaion. does imply he assumpion ha curren and pas spo prices conain all he relevan pricing informaion. Oher sylised examples migh include he announcemen ha a new plan will go online or non-ypical and severe weaher forecass or new regulaory legislaion. Summarising, he hisorical filraion is no sufficien when seing up a spo-forward relaionship for a non-sorable underlying. Building on Benh and Meyer-Brandis (29), we will herefore propose a new pricing relaionship for elecriciy forwards: F (, T )=E Q [S T G ] (.2) where G will be called he marke filraion. This filraion will conain all he informaion relevan and publicly available. Thereby, Equaion.2 will make sure ha he forward price will adjus o siuaions such as described in he sylised example above. Consequenly, we will define he informaion premium in Secion 4.2 as he difference beween he prices as calculaed by Equaion.2 and Equaion.. The informaion premium will be our main objec o quanify he influence of he addiional informaion. Summarising, we posulae ha for a valid spo-forward relaionship we need o consider no only he risk aiudes of marke paricipans (in erms of a risk-neural measure) bu also he larger informaion se ha is available o hem. In his hesis, and in paricular in he empirical chapers (for example Chaper 8), we will jusify and illusrae his informaion-based approach wih wo marke siuaions from he German EEX marke. Boh siuaions feaure significan price movemens due o forward-looking informaion. They will be inroduced and discussed nex..2.. The Beginning of he Second Phase of he EUETS Afer he firs, raher non-commial phase of he European Union Emissions Trading Scheme, he sricer second phase commenced in January 28. This phase lased unil 22. We refer o Secion. for a shor marke descripion. In Figure.2. forward prices observed on //26 are illusraed. The delivery period is depiced as he lengh of he horizonal lines which denoe he price. The ypical shape of prices in winer is readily observable: lower values for Ocober and April, highes prices in January and February. Slighly lower prices in December are due o clusering of bank holidays.

22 6. Inroducion and Summary of he Thesis Oc 6 Nov 6 Dec 6 Jan 7 Feb 7 Mar 7 Apr 7 Figure.2..: EEX forward prices on //26. Observed monh forward prices in Euro. Lengh of bars indicaes delivery period. As illusraed in Figure.2.2 (which is similar o Figure of Benh and Meyer- Brandis (29)), we are facing a differen siuaion in he subsequen year. Alhough all effecs seen in Figure.2. persis, he mos sriking feaure is he huge price increase beween he December and he January forwards. The amoun of he increase is more han 6 e, corresponding o some 34%, compared o an increase of 4.5 e (around 7.5%) in 26. Generally, here is a shif upwards in forward prices in 28. The spo price on //27 was around 45 e. Clearly, he price shif can be explained by he marke s anicipaion of he effecs due o he inroducion of he second phase of CO 2 cerificaes. The coss of hese cerificaes were obviously assumed o cause a major rise in elecriciy prices. We will provide a quaniaive analysis and furher qualiaive insighs in Secion Summarising, we claim ha we find forward prices anicipaing publicly available fuure informaion bu a he same ime experience a spo ha does no The German Aom Moraorium The Tōhoku earhquake occurred on March 2. The consequen sunami severely damaged several nuclear power plans, in paricular ha in Fukushima. Only hree days laer, on 4 March 2, he German governmen reevaluaed is energy policy and issued he so-called Aom Moraorium by which he seven oldes plans (eigh reacors wih a capaciy of more han eigh GW) were o be shu down immediaely (over he course of ha week). The measure was o las for a period of hree monhs and was o allow for a new evaluaion of he usage of nuclear power in Germany. Figure.2.3 shows he EEX spo price (seven days moving-average for clariy) for he period from November 2 unil mid Augus 2. Key daes are highlighed by verical bars: he earhquake on /3/2, he Moraorium on 4/3/2, he governmen s decision o permanenly shu down he power plans on 3/5/2 and he official end of he Moraorium on 5/6/2. We remark ha he verical

23 .3. Objecives and Conribuion of he Thesis Oc 7 Nov 7 Dec 7 Jan 8 Feb 8 Mar 8 Apr 8 Figure.2.2.: EEX forward prices on //27. Observed monh forward prices in Euro. Lengh of bars indicaes delivery period. bars will feaure hroughou he remainder of he hesis. I is surprising o see ha alhough eigh GW of cheap nuclear power were aken from he marke (and he meri order), spo prices did no increase or even move significanly. We will find reasons for his and provide a qualiaive discussion in Secion We also refer he reader o he official repor of he German BNA (Bundesnezagenur, he federal regulaory office for elecriciy) o he federal minisry of economics and echnology (Bundesnezagenur (2)). Sill, he effec of he Moraorium was a sharp increase in forward prices, no only of hose whose delivery fell ino he hree monhs of he Moraorium bu also of hose wih a laer delivery period. As an example, Figure.2.4 shows he evoluion of he price of he forward wih mauriy in May 2. The forward price had a mean value of e before he Moraorium and a e pos-moraorium mean price. This corresponds o an increase by more han e, i.e. almos 25%. For he second half of his ime series we also see ha prices remained more or less consan unil he las day of he delivery period. Clearly, his implies ha by hen also he spo had adjused o he increased price level. Again, we refer o Secion for a discussion and more insighs. Concluding, we find ha once more forward prices reaced o some fuure informaion which was publicly available bu he spo did no (insanly). There is an apparen asymmery in prices and a large informaion premium added o forwards by marke paricipans..3. Objecives and Conribuion of he Thesis Elecriciy markes have only been esablished relaively recenly and are sill ransforming. Quaniaive Financial Modelling of hese markes is a challenging ask. This is no only due o heir incompleeness and illiquidiy or due o regulaory issues ypical of such markes bu also because he underlying elecriciy is special.

24 8. Inroducion and Summary of he Thesis 65, EEX Spo (7 days MA) Earhquake (/3/2) Moraorium (4/3/2) Decision o shu down plans (3/5/2) End of Moraorium (5/6/2) 6, 55, 5, 45, 4, 35, 3, Figure.2.3.: EEX spo price 2/22. Seven days moving-average from November 2 unil mid of Augus 2. In Euro. 65 May 2 Forward Earhquake (/3/2) Moraorium (4/3/2) Decision o shu down plans (3/5/2) Figure.2.4.: EEX May 2 forward conrac. Lifeime from /2/2 unil 3/5/2. In Euro.

25 .3. Objecives and Conribuion of he Thesis 9 I is a flow commodiy ha canno be sored in relevan quaniies and ha needs o be consumed as produced. All ypes of marke players face enormous risks because he marke is very volaile and invesmens needed are usually huge and long-lasing. Forward and fuures conracs are he main building blocks of any hedging or rading sraegy and heir correc valuaion as well as heir relaionships o oher basic producs is absoluely essenial for he indusry. Hence, i is he objecive of his hesis o quaniaively invesigae he impac of he non-sorabiliy of elecriciy. In paricular, we wan o explore how he informaion asymmery ha was moivaed in he previous secion affecs he Financial Mahemaics and Financial Modelling of elecriciy markes. This very imporan aspec has unil now been almos compleely negleced in he academic lieraure. All relevan papers, may hey be on pricing, hedging or risk-managemen in elecriciy markes, have so far worked wih he radiional spo-forward relaionship. We hus formulae an exension o he classical relaionship of Financial Mahemaics and augmen he exising lieraure by discussing various analyical, concepual as well as empirical consequences ha arise from his new basic marke srucure. We are now going o presen he soluions and conribuions presened in individual chapers of his hesis one by one. Building on he reference paper Benh and Meyer-Brandis (29) and he moivaion of Secion.2 we propose he new spo-forward relaionship in Chaper 4. This relaionship does no only ake a risk-adjusmen (via a measure change) ino consideraion bu also accouns for he informaion asymmery menioned before (via an enlargemen of filraion). In Chaper 5 we hen exend he analyical resuls of he reference paper o he more realisic case of forward conracs wih a delivery period. In paricular, we redefine he objec of he informaion premium for hese conracs and calculae i using he mahemaical echnique of enlargemen of filraion. By ranslaing resuls from he lieraure on modelling of insider rading we also propose a simpler mehod o calculae he emerging informaion drifs making use of Malliavin calculus. Furhermore, we provide formulae for he informaion premium for various differen arrangemens of he ime axis wih one or more pieces of fuure informaion and for direc as well as for correlaed fuure informaion. We also recover he resuls of Benh and Meyer-Brandis (29) as limis of our more general formulae. The issue of pricing opions on elecriciy forwards wih delivery period in he presence of an informaion premium (i.e. under an enlarged filraion) is hen discussed in Chaper 6. We idenify prices for vanilla opions aking he fundamenal relaionship beween he wo filraions a hand ino consideraion. These prices differ from hose calculaed in he absence of informaion asymmery. Thereby, we show ha previous resuls from he lieraure do no ranslae auomaically o energy markes. Sill, we find ha differences in opion prices are no due o a modified volailiy srucure bu only due o iniial forward prices. Referencing he relevan lieraure we find ha his is a general resul ha also applies o more complicaed pay-off srucures. Moivaed by he ideas of he second reference paper Benh, Carea, and Kiesel (28a) we idenify indifference forward prices of marke agens wihin he above

26 . Inroducion and Summary of he Thesis informaion framework in Chaper 7. This implies calculaing he expeced uiliy of he spo under an enlarged filraion. In he exising (and mahemaically relaed) lieraure on insider rading, auhors have hus far always chosen a combinaion of a log-normal underlying and he logarihmic uiliy. This leads o a simplifying cancellaion and closed-form resuls. We make use of he disribuional properies of he Brownian bridge o obain resuls for a more general class of uiliy funcions. As o our bes knowledge his is a new approach. Chaper 8 is he empirical cenrepiece of he disseraion a hand. I conribues o he academic lieraure on elecriciy markes by providing he firs empirical invesigaion abou he informaion premium and also by showing is exisence (and hus he pracical relevance of his hesis) for he wo scenarios previously discussed in Secion.2. To his end we propose a saisical mehod ha ess for he properies of he informaion premium. This urns ou o be non-rivial. Wihou furher assumpions on he srucure of he enlarged filraion his problem ranslaes o showing ha he premium is non-measurable wih respec o he hisorical filraion. In oher words, we need o show ha he premium is he residual when projecing he forward price under he enlarged filraion ono he space spanned by he hisorical filraion. Therefore, we propose a mehod based on Hilber space represenaions and basis regressions in order o calculae and analyse expecaions under he enlarged filraion. This mehod can be applied more generally, and o he bes of our knowledge no oher es for non-measurabiliy has ye been proposed in he lieraure. Summarising, we add o he scienific lieraure by presening and exploring a new spo-forward relaionship for elecriciy and by providing an overview of how i affecs he differen aspecs and branches of Quaniaive Financial Modelling..4. Srucure of he Thesis The srucure of his hesis is moivaed by he objecives and conribuions brough forward in Secion.3. I is depiced graphically in Figure.4.5 which also illusraes he inerdependence of he individual chapers. Afer he general background informaion and moivaing houghs of his chaper, Chaper 2 will be devoed o he mahemaical heory applied laer. In Secion 2.2 he heory of enlargemen of filraions will be reaed. This will also be he basis for all heoreical chapers of he hesis. Secion 2.3 on he oher hand will very briefly remember some of he key ideas and resuls of Hilber space heory, in paricular when applied o spaces of square-inegrable sochasic processes. Then, we will presen a cerain spo model in Chaper 3. We will calculae forward prices wih delivery period for his model in Secion 3.4. Furhermore, we will lay he foundaions for empirical examinaions by describing how o fi he model o observed daa (Secion 3.5) and how o simulae price pahs (Secion 3.6). Secion 3.7 will provide a firs shor empirical sudy. Afer discussing spo-forward relaionships and he reference paper Benh and Meyer-Brandis (29) in Chaper 4, we will show how o apply he echniques of he second chaper in order o calculae he informaion premium for various seups in

27 .4. Srucure of he Thesis Chaper 5. Chaper 6 will discuss he issue of pricing opions on fuures wih addiional informaion. We will exend he ideas of Benh, Carea, and Kiesel (28a) and calculae indifference prices and marke power in Chaper 7. Finally, Chaper 8 will feaure he horough empirical sudy of he informaion premium, in paricular analysing he wo marke scenarios presened previously. This chaper will bring ogeher he mahemaical srucures of Chaper 2, he saisical echniques of Chaper 3 and he calculaions of Chaper Daa, Programming and Compuer Algebra The price daa used in his hesis was obained from various sources. EPEX Phelix spo and EEX Phelix fuures prices were obained parially from Bloomberg and Thomson Reuers erminals, parially from he EEX websie direcly. The EEX gas prices discussed in Secion 8.5. were aken from a Thomson Reuers erminal and so was he DAX ime series of he same secion. Daa processing and rearranging was done mainly using MS Excel and MS Visual Basic. More complicaed programmes such as he calibraion, simulaion and measure change procedures, regressions and saisical ess were wrien or implemened in Insighful S-Plus 6.. All he graphics in his hesis were prepared using MS Excel. Some complicaed analyical expressions were calculaed or double-checked making use of wo compuer algebra sysems, namely Wolfram Mahemaica 7 and WolframAlpha (online a hp:// Publicaions We have highlighed hose chapers ha are based on publicaions and working papers in Figure.4.5. A version of he discussion on opion pricing, which is Chaper 6 in his hesis, is going o be published as par of a Springer proceedings iled Quaniaive Energy Finance. The ile of he chaper will be Elecriciy Opions and Addiional Informaion and is reference here is Benh, Biegler-König, and Kiesel (23b). Pars of he empirical sudy of Chaper 8 will be published in volume 36 of he journal Energy Economics under he ile An Empirical Sudy of he Informaion Premium on Elecriciy Markes. In his hesis, his paper will be referenced as Benh, Biegler-König, and Kiesel (23a). Boh papers are also available online on SSRN and can be found a hp://papers.ssrn.com/sol3/cf_ dev/absbyauh.cfm?per_id=8729. Las bu no leas, he maerial presened in Chaper 7 is he basis of a working paper. Is preliminary ile is Fuure Informaion and a Broader Class of Uiliy Funcions and i will include an addiional secion abou he opimal porfolio of an insider on a sock exchange.

28 2. Inroducion and Summary of he Thesis Chaper Chaper 2.2 Chaper 3 Chaper 2.3 Chaper 4 Chaper 5 Chaper 8 Chaper 6 Chaper 7 Figure.4.5.: Srucure of he hesis. Chapers based on publicaions and working papers highlighed.

29 Chaper 2. Mahemaical Background 2.. Lieraure Overview and Summary In his chaper we will lay he mahemaical foundaions for he remainder of he hesis. The firs wo secions will discuss he heory of enlargemen of filraions or French grossissemen de filraions. Hisorically, research on enlargemen of filraions was iniialised by Iō inaconribuion o he proceedings Iō (978). There, he presened he following equaion in erms of a Brownian moion W, which he said should hold from he inuiive meaning of inegrals (Iō (978, page 95)): s W dw u = W (W W s ) (2.) He realised ha he inegral on he lef side is problemaic as W is random for <. Thus, in order for he equaion o make sense, Iō proposed o consider i under a differen filraion (or, as he wries, a differen reference se), namely he one generaed by W and W, <. Moreover, he found ou ha under his filraion he Brownian moion W is a semi-maringale, oo, and he provided is decomposiion. These wo resuls will be crucial in his chaper and we will formulae hem as Theorem Much scienific progress was hen made by French mahemaicians in he 98s. We will alk abou a number of relevan papers and proceedings laer on, in paricular in Secion 2.2.: Chaleya-Maurel and Jeulin (985), Jacod (979), Jacod (985), Jeulin (979), Jeulin (98), Jeulin and Yor (978), Yor (978), Yor (985). Chaper VI of Proer (25) is a comprehensive reference explaining boh branches of he heory: iniial and progressive enlargemen (his hesis will only feaure he former ype). A new moivaion o sudy he enlargemen of filraions was provided by Pikovsky and Karazas (996), who applied he heory o model insider rading on financial markes. Since hen, a muliude of papers has covered his idea and we will menion he mos imporan ones in he inroducion of Chaper 6. The srucure of Secion 2.2 will hen be as follows: Secion 2.2. will feaure imporan assumpions and resuls and we will concenrae on he maringale decomposiion of sochasic processes under enlarged filraions. These decomposiions will laer be used, for example in Chaper 5. Then, in Secion we will approach filraion enlargemen from a differen angle and we will explain how one can consider i as a change of measure. Our main reference for his approach will be he shor noe by Proer (989) as well as Föllmer and Imkeller (993). The docoral hesis by Ankirchner (25) also follows he measure change idea. We will uilise he findings of his secion in Chaper 6.

30 4 2. Mahemaical Background Furhermore, we will presen Imkeller s mehod and provide a brief overview of Malliavin calculus in Secion This mehod will grealy simplify our calculaions in Chaper 5 alhough is original moivaion was an exension of he French lieraure. Imkeller esablished his resuls in Imkeller (996, 23) as well as in Imkeller e al. (2). The main references on Malliavin calculus we will use are Øksendal (996) and Nualar (26). Ye anoher aspec of enlargemen of filraion will be explored in Secion 2.2.4: is connecion o linear sochasic differenial equaions. The well-known Brownian bridge will be used o exemplify and o prepare for he more complicaed calculaions of Chaper 7. The main reference of his secion will be Karazas and Shreve (99). Finally, in he las secion of his chaper, Secion 2.3, we will remind he reader of a number of sandard resuls abou Hilber space heory and is connecion o condiional expecaions. In paricular, we will esablish he exisence of complee orhonormal sysems for he space of square-inegrable random variables and cie resuls ha show ha condiional expecaions on his space possess funcional forms. These heoreical resuls will hen be uilised when proposing our empirical es in Chaper 8. Throughou he secion we will refer o sandard lieraure such as Royden (968) for Hilber space aspecs or Klenke (26) for noions of probabiliy heory Enlargemen of Filraion In Equaion.2 we proposed a new spo forward relaionship in erms of a filraion G including addiional fuure informaion. We will now define a general mahemaical framework o model his informaion approach as discussed in Secion.2. We remark ha his is he classical seup as in Jacod (985, page 5) or Amendinger (999, page 3). Le (Ω, F, F, P) be a filered probabiliy space. Throughou his hesis we will always assume ha he usual hypoheses hold for he spaces we consider and we refer o Proer (25, Page 3 and Secion I.5) for definiions and a discussion. Clearly, F will denoe he hisorical filraion of some underlying sochasic process, say L, i.e. F = σ(l s : s ). Definiion Enlarged filraion. Le G be an F-measurable random variable wih values in a Polish space (U, U). We inroduce he (iniially) enlarged filraion H = F σ(g), <T Υ where T Υ denoes he ime horizon. Furhermore, we define a filraion G ha includes non-precise addiional informaion abou G, i.e. F G H. We remark ha we will assume ha H TΥ = G TΥ = F TΥ, i.e. he addiional informaion expires a ime T Υ. In oher words, he addiional informaion concerns he fuure ime poin T Υ. Also, if G was no F-measurable, hen he enlarged filraion would be rivial wih respec o F -measurable processes and he following secions would be obsolee.

31 2.2. Enlargemen of Filraion Enlargemen of Filraion and Maringale Decomposiion In his secion we wan o answer he quesion wheher or no a semi-maringale under he hisorical filraion remains a semi-maringale under he enlarged filraion as defined in Definiion If so, we would also like o find is decomposiion under he new filraion. Before we answer hese quesions in general, hough, we will go back o Iō s original problem moivaed by Equaion 2.. The following famous resul provides he answer o boh of he above quesions for he special case ha we enlarge he hisorical filraion of a process by is fuure value. We sae here a version exended o Lévy processes, he proof will be following he one in Proer (25, Theorem VI.2.3). Theorem Iō s Theorem (exended o Lévy processes). Le L be a Lévy process and F is hisorical filraion. Le H = F σ(l TΥ ). Then L is also a semi-maringale wih respec o H.Also,ifE[ L ] <, hen he process is a H -maringale on [, ). TΥ L TΥ L s ξ = L T Υ s Proof. We will prove he saemen for T Υ =. We can easily generalise due o he scaling properies of Lévy processes. If he saemen holds for, i holds for > rivially because in his case F = H. For simpliciy (and wihou loss of generaliy) i will be assumed ha E[L ] = and hus L is a F -maringale. The proof now consiss of hree pars:. We will emporarily assume ha E[L 2 ] < for all >. Now he independen incremen propery of L is used. One defines auxiliary variables s< wih s = j n and = k n ( j, k n) as well as he random variables ds Y i = L i+ n L i n yielding he easy ideniies n k L L s = Y i, L L s = i=j Also, he lengh of he inerval s in erms of i, j, k, n is n n j n = n j n,ha of s is k j n. Thus, as he Y i are iid and inegrable, we calculae k n E[L L s L L s ]=E Y i Y i = n n k j Y i n j n i=j i=j i=j Y i = k j n j (L L s )= s s (L L s ) i=j

32 6 2. Mahemaical Background Clearly, E[L L s H s ]=E[L L s L L s ] for s<. Hence, verifying he maringale propery of he process ξ : [ L L s ] u E[ξ ξ s H s ]=E L u du L L L u s + u du H s [ ] L L u = E [L L s H s ] E s u du H s = s s (L E[L L u H s ] L s ) du u = s s (L L s ) = s s u u s (L L s ) du Now, we need he assumpion E[L 2 ] < for he case ha = because here is a poenial explosion, i.e. he process can be unbounded. I needs o be verified ha [ L L s E s ] ds < By using he converse of Jensen s inequaliy and he properies of Lévy processes one ges [ ((L E[ L L s ]=E L s ) 2) ] 2 E [ (L L s ) 2] 2 κ( s) 2 where κ is some consan. Thus, [ L L s E s and he saemen is proved. ] ( s) 2 ds κ s ds < 2. Now, we reduce he assumpion E[L 2 ] < o he weaker saemen E[ L ] < for >. Now L has càdlàg pahs and hus only a finie number of jumps bigger han. Hence, defining J = L s { Ls>} we can decompose <s J 2 = <s L s { Ls< } L = Y + J J 2 where Y is a L 2 -process and independen of J and J 2. Now, inroducing a new filraion H = F σ(y,j,j2 ) (which saisfies H H ), he process Y Y s ξ = Y ds s