Pricing Derivative Instruments On Emissions Allowances: Evidence From The European Union Emissions Trading Scheme

Transcription

1 Pricing Derivaive Insrumens On Emissions Allowances: Evidence From The European Union Emissions Trading Scheme Mads Basholm Kjærgaard Advisor: Francesco Violane Deparmen of Economics and Business Aarhus Universiy Absrac This research concerns modelling reurns and pricing opions on fuures in he EU ETS. The sparse amoun of sudies ino his subjec enables a preliminary sudy of he EUA spo and fuures markes and he models abiliy o model reurns, volailiy, and price fuures opions on EU allowances in phase III of he EU ETS. This research conducs a economeric analysis o highligh he main feaures of he allowance price and reurns processes and furher suppor and expand he previous findings, ha he EUA spo reurn series is heeroskedasic, exhibis ARCH effecs, and shows volailiy clusering. Furhermore, reurns is no normally disribued, bu exhibis heavier ails. I was also found ha he price process is non-saionary, and he relaionship beween spo and fuures prices, can be expressed by a cos-of-carry relaionship. This research uses he GARCH and Markov regime swiching GARCH mehodologies for modeling allowance reurns and pricing opions on fuures. The models are esimaed hrough maximum likelihood using he normal, suden-, and GED disribuion on EUA spo reurns and i is found ha boh single regime and regime swiching models improve he fi of spo reurns. The inclusion of suden- and GED disribuions also improve he fi of he single regime GARCH models, bu yields unconvincing resuls regarding he significance of a second regime when esimaed using suden- or GED disribuions. The volailiy forecasing performance of he developed models are examined and he IGARCH model ouperformed he oher models. The ou-of-sample analysis gave furher doub o he significance of a second regime when using suden- or GED disribuions. For pricing opions on allowance fuures a risk-neural and Mone Carlo frameworks are developed for he models and he models abiliy o price opions on allowance fuures examined. The analysis showed ha he added flexibiliy of he Markov regime swiching model proved o be significan when pricing opions on allowance fuures, furher jusifying he complexiy of his ype of model.

2 Table of conens Chaper 1 : Problem Formulaion And Mehodology : Inroducion : Problem Saemen : Limiaions : Research srucure : Lieraure review : Mehodology : Daa : Basic ideas and key conceps... 9 Chaper : The European Union Emission Trading Scheme : Kyoo Proocol and he Doha Amendmen... 1.: Flexibiliy mechanisms : The European Emission Trading Scheme : Resuls from Phase I and II : Allocaion and he use of allowances beween compliance periods and phases : Exchanges : Paricipans Chaper 3: Evidence From The Spo And Fuures Markes : Daase : Volailiy analysis : Disribuion analysis : Auocorrelaion, uni roo, and saionary ess : Relaionship beween spo and fuures prices... 8 Chaper 4: Modelling Allowance Spo Prices : Model specificaions : Maximum Likelihood Esimaion : Forecasing Volailiy : In-sample analysis : Ou-of-sample analysis Chaper 5: Pricing Opions On Fuures : Opion pricing using Black-Scholes, single regime GARCH, and MRS GARCH models : Mone Carlo simulaion : Empirical resuls Chaper 6: Final Remarks : Conclusion : Furher Research References... 61

3 Lis of Figures Figure.1: Traded volumes of EUAs Figure.: The price of EUAs during phase III wih commens on back-loading Figure 3.1: EUA spo prices Figure 3.: EUA Reurns Figure 3.3: -Days rolling volailiy Figure 3.4: Hisogram of reurns Figure 3.5: QQ-plo for reurns Figure 3.6: Correlogram of he price process Figure 3.7: EUA spo and December 014 fuures prices Lis of Tables Table 3.1: Descripive Saisics Table 3.: Resuls from he heeroskedasiciy and auocorrelaion ess Table 3.3: Resuls of normaliy ess Table 3.4: Resuls of he uni roo ess Table 3.5: Resuls of coinegraion ess Table 4.1: In-sample performance of single regime models Table 4.: In-sample performance of regime swiching models Table 4.: Ou-of-sample volailiy forecasing performance Table 5.1: Opion pricing accuracy 3

4 Chaper 1 : Problem Formulaion And Mehodology 1.1: Inroducion The European Union Emissions Trading Scheme (EU ETS) was launched on January 1s 005 as one of he main mechanisms of reducing greenhouse gases dicaed by he Unied Naions Framework Convenion on Climae Change (UNFCCC), also known as he Kyoo proocol. The Kyoo proocol was a response o he hrea of global climae change. The general convicion among scieniss and poliicians alike is ha an increase in greenhouse gas (GHG) emissions has caused he increase in emperaures over he las cenury. The larges growh in GHG emissions comes from energy supply, ransporaions, and indusry where carbon dioxide (co) is he mos imporan anhropogenic GHG and accoun for 77% of he global GHG emissions (IPCC, 007). The effecs of coninuous growh in GHG emissions may be serious on an environmenal, social, and economic level if no srong acion is aken (Sern, 007). To address his risk, he world agreed on an inernaional reay called Unied Naions Framework Convenion on Climae change (UNFCCC) and oday 195 counries have raified he convenion. The overall objecive of he convenion is o sabilize he concenraion of GHGs, such ha he level would preven dangerous anhropogenic inerference on he climae sysem. The reay was no legally binding which led paricipans signing an addiion o he reay in 1997, known as he Kyoo Proocol. The objecive of he Kyoo proocol was o reduce he GHG emission wih five percen, compared o he level of 1990, over he period of In December 01, EU signed he Doha amendmen, became effecive on January 1s 013, in which EU has agreed o reduce GHG emissions wih 0% compared o he levels of 1990 before 00 (UNFCCC, 014). As a response, he EU developed The European Union Emissions Trading Scheme (EU ETS) as a flexible and effecive insrumen o help companies o archive heir compliances. The EU ETS covers 45% of he oal emissions of he EU and including energy inensive secors and around insallaions (European Commission, 013). Emissions rading sared on January 1s 005 and he EU ETS is he larges emissions rading program in he world, and raded volumes keeps growing. Afer showing promising resuls in erms of efficiency and liquidiy, he EU ETS enered phase III in 013, where spo and fuures rading keeps growing and furhermore, derivaive insrumens have been available and used by marke paries since 006 (Daskalakis e al., 009; European Commission, 013). Wih he increase in paricipans and affeced secors, he demand for efficien and effecive risk managemen ools is increasing. However, mos of he research pu ino he EU ETS is aimed a exploring and deermine he efficiency and effec of he EU ETS 4

5 on climae change, and he research of he pricing of derivaives on emissions allowances is sparse. 1.: Problem Saemen The purpose of he research is o sudy he emission allowances marke and in paricular he mos liquid insrumen, he EU allowance (EUA) on carbon emissions, despie a growing marke wih promising poenial. The lieraure on his subjec is sparse and has no showed consisen resuls in modelling and idenifying he dynamics of he spo price on EUAs, and price derivaive insrumens on EUAs. The specific objecive of he presen research is o idenify characerisics of he spo dynamics, assess differen models abiliy o efficienly model hese characerisics and price opions on EUA fuures. The models have o work boh heoreically and empirically while efficienly and correcly price opions on EUA fuures. To find he opimal answer o he problem saemen, he following quesions mus be answered: Which feaures and characerisics have o be considered when choosing a model for modelling EUA spo and fuures prices and reurns? Which models implemen he observed feaures and characerisics for modelling reurns? How well do he models perform empirically for modelling reurns on emission allowances? How can he models be implemened for he pricing of opions on fuures and how do hey perform empirically? 1.3: Limiaions The focus of he presen research is on he European Union Allowance (EUA). Despie he relevance of oher emission allowances, like Cerified Emission Reducions (CER) and Emission Reducion Uni (ERU), hese insrumens and heir correlaion wih EUAs will no be aken ino accoun. The research will only focus on daa from he phase III. Despie he relevance of he price dynamics of iner- and inra-phase price dynamics of phase I and II, here is no prohibiion of banking and borrowing beween phase II and III, hence his problem is assumed no o be presen. However, o ensure he qualiy of daa and avoid oudaed regulaion o have an effec on he price, only daa from phase III is aken ino accoun. This research will only focus on inra-phase conracs, so he assumpion of cos-of-carry is preserved, bu he validiy of his assumpion is esed. I is beyond he scope of he presen research o assess he impac of he EU ETS on climae change. An in-deph discussion of rules and regulaion of he EU ETS will no be 5

6 provided, bu only key ideas and regulaion wih direc impac on he price iself is explained. I is assumed ha he reader of he research is familiar wih basic saisical and economeric ools for he analysis of ime series. Therefore, only a brief inroducion and no an in-deph discussion of saisical and economerical mehodology is provided. 1.4: Research srucure The srucure of he research is as follows. The res of chaper one will include a lieraure review, a discussion of mehodology, a discussion of he daa used in his research, and an inroducion o basic ideas and key conceps which are imporan for he sudy. Chaper wo is an inroducion o he EU ETS in which he background of emissions rading and he EU ETS is provided and he resuls from he firs wo phases are discussed. Moreover, he marke characerisics, paricipans, and key regulaion wih effec on he allowance price are highlighed and a brief discussion provided. Chaper hree is an economeric analysis wih he purpose of highlighing he main characerisics and feaures of he EUA price and reurn dynamics. This chaper will include descripive saisics, a disribuion analysis, a volailiy analysis, an analysis of saionariy, and an analysis of he relaionship beween spo and fuures prices. The economeric analysis is he foundaion for choosing he models for modelling EUA spo prices and pricing opions on fuures. Chaper four will include a discussion and descripion of he models for modelling EUA reurns. The models are presened in he following order: Firs he geomeric Brownian moion, second he generalized auoregressive condiional heeroskedasiciy (GARCH) model, and hird a Markov Regime Swiching GARCH model. Chaper four also provides an esimaion procedure for he models and an empirical analysis of he in- and ou-of-sample performance of he models for modelling EUA reurns and forecasing volailiy. In chaper five, a framework for pricing of opions on fuures hrough he differen models is presened. Chaper five ends wih an empirical invesigaion of he ou-of-sample performance for he pricing models on acual raded opions in order o deermine he bes model for pricing opions on EUA fuures. Finally, a conclusion and a discussion for furher research is is found in chaper six. 1.5: Lieraure review The academic works on carbon emissions rading and he EU ETS have increased in he pas years, bu usually wih high aenion owards he economics of emissions rading and is impac on climae change. The goal of hese sudies is o assess wheher he marke will effecively lead o a reducion in emissions, or if i will provide for he bes flexibiliy and cos abaemen for reducing emissions and reach compliance (Egenhofer, 007; Parker, 011; Vlachou, 013). Since i is beyond he scope of his research o explicily analyze he consequence and efficiency of emissions rading and allowance 6

7 allocaion. Even hough i has an effec on he price iself, an analysis of he bes srucure and efficiency of emissions rading is no conduced. Only a brief inroducion o he key elemens, paricipans, markes, and regulaion is provided. The amoun of financial sudies regarding he European Union Emissions Trading Scheme (EU ETS) has increased hese pas years, bu an exensive and consisen analysis of he Emission Uni Allowances (EUA) from he EU ETS, sill needs o be conduced. An exensive and conclusive sudy on he bes pricing model for EUA derivaives is no currenly presen, and analysis conduced on he EUA marke has concerned phase I and he early sages of phase II while a comprehensive sudy of Phase II and Phase III is sill needed. Benz & Trück (008) observe skewness, excess kurosis, and heeroskedasic volailiy in he reurns process and sugges he use of AR-GARCH and Markov regime swiching models wih auoregressive or normal disribuion processes in he regimes for modelling he price dynamics of EUAs. They find ha AR-GARCH and Markov regime swiching has a superior fi and have a beer forecasing performance han models wih a consan variance. Uhrig-Homburg & Wagner (009) examine he relaionship beween spo and fuures prices. They find ha spo and fuures prices are described by a cos-of-carry approach, which is suppored by Gorenflo (013) and Daskalakis e al. (009). These aricles also conclude ha iner-phase fuures and spo prices are described by a cos-of-carry approach. However, boh sudies conain daa from phase I and he beginning of phase II. Chevallier e al. (009) and Joyeux & Milunovich (010) also es he cos-of-carry relaionship beween spo and fuures prices and find ha he cos-of-carry relaionship is no significan beween and he rial period respecively. Byun (013) examines hree models for forecasing EUA fuures volailiy: implied volailiy, k-neares neighborhood, and a GARCH ype model and conclude ha GARCH models have he bes volailiy forecasing abiliy. Chevallier e al. (011) analyses he inroducion of opions in Ocober 006, during phase I, and conclude ha he inroducion of opions on fuures have an impac on he volailiy of EUA reurns. Daskalakis e al. (009) is closes o he presen sudy and invesigaes he EU ETS and he hree main markes of EUAs a he ime. They develop a heoreical and empirical valid framework for pricing opions on EUA fuures, as well as a framework for inraphase and iner-phase fuures opions and find ha borrow and banking prohibiion, beween phase I and II, has significan implicaions for pricing fuures and opions on fuures. Since here is no prohibiion of banking and borrowing in and beween phase II and III, i is only heir analysis of inra-phase opions ha is comparable o he presen research. Daskalakis e al. (009) concludes ha a jump diffusion model (JDM) proposed 7

8 by Meron (1976) has a superior fi on log-reurns boh in- and ou-of-sample and performs beer for opion on fuures pricing han a more simple model. Sudies on EUA fuures and fuures opions is sparse and hose conduced use daa from phase I and he sar of phase II. This research aims o conribue and furher expand he sudies on EUA fuures and fuures opions, while using daa from phase III, o reduce his knowledge gab. 1.6: Mehodology To invesigae and answer he problem saemen i is imporan o ouline and discuss he research mehod. This research is a quaniaive empirical invesigaion of EUA spo, fuures and opion on fuures prices. This research adops he mehodology as Daskalakis e al. (009) which is as follows: Firs, an economeric analysis will provide he foundaion for choosing which models is used for modelling EUA reurns and pricing opions on EUA fuures. Secondly, he models and heir esimaion procedure is explained and discussed. The models abiliy o model EUA reurn and forecas volailiy is examined in order o es heir abiliy for calculaing he expeced payoff of he opions. Thirdly, a risk-neural valuaion framework is developed and numerical resuls are obained hrough closed form soluions and Mone Carlo mehods. The obained heoreical opion prices is compared o prices on acual raded opions o assess he empirical performance of each model. On basis of he heoreical mehods and he numerical resuls, he heoreical opions prices are compared o he prices of acual raded opions. To choose which model is he bes a pricing opions on EUA fuures, he disance beween heoreical and observed prices is calculaed and invesigaed. Programming for implemening he models and calculaing he heoreical prices is done in MATLAB R014a including he opimizaion, economeric, and saisical oolpacks. All codes used in he presen research is found on he CD. 1.7: Daa For he inroducion and descripion of European emissions rading, repors and saemens from key governmenal and inernaional agencies and eniies are used o assure he mos objecive descripion of he European Union emissions rading scheme. The hisorical EUA spo, fuures, opions on fuures prices and he risk-free ineres rae, where he six-monh Libor is used, are rerieved hrough a Bloomberg Terminal. The lieraure for developing he heoreical frameworks is based on aricles from praised academic journals and exbooks in order o assure an academically praised and valid heoreical framework. 8

9 1.8: Basic ideas and key conceps This secion inroduces some basic ideas and key conceps used for improving he undersanding of he presen sudy : Fuures pricing In order o price fuures, some assumpions have o be made for some of he marke paricipans: The marke paricipans have no ransacion coss. The marke paricipans are subjec o no axes. The marke paricipans can borrow and lend money a he same risk-free rae. The marke paricipans ake advanage of arbirage opporuniies as hey occur. No resricions on shors-selling I is imporan o noe ha hese assumpions are no required o be rue for all marke paricipans, bu only need o be rue, or approximaely rue for key marke paricipans such as large derivaive and insiuional raders, as i is heir eagerness for aking advanage of arbirage opporuniies which deermines he relaionship beween spo and fuures prices. Wih hese assumpions for a fuures conrac on an asse wih price S 0 ha provides no income, where T is he ime o mauriy and r is he risk-free rae, han he fuures price F 0 is given as F rt 0 S0e (1.1) rt Oherwise, arbirage opporuniy exiss because of he fac ha if F0 S0e an arbirageur can buy he asse and shor he fuures conrac, and if F0 S0e arbirageur can ener ino a long posiion in a fuures conrac and ener a shor posiion in he asse, and in boh cases earn a risk free profi (Hull, 01; Wilmo, 007). 1.8.: Opion pricing A sandard opion, also known as a plain vanilla opion, is a financial conrac which gives he owner of he conrac he righ, bu no he obligaion, o buy or sell a specific asse a a pre-specified price (srike) and a a pre-specified ime (mauriy). In he presen research, he specific asse is EUA fuures. The opion can eiher be a call opion, in which he conrac s owner has he righ, bu no he obligaion o buy he underlying asse, or a pu opion in which he owner of he conrac has he righ o sell he underlying asse. The presen research only deals wih European opions. This means opions can only be exercised a mauriy, unlike American opions where he opion can be exercised a any ime unil mauriy. To buy an opion, he opion buyer have o pay an opion premium o he seller of he opion who is obligaed o buy or sell he rt an 9

10 underlying asse a he srike price, if he buyer decides o exercise he opion. This premium is he opion price. The payoff for a call and a pu opion, respecively, is expressed as payoff max(s K,0) (1.) payoff max(k S,0) (1.3) Where S is he asse price a mauriy and K is he srike price. Under he law of no arbirage, which saes ha here are no arbirage opporuniies in he marke, he value of an opion is saed as V e payoff (1.4) r( T ) S, E[ ] Where r is he risk-free rae and T is he ime o mauriy. (Hull, 01; Wilmo, 007) 1.8.3: Risk-neural valuaion One of he mos imporan conceps in he analysis of derivaives is he risk-neural valuaion approach. For a more inuiive explanaion of he risk-neural approach, some properies of he real world are defined: An invesor is very sensiive o risk and expec greaer reurn for aking on more risk. Invesors know abou hedging and risk eliminaion, and use hese ools acively. People ofen call he risk-neural world for a world where people do no care abou risk, so le define some probabiliies of he risk-neural world Invesors do no care abou risk and do no expec greaer reurn for aking on more risk Everyhing is priced using simple expecaions Since invesors in he risk-neural world have no risk preferences and do no expec a greaer reurn for aking on more risk, hen he expeced reurn of any asse is he riskfree rae. Furhermore, he presen value of cash flows in he risk-neural world is calculaed by discouning he expeced value wih he risk-free rae. The assumpion of a risk-neural world grealy simplifies he analysis of any derivaive conrac, and a derivaive wih a payoff a a paricular ime can be valued using risk-neural valuaion by using he following procedure: I is assumed ha he expeced reurn of he underlying asse is he risk-free ineres rae Calculae he expeced payoff from he derivae Discoun he expeced payoff wih he risk-free ineres rae 10

11 I is imporan o noice ha he risk-neural world is a device used o obain soluions o pricing models and ha he soluions are applicable in all worlds, boh he risk-neural and he real world. Two hings happen when moving from he risk-neural world o he real world. Firs, he expeced growh rae of any asse may no be he risk-free ineres rae. Secondly, he discoun rae used for any payoff from derivaives changes. However, hese wo changes always offse each oher exacly. Hence, he resul of risk-neural valuaion is applicable in all worlds (Hull, 01; Willmo, 007) : Law of one price and he no-arbirage principle The law of one price saes ha prices of equivalen insrumens should be he same no maer how and where he insrumen is being raded. Oherwise, arbirage opporuniies will emerge. An arbirage opporuniy is a zero cos and risk free profi opporuniy. The no arbirage principle saes ha if arbirage opporuniies emerge, hen arbirageurs would rush o and exploi he arbirage opporuniies unil marke prices adjus and arbirage opporuniies cease o exis. The law of one price and no arbirage principle is very imporan in order o develop a framework for correcly pricing derivaives (Hull, 01; Ma, 011) 1.8.5: Law of large numbers In probabiliy heory he law of large numbers saes ha as he number of idenically disribued random generaed variables increases, heir sample mean approaches heir heoreical mean. For example, le X be a random variable. Is expeced value can be wrien as A E[ X ]. If X,..., 1 X n wih n independen random variables wih he same disribuion, and hen he following approximaion is made n 1 n n i 1 A A X (1.5) When n i A A as n. The law of large numbers is cenral for Mone Carlo simulaions and is implemened when developing a Mone Carlo framework of pricing opions (Brandimare, 006). 11

12 Chaper : The European Union Emission Trading Scheme The purpose of his secion is o inroduce he European Union Emission Trading Scheme (EU ETS) and EU Allowances (EUAs). Firs, i is necessary o inroduce he Kyoo Proocol and he hree esablished flexibiliy mechanisms, because he European Union Emissions Trading Scheme (EU ETS) is a consequence of one of hese mechanisms. This secion will include a presenaion of essenial regulaion, exchanges, paricipans and, mos imporanly, he financial insrumens of carbon asse rading in he EU ETS..1: Kyoo Proocol and he Doha Amendmen The Kyoo proocol was he inernaional response o climae change. I was approved by he 3 rd conference of he paries of he Unied Naions Framework Convenion on Climae Change (UNFCCC) in December The Kyoo proocol was a legal binding commimen o reduce he emissions of greenhouse gases (GHGs) wih a leas 5% of he emissions in 1990 wih a commimen period from 008 o 01. The Kyoo Proocol did no became effecive unil February 005, because he Proocol had o be raified and approved by a leas 55 paries o he convenion, including developing counries represening 55% of he oal emissions in The Proocol only became effecive when Russia decided o follow 140 oher counries and raified he agreemen. I should be noed ha he larges emier of GHGs, he USA, which represen 5% of he oal emissions and 40% of he emissions from developed counries, did no raify he proocol (Mansane-Baaller & Pardo, 008). Wih he raificaion of he Kyoo Proocol, he paries commied hemselves o reduce GHG emissions wih 5% of he emissions of 1990 in he commimen period from 008 o 01. The GHGs lised in Annex A of he proocol is carbon dioxide (CO), mehane (CH4), nirous oxide (NO), hydrofluorocarbons (HFCs), perfluorocarbons (PFCs) and sulphur hexafluoride (SF6). The reference gas in he Kyoo Proocol is CO and a CO equivalen measure was developed o indicae he global warming poenial of each GHG. Therefore, CO is he reference gas for assessing he reducion arges as well as he quaniies in he specificaion of financial conracs (Mansane-Baaller & Pardo, 008). I is imporan o noe ha he EU-15 counries was reaed as a single eniy. Consequenly, he reducion arge of 8% assigned o he EU-15 for he period was disribued among is members hrough an allocaion mechanism. The Kyoo Proocol gave posiive resuls in he firs commimen period from 008 o 01, and effecively reduced CO emissions (Aichele & Felbermayr, 013). On December 8 h 01 he Doha amendmen o he Kyoo Proocol was adoped. The Doha amendmen esablished a second commimen period saring from 013 o 00 in which he paries agreed o reduce heir emissions of GHGs wih 18% compared o

13 levels (UNFCCC, 01). I is imporan o noe ha USA sill has no raified he Kyoo Proocol and during he firs commimen period Russia, Canada and Japan wihdrew from he Kyoo Proocol. Furhermore, China and India do no have o reach heir reducion arges, because of heir saus as developing counries. The EU wans o mainain is role as he global leader in environmenal issues, consequenly implemening he plan. Wih he plan, he EU is commied o reducing is CO emissions wih 0% compared o 1990 levels, raising he share of EU energy consumpion produced by renewable resources o 0% and improve he energy efficiency in he EU wih 0%. Furhermore, EU is offering o increase is emission reducion arges o 30%, if oher major economies in he developed and developing world commi o underake heir fair share of he global emission reducion effor (European Commission, 01)..: Flexibiliy mechanisms In order o achieve heir emission reducion arges, paricipans of he Kyoo Proocol have o implemen climae and environmenal policies ha have a miigaing effec on climae change. The Kyoo Proocol esablishes hree flexibiliy mechanisms which allow for minimizaion of he overall coss of achieving he emission arges. The Three flexibiliy mechanisms are he Join Implemenaion Mechanism (JI), The Clean Developmen mechanism (CDM), and Inernaional Emissions Trading (IET) (Mansane- Baaller & Pardo, 008)...1: Join Implemenaion Mechanism The Join Implemenaion Mechanism (JI) consiss of he realizaion of emission reducion projecs by an annex I counry in anoher annex I counry. In reurn, he JI projecs are awarded wih credis called Emission Reducion Unis (ERUs) which can be used by he pary promoing he projec o mees is own emission arges. Hence, he JI mechanism provides a flexible and cos efficien alernaive o reduce emissions domesically by aking advanage of invesing in counries wih lower conversion coss. To avoid double accouning he ERUs are deduced from he hos counries assigned amoun and on he oher hand, he hos counry benefis from foreign invesmens and echnology ransfer (UNFCCC, 005)...: Clean Developmen Mechanism The purpose of he Clean Developmen Mechanism (CDM) is he possibiliy for annex I counries o inves in non-annex I counries. The goal is o achieve susainable developmen and conribue o he objecive of he Kyoo Proocol while assising he annex I counries in achieving heir goals. The annex I counries receive credis called Cerified Emission Reducions (CERs) by invesing in projecs and aciviies ha reduce emissions in developing counries. The CDM gives he same advanage as JI by providing an alernaive o expensive domesic emission and by invesing in emissions reducing 13

14 projecs and aciviies in developing counries wih lower coss (Mansane-Baaller & Pardo, 008). I is imporan o noe ha he Kyoo Proocol does no impose emission reducion commimens on developing counries. However, he developing counries accoun for 59% of global emissions and play a crucial role in global emissions reducion (NEAA, 013)...3: Inernaional Emissions rading The hird flexibiliy mechanism is Inernaional Emission Trading which allows annex I counries o rade Assigned Amouns Unis (AAUs) wih oher annex I counries. Paries can ake advanage of he difference beween reducion coss and AAU prices. All ypes of unis are raded o achieve compliance. In addiion o ERUs and CERs, oher ypes of unis can be raded, such as AAUs, Removal Unis (RMUs) and Verified Emissions Reducions (VERs). However, i is beyond he scope of his research o describe each of hese unis, so an in-deph descripion is no provided. The EU ETS has is background in IET and is an addiional mechanism allowed by he UNFCCC. The following secion will describe and presen he EU ETS, key regulaion, markes and paricipans (Mansane-Baaller & Pardo, 008)..3: The European Emission Trading Scheme The European Union Emissions Trading Scheme (EU ETS) is he cornersone of EUs effor o achieve compliance wih he Kyoo Proocol. I is a cap-and-rade sysem wih an EU wide cap on emissions. The EU ETS covers more han power saions and manufacuring plans in he 8 member saes of EU as well as Iceland, Liechensein and Norway. Around 45% of oal emissions of he EU is covered by he EU ETS. Through a process called surrendering, insallaions ha are covered by he EU ETS are required o surrender emission allowances equivalen o heir emissions in previous years before April 30 and hese surrendered allowances are cancelled a June 30. For every one of emissions ha he company has no covered by allowances, he company receives a fine of 100. The company is sill required o surrender he necessary allowances he following year. The scheme officially sared on January 1, 005 and is so far divided ino hree phases (European Commission, 013) Phase I corresponds o he period January 1, 005 o 31 s December 007. Phase I was a hree year pilo period of learning by doing during which he purpose was o prepare for phase II, as ha is when i would need o funcion effecively o help he members of EU o reach heir reducion arges. Phase I succeeded in esablishing free rade of emission allowances across EU and he necessary infrasrucure monioring, reporing, and verifying emission allowances from businesses and paricipans (Mansane-Baaller & Pardo, 008; Erupean Commission, 013). 14

15 Phase II coincides wih he commimen period of Kyoo Proocol, and lased from January 1, 008 o December 31, 01. Iceland, Liechensein, and Norway joined he EU ETS a he beginning of phase II which means ha he scheme now covered 31 counries. Businesses was allowed o rade JI and CDM credis which enlarged he range of available cos effecive emission miigaion opions. The EU ETS became he bigges source of demand for CDM and JI credis, making hese credis he main driver behind he inernaional carbon marke and he main provider of clean energy invesmens in developing counries (European Commission, 013) Phase III sared a January 1, 013 and is supposed o end a December 31, 00. Before phase III, he cap of allowances was allocaed o member saes hrough naional allocaion plans. Phase III inroduced an EU wide cap on allowances which should be disribued hrough aucions insead of free allocaion as in Phase I and II. The cap for 013 i se o ons of CO, and is expeced o decrease wih 1,74% a year unil 00 which means ha he emissions from fixed insallaions are reduced wih 1% (EU, 014).4: Resuls from Phase I and II In Phase I of he EU ETS, emissions were reduced by an esimaed -5%, bu prices were very volaile. A he sar of phase I he price was 8 /on, bu hen i increased and he price exceeded 30 /on in early 006, only o fall back o he level of 8 /on in April 006. This volailiy was aribued o he lack of precise emissions daa, ransparency, and he lack of banking beween he firs and he second phase (IETA, 013). Emissions during phase II was below he cap, bu allowance prices were sill volaile. The price sared very high and rose o above 0 /on in he firs half of 008, and reached an average of /on in he second half of ha year. However, prices began o fall by he sar of 009 and a he end of phase II, he prices were around 7 /on (IETA, 013; Bloomberg Terminal). 15

16 Figure 1.1: Traded volumes of EUAs. Source: EU, 013 Figure 1.1 shows he raded volumes and he plaform of exchange of EUAs in phase I and II. The figure shows ha he raded volumes have increased during he life of he EU ETS. Especially he ransiion beween phase I and phase II increased he raded volumes significanly. The majoriy of he rading is conduced hrough exchanges, bu i is also imporan o noe he very large par of rading, almos as large as he exchangeraded par, which is done over-he-couner (OTC). The figure also shows he amoun of raded EUA s during aucion is increasing, his is discussed laer in his secion (European Commission, 013; World bank, 014)..5: Allocaion and he use of allowances beween compliance periods and phases In he firs wo phases, he vas majoriy of allowances was given away for free hrough free allocaion. In Phase III, aucioning is he main mehod of allocaing allowances; his means ha businesses, covered by The EU ETS, have o buy an increased amoun of heir allowances a aucions. I is a goal for he EU o phase ou free allocaion compleely by 07, because aucioning is he mos ransparen mehod and pus ino pracice he principle ha he one who emis GHGs has o pay (European Commission, 013). Anoher imporan feaure of he EU ETS is banking and borrowing across years and phases. Banking refers o he possibiliy o save unused allowances for fuure use, and borrowing refers o he possibiliy o borrow allowances from fuure allocaions, for use in he curren period. Banking and borrowing reduce he overall compliance coss by allowing ineremporal flexibiliy. Banking is allowed beween compliance periods and 16

17 beween phase II and III. This means ha un-surrendered allowances are sill valid for compliance he following years. Borrowing is also allowed o some exen. Businesses covered by he EU ETS can cover evenual shorages wih allowances issued in he nex year, since he emissions from period have o be surrendered afer allowances for +1 have been released (Chevallier, 01). Since he financial crisis in 009, he EU ETS has experienced a growing surplus of allowances. The surplus of allowances doubled in he period beween early 01 and early 013 o almos wo billion allowances, and i has grown furher o over,1 billion allowances. The growing surplus is expeced o sop by he end of 014, bu he overall surplus is no expeced o decline during phase III. The surplus may preven he EU ETS o funcion correcly and if no addressed, he imbalances will affec he EU ETS abiliy o mee more demanding reducion arges in a cos effecive manner. On he shor erm, he European Commission addresses his imbalance by posponing he aucioning of 900 million allowances unil he end of phase III, in order for he demand o pick up. This measure is called back-loading and i does no reduce he overall amoun of allowances, bu only how he allowances are disribued during phase III. I is expeced ha he amoun of aucion volume of allowances will be reduced wih 400 million in 014, 300 million in 015 and 00 million in 016. I is assessed ha back-loading can rebalance supply and demand while reducing price volailiy wihou an impac on he compeiiveness, and increase he governmen revenues in he early sages of phase III. However, i is imporan o noice ha back-loading is only a shor erm measure, and a susainable soluion requires srucural and regulaory changes. In order o address he allowance surplus on a long erm, he European commission has proposed o esablish a marke sabiliy reserve in he beginning of phase IV in 01. The sabiliy reserve will improve he EU ETS resilience o major shocks by coninually adjusing he supply of allowances and address he surplus of allowances. 17

18 Figure 1.: The price of EUA s during phase III wih commens on back-loading. Source: world bank, 014 Figure 1. shows he price of spo EUA s, wih commens on discussion of back-loading in he European parliamen. Figure 1. shows ha back-loading negoiaions have had a huge influence on he price of EUA s during phase III and is a large conribuor o he volailiy of EUA s (World Bank, 014)..6: Exchanges A significan change beween phase II and phase III is he increasing use of aucioning as he main mehod of allocaion in phase III. I is expeced ha 40% of allowances in he EU ETS is aucioned in 013, and he amoun of aucioned allowances is expeced o increase he following years. The increase emphasis on aucioning is caused by he belief ha aucioning will suppor and srenghen he price signal of allowances as a simple, ransparen, and efficien mehod of allocaing allowances. The main exchanges for aucioning are he European Energy Exchange (EEX) and he InerConinenal Exchange (ICE). ICE is by far he larges wih 98,8% of he raded volume beween and However, as seen in figure 1.1, a large amoun of he oal raded volumes is no recorded, because i is raded OTC, as discussed in secion.4. The average monhly rading volume is conracs on he ICE while only on he EEX. However, spo rading is only on he EEX while he closes o spo rading on ICE is rading on fuures wih expiry on ha paricular day. Acually, only 3,3% of he rading during ha period was spo rading, he res was fuures rading. I is usual ha he fuures marke for commodiies is much more liquid hen he spo marke, because i is easier and more convenien o rade fuures insead of he commodiy iself. I is he same wih EUAs, because EUA fuures does no necessarily lead o he delivery of EUAs, as in mos circumsances he conrac is closed ou prior o delivery. Therefore, opions 1 All volume informaion can be found on he CD 18

19 on EUAs can only be raded hrough opions on EUA fuures ha are only available on he ICE (European Commission, 013; Hull; 01; Bloomberg Terminal)..7: Paricipans Secors covered by he EU ETS include he power and hea generaion secor, combusion plans, oil refineries, coke ovens, iron and seel plans, and facories making cemen, glass, lime, bricks, ceramics, ec. Since 01 emissions from all flighs from, o and wihin he 8 EU member saes plus Iceland, Liechensein, and Norway are included in he EU ETS. However, o allow ime for negoiaions and developmen of a global marke measure applying for emissions from aviaaions, he EU ETS requiremens were suspended for flighs o and from he EU. For he period 013 o 016, only emissions from flighs wihin he EU are covered by he EU ETS. The Inernaional Civil Aviaion Organizaion agreed in 013 o develop a global markebased mechanism addressing emissions from aviaion by 016 and apply i by 00. However, no all paricipans in rading of EUAs are subjec o he EU ETS; paricipans may also include brokers, financial insiuions, funds, or NGOs which uses he EUAs as an asse class (European Commission, 013). This chaper has inroduced and described he essenials of he EU ETS and he carbon emissions marke. This research focuses on he emissions allowances insrumens raded under he EU ETS. I has been imporan o define hese insrumens, markes, exchanges, and paricipans for a beer undersanding of emission allowances. The EU ETS is based on he Kyoo proocol, as he EU ETS is a consequence of he flexibiliy mechanisms allowed by he proocol. Some specific characerisics of he EU ETS and heir impac on he prices of emissions allowances, such as back-loading, banking, and borrowing and allocaion of allowances, have been highlighed and heir impac on prices discussed. The raded fuures volume is found o be much larger han he volume of spo rading which is why his research focuses on fuures and opions on fuures. 19

20 Chaper 3: Evidence From The Spo And Fuures Markes This secion conains an analysis of he spo price dynamics in order o idenify he main characerisics of he spo price ime series. I is necessary o perform an economeric analysis on he ime series of he spo price and reurns, o idenify he main feaures of he spo price and reurns dynamics before choosing pricing models. These feaures are hen implemened in models for modelling EUA reurns and pricing opions on EUA fuures. The economeric analysis will include a presenaion of he daase followed by a descripion of he range of reurns. An analysis of he volailiy of reurns, disribuion of reurns, auocorrelaions, and uni roo ess o es for saionariy. The las par of he economeric analysis is an analysis of he relaionship beween spo and fuures prices o asses if hey are coinegraed and follows he cos-of-carry relaionship. 3.1: Daase Spo rading in he EU ETS is performed hrough he European Energy Exchange (EEX) in Germany. Spo rading of EUAs in phase III sared on Fuures rading sared as well on and is conduced on boh EEX and he ICE. Volumes raded a spo prices is only averaging of 6,7 ons per day while fuures rading is rading a a volume of , ons per day. The daase for phase III consiss of closing prices from o which gives 431 observaions. The hisorical prices are obained hrough a Bloomberg erminal which gives he daa high credibiliy. The dynamics of he spo price and he logarihmic reurns, denoed reurns from here on, are shown in figure 3.1 and 3.. r s ln, where s is he EUA price a ime. s 1 0

21 Spo Figure 3.1: EUA spo prices Reurns 0,5 0,15 0,05-0,05-0,15-0,5-0,35-0, Figure 3.: EUA Reurns Figure 3. clearly shows ha he reurn process exhibis large spikes; herefore, i is necessary o invesigae if hese spikes are a naural par of he process, or if are hey an exraordinary even caused by exernal facors. If hese spikes are caused by an exraordinary exernal even, hey are removed from he sample. The larges spike occurred on he when he price dropped from 4,73 o 3,14 which is a negaive reurn of -40,97%. Figure 1. in chaper shows ha on he he EU parliamen had a negaive voe agains back-loading which mus be defined as an exraordinary exernal even, ha is no a par of he reurn dynamics and is herefore removed. 1

22 Two large posiive spikes wih a reurn of 0,17% and 0,19% occurred on he and , respecively. These spikes are no a resul of any exernal even and is no removed. The large spike of -18,61% on is also no an exernal even and is kep as a par of he reurn process. Some descripive saisics of he filered reurns is given in able 3.1 # Obs. Mean Median Max Min Sd. Dev. Skewness Kurosis 49 0,069% 0,000% 0,187% -18,615% 4,91% 0,118 6,601 Table 3.1: Descripive saisics The maximum reurn is 0,187% and he minimum is -18,615% wih a mean of 0,069%. These findings combined wih figure 3. indicae ha he reurn process may be very volaile, especially in he beginning of 013. The volailiy in he beginning of 013 was due o policy changes and uncerainy regarding he back-loading plans in he EU. Besides policy changes, he main price drivers of EUA prices is he elecriciy, gas, and coal prices, equiy index, he economic growh, and weaher condiions (Rickels, 010, Aaola e al, 013; Crei e al, 01). Since he power secor is one of he main operaors in he EU ETS here is a srong link beween EUA and elecriciy prices. 3.: Volailiy analysis The daily hisorical sandard deviaion of he reurns was 4,91% which gives an annualized sandard deviaion of 67,95%. I is imporan o conclude if he reurns are characerized by homoscedasiciy, consan volailiy, or heeroskedasiciy, non-consan volailiy, as i has a huge influence on he choice of models. To invesigae if he reurns are heeroskedasic, a rolling window of he sandard deviaions is calculaed wih a window of rading days. The resul is shown in figure ,000% 1,000% 10,000% 8,000% 6,000% 4,000%,000% Rolling Window 0,000% Figure 3.3: -days rolling volailiy

23 Figure 3.3 shows signs of heeroskedasiciy in he reurns, hence he variance is no consan and flucuaes significanly in he observed period wih a minimum sandard deviaion of 1,175% and a maximum of 11,88% compared o he hisorical sandard deviaion of 4,91% which can be a sign of volailiy clusering and heeroskedasiciy in he reurns process. To furher es for volailiy clusering and condiional heeroskedasiciy, also known as ARCH (Auoregressive condiional heeroskedasiciy) effecs, a Ljung-Box Q-es and an Engle s ARCH es is conduced. Boh ess for condiional heeroskedasiciy and auocorrelaion in he residuals (Ruey, 00; Verbeek, 01). The es saisics are as follows: Q m Where n i T( T ) (3.1) T i funcion. i1 Q m is he Ljung-Box es saisic, T is he ime, and is he auocorrelaion Engle s Arch es uses he Lagrange muliplier saisic TR where T is he ime and R is he coefficien of deerminaion from fiing he ARCH model via regression (Verbeek, 01). Tes-Saisics Criical Value P-Value null hypohesis Ljung-Box Q-es 71,18 31,410 0,000 Rejeced Engle's Arch-es 11,917 3,841 0,001 Rejeced Table 3.: Resuls from he heeroskedasiciy and auocorrelaion ess. Table 3. conains he resuls from he Ljung-Box Q-es and Engle s ARCH es. The Ljung-Box Q-es and Engles Arch-es boh srongly rejecs he null hypohesis of homoscedasiciy and no auocorrelaion. The resuls conclude ha he ime series is heeroskedasic and exhibis auocorrelaion in he residuals hence, he ime series shows volailiy clusering and ARCH effecs. The volailiy analysis shows ha he variance of he reurns is no consan or heeroskedasic and he residuals exhibis auocorrelaion. This means ha reurns conain volailiy clusering which implies ha he reurns conain periods of high volailiy and periods wih low volailiy. These characerisics needs o be aken ino accoun when choosing a pricing model. 3.3: Disribuion analysis The disribuion of reurns is very imporan for he choice of models and heir performance. Therefore i is imporan o analyze wheher he disribuion is normal or no and which characerisics describes he disribuion. 3

24 -0% -18% -16% -14% -1% -10% -8% -6% -4% -% 0% % 4% 6% 8% 10% 1% 14% 16% 18% 0% The skewness and kurosis values imply ha he reurns disribuion is posiively skewed and lepokuric. The posiive skewness of 0,118 indicaes a disribuion ha is skewed o he righ. The posiive kurosis value imply a disribuion where he reurns are lepokuric meaning he observaions are cenered around he peak. This gives he disribuion a higher peak and exreme observaions are more likely o appear, resuling in faer ails. The kurosis for he normal disribuion is 3, and he kurosis for he disribuion of reurns is 6,601 which Implies ha he disribuion of reurns has a higher peak and longer ails, han he normal disribuion Hisogram Figure 3.4: Hisogram of reurns Figure 3.4 shows a hisogram and ogeher wih he measures for skewness and kurosis, indicae a disribuion of reurns, which is skewed o he righ, have heavier ails, and a higher peak. This indicaes ha he disribuion of reurns are no normally disribued (Jorion, 007; Verbeek, 01). To check if reurns are normally disribued, a Quanile-Quanile plo (QQ plo) and a series of ess are conduced. To visually demonsrae he deparure from he normal disribuion, a QQ plo is used. A QQ plo compares he acual probabiliies of he reurns wih he expeced probabiliies of he normal disribuion. The QQ plo is a scaer plo where he sandardized reurns or Z-values are ploed agains he heoreical values of he normal disribuion. The Z-values is compued as Z i R i (3.) 4

25 where R is he reurn, µ is he mean of reurns, and σ is he sandard deviaion of reurns. If he reurns are normally disribued, hen he QQ plo should be a sraigh diagonal line. The QQ plo is showed in figure 3.5. Figure 3.5: QQ-plo for reurns The hick blue line in figure 3.5 represens he acual probabiliies, and he black line shows he expeced probabiliies of he normal disribuion. If he disribuion of reurns is normal, hen he acual probabiliies should lay as a sraigh line on op of he expeced probabiliies. Figure 3.5 clearly shows ha he reurns are no normally disribued, because he acual probabiliies are no in a sraigh line. The QQ plo also shows ha he lef end of he paern is below he line and he righ end is above he line. This indicaes long ails a boh ends of he disribuion which means ha exreme values are more likely o appear han in he normal disribuion. To furher es if he reurns are normally disribued, hree ess are used on he reurns. The hree ess are a Jarque-Bera es (Jorion, 007), an Anderson-Darling es (Anderson & Darling, 195), and Lilliefors es (Verbeek, 01). The ess are performed o assess he null hypohesis of normaliy of reurns. The es saisics are compued as follows: ( 3) JB T 6 4 (3.3) Where JB is he es saisic for he Jarque-Bera es, is he sample skewness, and is he sample kurosis. 5

26 i 1 A n [ln( F (x )) ln(1 F (x )] (3.4) n n i1 n i n 1 i Where A is he Anderson-Darling es saisic, n is he number of reurns, n {x... x } is he ordered reurns, and F is he cumulaive disribuion funcion for i n he normal disribuion. D max F(x) G(x) (3.5) x Where D is he es saisic of he Lilliefors es, F(x) is he empirical cumulaive disribuion funcion for he sample daa, and G(x) is he cumulaive disribuion funcion for he normal disribuion. Tes-Saisics Criical Value P-Value null hypohesis Jarque-Bera 3,733 5,836 0,001 Rejeced Anderson-Darling 6,781 0,751 0,001 Rejeced Lilliefors 0,083 0,043 0,001 Rejeced Table 3.3: Resuls of he normaliy ess. Tabel 3.3 conains he resuls from he normaliy ess, and i is very clear ha he reurns are no normally disribued, because he null hypohesis of normally disribued reurns is rejeced by every es. Furhermore, he P-values are 0,001 which means a srong rejecion of he null hypohesis. The disribuion analysis leads o a srong rejecion of he assumpion of normally disribued reurns. Moreover, he skewness and kurosis measures indicaing a disribuion wih a higher peak and heavier ails han he normal disribuion. This mus be considered when choosing a pricing model. 3.4: Auocorrelaion, uni roo, and saionary ess The assumpion of saionariy or non-saionariy has huge impac on he choice of pricing model. Therefore, i is imporan o invesigae if he price process is saionary or no. Firs, he auocorrelaion coefficiens is exanimaed in figure

27 Figure 3.6: Correlogram of he price process Figure 3.6 is a correlogram showing he auocorrelaion funcion (ACF) a differen lags. If he price process is saionary, hen he ACFs would decrease drasically or even cu off afer only a few lags. However, he ACFs of he price process are highly persisen and decrease very slowly, indicaing ha he price process is non-saionary. To furher invesigae if he price process is non-saionary, a series of uni roo ess is conduced. If he ime series conains a uni roo hen i is non-saionary. The uni roo ess ha are conduced are an Augmened Dickey-Fuller (ADF) es and a Kwiakowski-Phillips- Schmid-Shin (KPSS) es (Verbeek, 01). The ess are performed as follows. Any AR(p) process can be wrien as: Y Y c T... c Y (3.6) p1 p1 Then a simple Dickey-fuller es is performed on π wih he es saisic: DF (3.7) se ( ) Where se(π) is he sandard error of π. The KPSS es uses a srucural model defined as: y c u, (3.8) 1 c c u (3.9) 1 is he rend coefficien, u1 is a saionary process, and u is an iid process wih zero mean and variance of. Then he KPSS es saisic can be formulaed as follows: 7

28 T / 1 KPSS T S (3.10) Where KPSS is he es saisic, T is he sample size, is he newey-wes esimae of he long-run variance, and S e1 e... e. The resuls of he wo ess are repored in able 3.4 Tes-Saisics Criical Value P-Value null hypohesis ADF -0,66-1,941 0,4 Keep KPSS 1,590 0,146 0,010 Rejec Table 3.4: Resuls of he uni roo ess. The ADF es keeps he null hypohesis of uni roo which means ha process is nonsaionary. The KPSS es rejecs he null hypohesis of rend saionary process. This leads o he conclusion ha he price process of EUA is a non-saionary process. This conradics he common findings of mean reversion in commodiies energy asses. The non-saionary feaure has o be considered when choosing models. 3.5: Relaionship beween spo and fuures prices The relaionship beween spo and fuures prices is a very imporan issue ha needs o be covered. I is crucial for he pricing of fuures and opions on fuures ha here is no advanage of holding a fuures posiion insead of a spo posiion, hence spo and fuures are seen as subsiues. To apply a pricing mehodology, where derivaives wih fuures as underlying are priced wih a spo price model. I is imporan o jusify he assumpion ha spo and fuures prices follow he same sochasic process, as i is necessary for he no arbirage condiion hold and he fac ha spo and fuures prices converge a expiry dae, as shown in figure 3.7. (Hull, 01) 8

29 Relaionship beween spo and fuures prices Fuures Spo Figure 3.7: EUA spo and December 014 fuures prices To analyze he relaionship beween spo and fuures prices, a coinegraion analysis is performed firs o see if he spo and fuures prices follow he same sochasic drif and herefore is seen as subsiues. The cos-of-carry can summarize he relaionship beween spo and fuures prices. The cos-of-carry relaionship is expressed as follows: F r( T ) 0 S0e (3.11) Where F 0 is he fuures price, S0 is he spo price, and r is he risk-free rae. The cosof-carry relaionship suggess ha he law of no arbirage would hold in he long run, bu no necessarily in he shor run due o he imperfec marke, his was discussed in secion (Hull, 01). To es if spo and fuures prices follow he same sochasic rend and if he cos-of-carry relaionship holds in he long run, a coinegraion es is performed. The coinegraion es is an Engle-Granger coinegraion es. The ess assess he null hypohesis of no coinegraion among he wo ime series. The es is performed in wo seps where he firs sep is peforming he following regression: F S u (3.1) Then an ADF es is performed on he residuals u as done in equaion 3.6 and 3.7. The resuls of he Engle-Granger es are in able 3.5 9

30 Tes-Saisics Criical Value P-Value null hypohesis Engle-Granger -5,568-3,351 0,001 Rejec Table 3.5: Resuls of coinegraion ess The Engle-Granger es srongly rejecs he null hypohesis of no coinegraion, hence he spo and fuures prices follow he same sochasic rend and he cos-of-carry relaionship holds in he long run. This chaper has performed an economeric analysis on he EUA spo and fuures prices o highligh he main feaures and characerisics, which drives he price dynamics. The empirical analysis shows ha he disribuion of spo reurns is no normal, bu have heavier ails where more exreme values appear more frequenly and are skewed o he righ. The analysis has also shown ha he volailiy of he EUA spo prices is heeroskedasic, exhibis ARCH effecs, and shows clusering over ime. Examinaion of he auocorrelaion, uni roos, and saionary ess revealed ha he price dynamics process is non-saionary and no mean revering as observed wih oher energy commodiies. The coinegraion analysis showed ha spo and fuures prices are coinegraed and follow he same process and he cos-of-carry relaionship holds in he long run. These findings are almos equivalen o he findings of Benz & Trück (009), and Daskalakis e al. (009). However, boh finds a larger kurosis and a he disribuions is skewed o he lef insead of skewed o he righ as found in he presen research. 30

31 Chaper 4: Modelling Allowance Spo Prices 4.1 : Model specificaions The economeric analysis in chaper hree highlighed he main feaures and characerisics of he hisorical price process. These feaures need o be aken ino accoun when choosing a pricing model. Therefore, a pricing model for phase III EUAs needs o reflec he following feaures: Non-normaliy Volailiy clusering Heeroskedasiciy Heavier ails Non-saionariy The findings have led o he following hree mehodologies: The firs model is he Geomeric Brownian Moion (GBM), however he model assumes normally disribued reurns and a consan variance, bu i is a widely acceped model and will work as a base model or benchmark for he oher models. The second model is he GARCH mehodology. The GARCH models incorporae he feaures of heeroskedasiciy and volailiy clusering while also being able o use oher disribuions han he normal disribuion. The hird and las model is a regime swiching GARCH model which has he same feaures as he sandard GARCH models, bu incorporaes a regime swiching feaure ha allows for more flexibiliy, hen he sandard GARCH models, hence he sandard GARCH is also seen as a single regime model : The Geomeric Brownian Moion The firs model is he Geomeric Brownian Moion which is a sochasic process. I is a coninuous ime model of an asse price and i is he mos widely acceped model for equiies, commodiies, currencies, and indices (Willmo, 007). The GBM is specified as follows: ds S d S dw (4.1) Where S is asse price a ime, µ is he consan drif or he expeced reurn, and σ is he random variaion around he drif or volailiy of he asse. W is a sandard Brownian moion, also known as a wiener process, which is a special diffusion process ha is characerized by independen idenically disribued (iid) incremens ha are normally disribued wih zero mean and a sandard deviaion, equal o he square roo of he ime sep. The economeric analysis rejeced he assumpion of normally disribued reurns and a consan volailiy, bu he GBM is widely acceped and will work as a base model or benchmark o which he oher models is compared. The model is a markov process, implied by he independen incremens ha is a process in which only he 31

32 curren asse price is relevan for compuing probabiliies of evens involving fuure asse price. (Brigo e al, 007). The properies of he iid incremens are very imporan and are used in he esimaion and simulaion of he random behavior of asse prices. Through basic sochasic calculus and Iö s lemma, equaion (4.1) can be wrien as: d ln S ( ) d dw (4.) Since µ and σ are consan, equaion 4. implies ha ln S follows a generalized wiener process wih a consan drif rae of and a consan volailiy of σ. This means ha he change in log S 0 and log S is normally disribued wih mean variance of σ. This means ha: ln S ln S0 N[( )(T ), ( T )] and or (4.3) (4.4) ln S N[ln S0 ( )(T ), ( T )] Where S is he fuure value of he asse, S 0 is he asse price a ime 0, and N(m,v) is he normal disribuion wih mean m and variance v. Since he naural logarihm of he asse price is normally disribued, he asse price a ime T given he asse price a ime 0 is lognormal disribued. The sandard deviaion of he logarihm of he asse price is ( T ), which means i is proporional o he square roo of ime ahead. (Hull, 01; Brigo e al, 007) 4.1.: Single regime GARCH models The economeric analysis of he EUA spo prices showed ha he volailiy was no consan, bu heeroskedasic; he ails were heavy and he reurns exhibi volailiy clusering. These findings causes problems for he GBM, because i works under he assumpion of consan variance and normally disribued reurns. One way o deal wih his is o le he variance of residuals depend upon is hisory. Engle (198) who proposed he auoregressive condiional heeroskedasic (ARCH) process performed he seminal work in his area. The ARCH process allows he condiional variance o change over ime, as a funcion of pas errors, which leaves he uncondiional variance o be consan. The ARCH models have proven o be useful in modelling several differen economic and economeric phenomena (Engle, 198). In empirical applicaions, he 3

33 model has a relaive long lag in he condiional variance equaion, and o avoid negaive variance esimae,s a fixed lag srucure is usually imposed : The Sandard GARCH Bollerslev (1986) proposed a more general class of processes called he Generalized Auoregressive Condiional Heeroskedasic (GARCH) model. The idea for he model is o allow for a longer memory and a more flexible lag srucure. Le ε denoe a discreeime sochasic process wih real-values and I denoes he informaion se of all informaion hrough ime. Then he sandard GARCH(p, q) process is given as follows: S y ln µ S1 I 1 N(0,h ) (4.5) hz 1 h p q 0 ii ih i i1 i1 Where p 0 and q z I N(0,1) (4.6) (4.7), 0 for i 1,..., q, 0 for i 1,...,p i i If p=0 hen he GARCH is reduced o a simple ARCH(p) process and if p=q=0 hen is whie noise. Empirically he GARCH(1,1) ofen performs very well and is he specificaion used hroughou he res of he presen research. The GARCH(1,1) process is given as h h (4.8) 0 i The GARCH(1,1) is saionary when 1 1 1, his is examined in secion 4.3. If he value of 1 1 is close o 1 hen volailiy is highly persisen. The GARCH model of Bollerslev (1986) spawned a large amoun of research ino his area and a lo of differen GARCH models have been developed. I is beyond he scope of his research o access every GARCH model ou here. However, i is necessary o examine he mos popular GARCH ype models : GJR GARCH Glosen e al. (1993) observed an asymmeric effec of pas innovaions on condiional variance. Glosen e al. (1993) observed ha negaive shocks have a larger impac on volailiy hen posiive shocks. This asymmeric effec is also inerpreed as bad news as i has higher impac on volailiy han good news. To capure his effec, Glosen e 33

34 al. (1993) proposed he GJR GARCH model which has a simple exenion o he sandard GARCH model. The GJR GARCH(1,1) model is specified as follows: h (4.9) 0 i 1 i i I 1 i1 Where 1 1 when and oherwise zero. This is called he leverage effec, I 1 0 because he increase in risk was believed o come from he increased leverage induced by negaive shocks. The GJR GARCH is very similar o he Treshold GARCH (TGARCH) which is he same as he GJR model, bu applied o he sandard deviaions insead : Inegraed GARCH and RiskMerics TM When esimaing he sandard GARCH models, he value of 1 1 is ofen close o he value of 1. If he Inregraed GARCH (IGARCH) of Engle & Bollerslev (1986) is obained. In his model, he volailiy shocks have a permanen effec. Inegraed refers o he fac ha here migh be a uni roo problem which could lead o a saionary version of he sandard GARCH process o be non-exisen. However, his is no he case wih he IGARCH. Bougerol & Picard (199) show ha he IGARCH has a unique sricly saionary soluion. The IGARCH is specified as follows h (4.10) 0 1 i1 1 1 Where J.P. Morgan uses he IGARCH model for heir famous RiskMerics model, because i is acually an IGARCH wihou an inercep. The RiskMerics model uses an exponenial weighed moving average (EWMA). The RiskMerics model is given as (1 ) r (4.11) 1 1 is referred o as he decay facor and assigns he weighs ha are applied o he squared reurns and pas volailiy. I can be seen ha he EWMA used by RiskMerics is in fac he same as an IGARCH where 0 0, 1 (1 ). The RiskMerics, 1 model is an exensively used and empirically validaed model and he IGARCH version is used hroughou he res of he presen research (J. P. Morgan, 1996) : Markov regime swiching model The hird class of models o be inroduced is he Markov Regime Swiching (MRS) model following he idea of Hamilon (1989) who inroduced regime swiching models for financial ime series. To address he issue of volailiy clusering, spikes or drops, and heavy ails, a (MRS) model is used. These models have araced a lo of aenion, and have been applied in large number of areas including speech recogniion, populaion dynamics, river-flow analysis, and raffic models. The models are also very popular and 34

35 are used exensively for modelling elecriciy spo prices. The idea behind he MRS is o represen he observed sochasic behavior of EUA reurns a a specific ime by wo or more separae regimes. The regimes can have differen sochasic processes and he swiching mechanism of he regimes is assumed o be an unobserved Markov chain called R. I is described by he ransiion marix P which conains he probabiliy pij = P(R+1= j R= i) of swiching beween regime i a ime o regime j a ime +1. The presen research uses a sandard GARCH(1,1) model in each regime, bu i is easy o implemen differen GARCH specificaions in each regime. This research will only focus on a wo regime GARCH model, bu i is sraighforward o generalize he model ino hree or more regimes. However, i is imporan o noe ha when expanding he model o include more han wo regimes, a very large sample is required o obain correc parameer esimaes. Wih wo regimes he ransiion marix P is given as: P ( ) p p p (1 q) p p (1 p) q 11 1 pij 1 (4.1) The curren sae R a ime depends only on he pas, hrough he mos recen value R-1, because of he markov propery. The ergodic probabiliy or he uncondiional probabiliy of being in sae s 1 is given by 1 (1 p) / ( p q) (Marcucci, 005). In he mos general form he MRS GARCH model is expressed as: r I ~ (1) f ( ) 1 () f ( ) wih probabiliy p 1, wih probabiliy p1, (1 ) Where f (.) represens he assumed condiional disribuion. In his case, he () i condiional disribuion is assumed o be a normal disribuion. denoes he vecor of parameers in regime i and p 1, Pr[ s 1 I 1 ] is he ex ane probabiliy and I 1 is he informaion se a ime 1. The vecor of ime-varying parameers can be decomposed ino hese hree componens. (i) i i i ( µ, h, v ) (4.13) i i Where µ E( r I 1) is he condiional mean, h Var( r I 1) is he condiional variance, and i v is he shape parameer of he condiional disribuion, which means ha he densiy funcion of r is a locaion-scale wih ime-varying shape parameers (Marcucci, 005; Klaassen, 00; Gray, 1996). 35

36 The MRS-GARCH herefore consiss of four elemens: The condiional mean, he condiional variance, he regime process, and he condiional disribuion, in his case he normal disribuion. The sandard GARCH(1,1) model as given in secion 4.1. can be seen as a single regime GARCH model, herefore he condiional mean and variance equaion are almos he same as in equaion (4.7) and (4.9), bu wih he noaion being a lile differen. In he wo regime swiching GARCH model boh regimes are GARCH(1,1) models, bu wih differen parameers in each regime. Then he mean equaion of he MRS-GARCH is expressed as follows: y S ln µ () i S 1 (4.14) Where i 1, and hz where z is normally disribued wih zero mean and uni variance. The condiional variance given he whole pah of he regime s ( s, s 1,...) is given as h V[ s, I ]. The condiional variance follows a GARCH(1,1) process similar o () i 1 he one used in secion 4.1., bu where he parameers swich, depending on he regime. This yields he following expression for he condiional variance: h h (4.15) ( i) ( i) ( i) () i 0 1 i1 1 1 h is a sae-independen average of pas condiional variances, bu in a regime 1 swiching conex a GARCH model wih sae-dependen pas condiional variance would be infeasible. Insead of depending only on he informaion se I 1 and he curren regime s, which deermines he parameers of he mean and variance, he condiional variance would also depend on all pas saes s 1. This is a problem, because i would require he inegraion over a number of unobserved regime pahs. This would make he regime pahs grow exponenially wih he sample size, which will make he model inracable and herefore almos impossible o esimae (Marcucci, 005, Klaassen, 00, Gray, 1996). A soluion o his problem is proposed in secion : Maximum Likelihood Esimaion The models are esimaed hrough maximum likelihood. The base assumpion and saring poin of Maximum Likelihood Esimaion (MLE) is ha he condiional disribuion of an endogenous variable is known, excep for a finie number of unknown parameers. The idea behind MLE is o find he parameers esimaes ϴ, for an assumed or known probabiliy densiy funcion, ha will maximize he likelihood of having observed he given daa sample (Verbeek, 01). 36

37 4..1: The geomeric Brownian moion In sochasic processes he Markov propery is in general sufficien o wrie he likelihood funcions. Here i is key o noice ha for a Markov process x, he probabiliy densiy of he random sample is wrien as: L( ) = f = X( 0 ),X( 1),...,X( n ); f f f f X( n ) X( n-1 ); X( n-1 ) X( n- ); X( 1) X( 0 ); X( 0 ); (4.16) However, in special cases like he GBM, where he observaions are iid, he problem of finding he likelihood is simplified, since f ( x x ); ( ); i f x i1 and he likelihood, i wihou any condiioning, becomes he probabiliy densiy of each daa poin in he sample. In he case of he GBM, his happens when logarihmic reurns are used. The likelihood funcion for iid variables in general becomes: L( ) f ( x, x,..., x ) f ( x ) 1 n n (4.17) i1 i When maximizing he likelihood funcion, he produc of he densiy values can become so small ha i causes numerical problems. Therefore, he likelihood funcion is usually convered ino he log-likelihood. For he iid case he log-likelihood is n * L f xi i1 ( ) log ( ) (4.18) Usually he maximum of he log-likelihood funcion has o be found numerically hrough an opimizaion algorihm. However, he GBM uses logarihmic reurns where he incremens form normally disribued iid random variables, each wih a well known densiy of f ( x) f ( x; m; v) which is deermined by he mean and variance. This means ha he MLE mehod provides closed form expressions for µ and σ. From equaion (4.3) and (4.4) he mean and variance are known. The probabiliy densiy funcion (pdf) of he normal disribuion is also needed and is given as: m ( )( T ), v ( T ), ( x µ ) 1 pdf ( x, µ, ) e By differeniaing he pdf wih respec o µ and σ, and hen seing he derivaives equal o zero, gives he well known closed form expressions for he sample mean and variance of he reurns: (Brigo e al., 007). n m x / n and v ( x m) / n i1 i n i1 i 37

38 4..: The GARCH model The GARCH model, as given in secion 4.1., is a condiional model and herefore he MLE process is a lile bi differen, hen he case of he GBM. For a condiional variance model, he innovaions are z where z follows a sandard normal disribuion and he innovaion variance follows a GARCH(1,1) model wih condiional variance. Rearranging equaion (4.5) gives he following expression: y µ (4.19) I is assumed ha he values of is normally disribued wih a mean of zero and a consan variance. Then he likelihood of any realizaion of is: L 1 exp (4.0) However, he realizaions of are independen, and herefore he likelihood of he join realizaion of 1,,..., T is he produc of individual likelihoods which is he same as equaion (4.17) and as wih he GBM, i is more convenien o work wih sums insead of producs, so he naural logarihm is used. Subsiuing equaion (4.0) ino equaion (4.18) o obain he following log likelihood funcion for he join realizaions of : T T 1 ln L ln( ) ln ( ) (4.1) T 1 However, he condiional variance of is no consan, bu has he condiional variance h as described in equaion (4.8). I is hen necessary o modify he equaion above, his yields he following resul: (4.) T ln L ln( ) ln( h ) ( / h ) 1 Where y µ and h is one of he GARCH models presened in secion 4.1. (Enders, 010; Lükepohl & Kräzig, 004). 4..3: The MRS GARCH model As menioned in secion 4.1.3, he MRS GARCH model has a problem wih pah dependence ha makes i almos impossible o esimae using rue maximum likelihood. Therefore Bauwens, Preminger & Rombous (010) adap he Paricle Markov Chain Mone Carlo (PMCMC) algorihm of Andrieu, Douce & Holensein (010) 38

39 o deal wih his pah dependence issue. However, he algorihm is beyond he scope of his research and is boh ime consuming and very difficul o implemen. Consequenly, an approximaion o he MLE esimaion is needed. To avoid he problem of he condiional variance o be a funcion of all pas saes, a simplificaion is needed. Cai (1994) and Hamilon & Susmel (1994) were he firs o address his problem of combining regime swiching and ARCH models, hence eliminaing he GARCH erm. Boh Cai (1994) and Hamilion & Susmel (1994) also realize ha many lags are needed for he processes o be sensible. To avoid he problem of pah-dependence, Gray (1996) inegraes ou he unobserved regime pah s 1 in he GARCH erm of (4.15) by using he condiional expecaion of he pas variance. Gray (1996) uses he informaion observed in o inegrae ou he unobserved regime. Klaassen (00) uses he condiional expecaion of he lagged condiional variance, same as Gray (1996), bu wih a broader informaion se han Gray (1996), o inegrae ou he pas regimes by considering he curren one. Klaassen (00) adops he following expression: p q ( i) ( i) ( i) ( i) 0 1 i i1 i1 () i E 1 (4.3) h h s Where he expecaion E 1 () i h 1 s ( i) ( i) ( i) E 1 h 1 s p ii, 1 µ h 1 is: [( ) ] p [( µ ) h ] [ p µ p µ ] ( j) ( j) ( i) ( j) ji, 1 1 ii, 1 1 ji, 1 1 And he probabiliies is compued as: (4.4) p Pr( s j ) p p pji, Pr( s j s 1 i, I 1) Pr( s j ) p ji 1 ji j, 1 1 i, 1 (4.5) Where i, j 1, When performing MLE of MRS-GARCH model, an essenial ingredien, beside he probabiliy densiy funcion, is he probabiliy of being in he firs regime given he informaion se. This is he ex-ane probabiliy a ime given he informaion se in -1 which is compued as: 39

40 p Pr[ s 1 I ] 1, 1 f ( r 1 s 1 )(1 p1, 1) (1 q) f ( r 1 s 1 1) p1, 1 f ( r 1 s 1 )(1 p1, 1) f ( r s 1) p ( 1) ( )(1 ) 1 1 1, 1 p f r 1 s 1 p 1, 1 f r 1 s 1 p 1, 1 (4.6) Where p and q are he ransiion probabiliies as given in equaion (4.1) and f (.) is he likelihood funcion, hen he log-likelihood funcion is expressed as Tw 1, 1 1 1, 1 1 (4.7) Rw1 ln L log[ p f ( r s 1) (1 p ) f ( r s )] Where w , n and he likelihood funcion is he densiy funcion of he normal disribuion which is expressed as: f 1 () i h (. s ) exp i () i h (4.8) Where h is expressed in equaion (4.3) and is obained by rearranging equaion () i (4.14) which yields he following resul: y µ (4.9) () i The main advanage of Klaassens (00) regime swiching GARCH is ha i allows for higher flexibiliy in capuring he persisence of volailiy shock. A shock can be followed by a volaile period, bu wih differen parameers of he wo regimes and he GARCH effecs, he model is able o capure he pressure-relieving effec of large shocks. However, a problem wih he MRS GARCH is he large se of parameers ha needs o be esimaed. The MRS GARCH conains en parameers and herefore i needs a sufficienly large sample o idenify he regimes and esimae he parameers properly (Marcucci, 005; Klaassen, 00; Gray, 006). 4..4: Disribuions As discussed in chaper 3, he disribuion of EUA reurns is no normal disribued as assumed in he secions above, bu have a higher peak and faer ails. To accommodae for hese findings oher disribuions mus be considered when esimaing he models. The disribuions considered, besides he Normal disribuion, is he suden- disribuion and Generalized Error Disribuon (GED). Boh disribuions are symmeric, bu allow for a higher peak and faer ails han he normal disribuion. 40

41 Suden- Disribuion The suden- disribuion is also known as simply he -disribuion. If a random variable is -disribued wih v degrees of freedom ( v ) he densiy funcion of he - disribuion is expressed as v 1 v f ( I, v) v v ( v ) h ( v ) h v v/ v1 v (4.30) Where (.) is he gamma funcion, i1 x i x e dx (), i 0. Densiy funcion for he - 0 disribuion coincides wih he densiy of he normal disribuion when v. Using he same approach as in secion 4.., hen he loglikelihood funcion is found as: T v/ v 1 v ( v ) h v 1 v ln L ln v ln ln v 1 v ( v ) h (4.31) (Enders, 010; Lükepohl & Kräzig, 004) Generalized Error Disribuion The GED wih v degrees of freedom ( v 0 ) has he following densiy funcion: v v v f ( I, v) vexp h h v (4.3) Where is defined as: 1 v 3 v v 1 (4.33) When v, he densiy becomes he same as N(0, h ) and when v he disribuion is lepokuric. Following he same procedure as above he loglikelihood funcion is expressed as: v v T ln L ln v ln ln( h ) 1 h v v (4.34) (Enders, 010; Lükepohl & Kräzig, 004). 41

42 4.3: Forecasing Volailiy The opions analyzed laer in his paper have mauriies up o ¾ of a year and herefore i is very imporan o analyze he volailiy forecasing abiliies of he models. As menioned earlier, he GBM is lognormal disribued wih a consan volailiy, and hen he l ime ahead volailiy forecass is compued as l. The RiskMerics IGARCH also uses his relaion. The one period ahead volailiy forecas of he IGARCH is: h 1 1 1h (4.35) Where I is imporan o remember ha he IGARCH does no have an uncondiional variance and since E{ 1} E{ } and he IGARCH does no conain an inercep hen he l ime ahead variance forecass, according o (J. P. Morgan, 1996) is expressed as: h l lh (4.36) 1 The variance for he sandard GARCH model is dependen on he innovaions and variance of ime. Therefore, he 1-day ahead variance forecas for he sandard GARCH(1,1) is given as: h h (4.37) Noing ha under saionariy wrien as: E{ } E{ } 1 hen he uncondiional variance is (4.38) This expression is he same as obained as follows: (1 ) 0 hen forecas for he nex periods is 1 1 h h h E[ I ] h h h h h h ( 1 1)( h ) ( 1 1)( h 1 ) ( ) h h E[ I ] E[ h I ] h h h ( ) h 0 )( ( 11 h ) 4

43 When performing his derivaion recursively he l sep ahead forecass for he sandard GARCH(1,1) is obained yielding he following resul: l h l ( 11) ( h ) (4.39) The above equaion (4.41) implies ha when h l as l (Verbeek, 01). The same approach is used for he GJR GARCH o obain mulisep ahead forecass. The disribuion of innovaions is assumed o be symmeric, hen he uncondiional variance is expressed as: (4.40) 0 This can also be wrien as. Following he previous derivaion, 1 (1 1 1) hen he mulisep ahead forecass of he GJR GARCH is expressed as: 1 ) l h ( ( ) l 1 1 h (4.41) The improvemens made for he MRS GARCH made by Klaassen (00) allows for a sraighforward expression for he muli-sep-ahead volailiy forecass which is calculaed recursively as wih he GARCH(1,1) model. Then he l - sep ahead volailiy forecass a ime 1is given as: l l () i T, T l T, T Pr( T 1 ) T, T 1 1 i1 (4.4) hˆ hˆ s i I hˆ Where hˆt, T h is he ime T aggregaed volailiy forecass for he l-sep horizon and () h ˆ i TT, is he -sep-ahead volailiy forecas in regime i a ime T, as wih he GARCH model, i is calculaed recursively: ( ) ( ) ( ) ( ) ( ) T, T T T, T T hˆ a ( a ) E hˆ s (4.43) I is seen ha he muli-sep-ahead volailiy forecass are compued as weighed average of he muli-sep-ahead forecass for each regime, where he weighs are equal o he predicion probabiliies. The volailiy forecass of each regime is obained in he same way as for he GARCH(1,1) where he expeced volailiy of he previous period is he volailiy in each regime in he previous period weighed by he probabiliies given in equaion (4.5) (Marcucci, 005; Klaassen, 00). 43

44 4.4: In-sample analysis In his secion, he models are esimaed on he whole sample of 49 observaions. When aemping o compare single regime GARCH ype models wih MRS-GARCH models, a problem arises and sandard economeric ess for he model specificaion may no be appropriae, because some parameers may be insignifican under he null. To avoid his problem Hansen (199) proposes simulaion based ess, bu he focus in his research is on opion pricing and herefore on he predicive abiliy of EUA spo prices which makes i beyond he scope of his research o perform hese ess. Insead hree saisics are presened; he maximized value of he log-likelihood funcion (LLF) and he wo goodness-of-fi saisics, he Aikaike Informaion Crierion (AIC) and he Schwarz s Bayesian informaion Crierion (BIC) are provided as a model selecion crierion (Marcucci, 005; Enders, 010). The AIC and BIC are expressed as follows: AIC LLF k (4.44) BIC LLF k ln( n) (4.45) Where k is he number of parameers in he model and n is he sample size. When comparing models using he AIC and BIC hen he model wih he lowes AIC or BIC is he bes. All of he single regime GARCH models and he MRS GARCH models are esimaed using boh normal, suden-, and GED disribuions. Resuls of he esimaion on single regime GARCH models are in able 4.1, and resuls from he MRS-GARCH models are in able 4.. Table 4.1:Esimaed parameers, Log-likelihood, AIC and BIC from he single regime GARCH models using Normal, Suden, and GED disribuions. T-saisics using asympoic sandard errors in he parenhesis 44

45 Table 4.1 shows ha all of he single regime GARCH models ouperform he GBM and is consan volailiy. Table 4.1 also clearly shows ha he daa are no normally disribued, boh he suden- and he GED conribue significanly o he in-sample performance of all of he models, bu he GED is jus a iny bi beer han he - disribuion. For all of he models, he variance is highly persisen and a problem arises wih he suden- disribuion where he condiion for he GARCH and for he GJR, is no saisfied, making he processes non-saionary and explosive. This means ha if he reurn process is esimaed using he suden- disribuion, i is necessary o use he IGARCH models, because i forces saionariy. When comparing he individual models, he leverage parameers of he GJR GARCH do no conribue o a beer in-sample fi and he leverage parameers are insignifican for all of he disribuions. The insignificance of he leverage parameers means ha negaive reurns do no have a larger impac on volailiy han posiive reurns. When comparing he models using he AIC and BIC, he GARCH wih GED innovaions has he larges AIC and he IGARCH wih GED innovaion has he larges BIC, because he BIC gives a larger penaly on he number of parameers. Table 4. Esimaed parameers, Log-likelihood, AIC, and BIC from he Markov Regime Swiching GARCH models using Normal, Suden, and GED disribuions. T-saisics using asympoic sandard errors in he parenhesis 45

46 Table 4. shows ha he suden- and GED disribuions do no conribue as much as wih he single regime GARCH models. Furhermore, i is imporan o noice ha he parameer Q is insignifican and ha he loglikelihood for boh he suden- and GED is almos he same as wih he single regime GARCH models which quesions he significance of a second regime under hese wo disribuions. However, he second regime is clearly significan under he normal disribuion Overall, he bes model according o he loglikelihood is he MRS GARCH wih GED innovaions, bu i is only marginally beer han he single regime GARCH wih GED innovaions, which is also he bes model according o he AIC. The MRS GARCH models uses a large se of parameers which causes models o perform badly, according o he BIC. On he oher hand, he IGARCH wih suden- disribued innovaions and is hree parameers is he bes performing model according o he BIC. This shows ha he added complexiy by adding oher disribuions han he normal disribuion for he single regime GARCH models and adding a second regime for he normal disribuion conribues significanly for modelling EUA reurns. 4.5: Ou-of-sample analysis The GBM, GARCH, GJR, IGARCH, and MRS-GARCH models have been developed using boh normal, suden-, and GED disribuions. The in-sample analysis showed ha he MRS-GARCH and single regime GARCH wih GED innovaions had he bes in-sample fi. However, for opion pricing, he models are used for forecasing and especially he volailiy has a huge impac on opion price. Therefore, an ou-of-sample volailiy forecasing analysis is performed in order o asses forecasing performance of each model. The volailiy forecas of ineres is he forecas of he EUA spo volailiy over a l -day horizon. Four horizons are evaluaed which is a one day ( l 1), one week ( l 5 ), wo weeks ( l 10 ) and one monh ( l ). To improve he generaliy, an exensive ou-ofsample period is needed. Therefore, he sample of 49 observaions is spli ino wo wih 80 observaion in he firs sample which is used as an esimaion period and 149 observaions in he second sample, which is used for he ou-of-sample analysis. A rolling window approach is used where he model is re-esimaed and volailiy forecass V 1 s, l are generaed for each observaion. To assess he performance of he volailiy, a measure of he realized volailiy is needed. Since V 1 s, h E 1 ( s, h l µ ) squared change, i would be obvious o use he square roo of ( s l µ ) h, as a measure of he realized volailiy. However, Andersen & Bollerslev (1998) argue ha he squared change is an unbiased, bu noisy indicaor of he realized volailiy and insead propose reducing he noise by aking he sum of squared inra-period changes. Since he sample consiss of daily daa, he 46

47 l realized volailiy is calculaed as v, l (r i µ ˆ) where ˆµ is he mean of he reurns. i1 However he reurns are expeced o have zero mean. Therefore, he realized volailiy is calculaed as v, l l (r i ) i1 To evaluae he volailiy forecasing performance, he mean absolue error (MAE) and he roo mean squared error (RMSE) are used. The MAE and RMSE for he l-sep-ahead forecas are calculaed as: 1 MAE n v n (4.46) N ( ) ˆ, l ( ) N 1 1 RMSE( n) [ v ( n)] N Where N ˆ, l (4.47) 1 ˆ is he volailiy forecas. The single regime GARCH and GJR GARCH, wih suden- innovaions, showed non-saionariy in he in-sample analysis making he reurn processes explosive and unpredicable. Therefore, hese wo models are no used in he ou-of-sample analysis. Furhermore, when conducing he ou-of-sample analysis on he GJR-GARCH wih GED innovaions, a large amoun of he parameers obained leads he model o become non-saionary which also leads his model o be excluded from he ou-of-sample analysis. The resuls of he ou of sample analysis is in able 4.. One Day One Week Two Weeks One Monh MAE RMSE MAE RMSE MAE RMSE MAE RMSE GBM 0,0567 0, ,0700 0, , , , ,18739 GARCH-N 0, , , , , , , ,11947 GARCH-GED 0, , , , , , , ,1171 GJR-N 0, ,094 0, , , , ,0979 0,11933 IGARCH-N 0,0178 0,0840 0,076 0,0440 0,0340 0, , ,06573 IGARCH-T 0, ,0831 0,079 0,0435 0, ,0503 0, ,06531 IGARCH-GED 0, ,084 0,0758 0,0436 0, , , ,0658 MRS GARCH-N 0, ,090 0, , , , ,0939 0,10347 MRS GARCH-T 0,0569 0,0347 0, , , , , ,1984 MRS GARCH -GED 0,0856 0, , , ,145 0,3051 0,36 0,40986 Table 4.: Ou-of-sample volailiy forecasing performance From able 4. i is clear ha all of he single regime GARCH models ouperform he GBM on all horizons on boh MAE and RMSE. However, he regime swiching models wih GED and suden innovaions have rouble ouperforming he GBM on all 47

48 horizons. The regime swiching GARCH model wih normal innovaions is he only model ha ouperform he GBM on all horizons. When comparing he single regime GARCH models wih he regime swiching GARCH models, hen he single regime GARCH models ouperform he MRS GARCH models on all horizons. Only he MRS GARCH wih normal innovaions ouperform he single regime GARCH and GJR GARCH models wih normal innovaions, according o he RMSE, his leads o conclude ha he some of he errors of he GARCH and GJR GARCH models are larger, because RMSE gives a larger penaly o larger errors. The leverage feaure of he GJR model wih normal innovaions also seem o add a lile bi of explanaory power, because i ouperforms he sandard GARCH model wih normal innovaions on all horizons and boh MAE and RMSE, exep on MAE on he one monh horizon. Table 4. also shows ha performance of he models increase a lile bi when using oher disribuions han he normal disribuion. For he sandard GARCH models, he GED disribuion ouperform he normal disribuion on all horizon, han he one day horizon. When comparing he IGARCH models he difference beween he differen disribuions is very lile he normal disribuion only ouperforms he oher disribuions on he wo weeks RMSE. The MRS GARCH models wih suden- and GED innovaions perform very poorly and is ouperformed by all of he single regime GARCH models and he MRS GARCH wih normal disribued innovaions. This furher indicaes he insignificance of he second regime when using suden- and GED disribuions. Overall, he bes performing model is he IGARCH model. As menioned before he inclusion of oher disribuions hen he normal disribuion only helps wih very lile explanaory power when forecasing volailiy. This indicaes ha he added complexiy of MRS GARCH model and he added complexiy of oher disribuion han he normal disribuion do no conribue significan o he forecasing performance of EUA reurn volailiy. The economeric analysis in chaper 3 highlighed he main feaures and characerisics of he reurns and price processes. Based in he economeric analysis he GBM, GARCH, GJR GARCH, IGARCH and MRS GARCH models was chosen as he models for pricing opions on EUA fuures. The Models was presened and a framework for esimaing and forecasing volailiy was developed, using normal, suden- and GED disribued innovaions. An in-sample analysis was performed on he full sample where he MRS GARCH models had problems ouperforming he single regime GARCH models. The MRS GARCH wih GED innovaions only marginally ouperformed he single regime GARCH wih GED innovaions, which on he oherhand had he larges AIC, while according he IGARCH wih GED innovaions was he bes model according o he BIC. The in-sample analysis also quesioned he presence of wo regimes under he suden- and GED. An 48

49 ou-of-sample analysis was performed showing ha he IGARCH model was he bes a forecasing volailiy and he assumed disribuion did no have a significan effec on he forecasing abiliy. The ou-of-sample analysis also quesioned he applicabiliy of he MRS GARCH wih suden- and GED innovaions. These findings conribues furher and suppors he findings of Benz & Trück (009) and Byun (013) who finds ha an GARCH ype model and regime swiching models ouperform he consan volailiy and he added complexiy of hese models is jusified and should be implemened. 49

50 Chaper 5: Pricing Opions On Fuures I is naural o ask why he people choose o rade opions on fuures insead of opions on he underlying. The reason is ha, in many circumsances, he marke for fuures is more liquid han he underlying iself which is also observed in he allowance marke. EUA opions have been available on he exchanges since 006 where he underlying of hese conracs is eiher he monhly or he December EUA fuures. Opion volumes have been increasing, bu sill remains a low levels which may be a reason for he sparse lieraure on his subjec. In his secion, he models inroduced in chaper four is implemened for he pricing of opions on fuures. The frameworks for pricing opions on fuures are presened and since here does no exis any closed form soluion for he pricing of opions wih GARCH and MRS GARCH models, a Mone Carlo framework is consequenly developed in order o approximae opion prices. A he end of his chaper, he models is esed on acual raded opions on EUA fuures, and commenaries and conclusion on he bes model for pricing opions on EUA fuures are provided. 5.1: Opion pricing using Black-Scholes, single regime GARCH, and MRS GARCH models When i comes o opion pricing, here is a huge amoun of lieraure available. Fischer Black, Myron Scholes, and Rober Meron, who were awarded wih he Nobel Prize in economics for heir work in pricing and hedging derivaive, have done he mos famous work in his area. They developed he Black-Scholes-Meron model ha is based on he Geomeric Brownian moion. Their work spawned an immense amoun of research ino he subjec of opion and derivaives pricing (Hull, 01; Willmo, 007) 5.1.1: Opion pricing using Black-scholes framework The opion pricing using he GBM is done hrough Black s (1976) model which is based on he Black-Scholes-Meron framework. The difference beween Black s model and he Black-Scholes-Meron model is ha Black s model is specifically designed for he pricing of opions on fuures. A very good feaure wih Black-Scholes-Meron and Black s model is ha hey provide a closed form soluion for he opion price. An observaion abou he risk-neural world is ha he fuures price behaves he same way as a sock paying dividend yield a he risk free rae. To formally prove his observaion, he drif of a fuures price in a risk-neural world is calculaed. The fuures price a ime is defined as F and he selemen daes are supposed o be a imes 0,,.. han he value of a fuures conrac a ime is equal o zero. A ime, he fuures conrac provides a payoff of F F0 and if r is he very shor erm ineres rae is 0, hen he value of he fuures conrac hrough risk-neural valuaion is 50

51 r e E( F F ) 0 (5.1) Where E denoes he expecaions in a risk-neural world. Therefore, we have ha r e E( F F ) 0 0 (5.) Showing ha E( F ) F, similar 0 E( F ) F and puing many resuls like hese yields, he following expression for any ime T E( FT ) F (5.3) 0 Then he drif of a fuures conrac in a risk-neural world is zero and when using he usual assumpions made for he GBM, he process for he fuures price in a risk-neural world is df FdW (5.4) Black s model is used as benchmark for pricing opions on EUA fuures, because i is widely acceped and implemened. Black s model yields a closed form soluion of boh call and pu opions wih srike price K and expiry a ime T. The closed form soluion is rt c e [ F N( d ) KN( d )] (5.5) 0 1 rt p e [ KN( d ) F N( d )] (5.6) Where d d ln( Fo / K) T / (5.7) T ln( Fo / K) T / d1 T (5.8) T For an in-deph explanaion and derivaion of he closed form soluion o Black s model, see Hull (01). 5.1.: GARCH opion pricing The GARCH process and is varians have gained increased aenion since Bollerslev (1986) firs inroduced he model. Today, i is a very prominen model for modelling financial ime series. Chaper 4 shows ha GARCH processes are beer han he GBM a modelling EUA reurns and beer han MRS GARCH and GBM a forecasing volailiy. The increased performance of he GARCH in modelling reurns spawned a lo of research in using he GARCH process o sudy opions. The firs aemp o develop a rigorous heoreical framework was Duan (1990), however he risk-neural valuaion relaion was incorrecly applied. Sachell & Timmermann (199) and Amin & Ng (1993) boh 51

52 proposed opion pricing models using he GARCH framework, bu boh resuls invalidaed he risk-neural validaion relaionship. Duan (1995) was firs o develop a valid heoreical framework for pricing opions on asses whose coninuous compounded reurns follow a GARCH process. Duan, Gauhier & Simonao (1999) provide an analyical approximaion for opion prices following Duan s (1995) model. The approximaion performs very well for shor-mauriy opions, bu is less reliable for long-mauriy opions. Heson & Nandi (000) developed a closed-form soluion for pricing opions in he GARCH framework using an affine dynamic specifically designed o yield he closed-form soluion. Hsieh & Richken (005) compare Duan s (1995) model wih Heson & Nandi s (000) closed-form soluion, and find ha Duan s (1995) model is superior in removing biases from pricing residuals for all moneyness and mauriy caegories. Therefore, he presen research will use he mehodology Duan s (1995) for pricing opions using he GARCH framework. Three disinc feaures characerize he GARCH opion-pricing model. Firs, he GARCH opion prices is a funcion of he embedded risk premium in he underlying asse. Second, he GARCH opion pricing model is non-markovian, excep for he GARCH(1,0) or ARCH(1). Third, he GARCH opion pricing model is able o explain some welldocumened sysemic biases such as underpricing of ou-of-he-money opions, underpricing of low-volailiy models, underpricing of shor-mauriy opions, and he implied volailiy smile. Duan s (1995) GARCH opion pricing model consider a discreeime economy where he reurns follow he following process S 1 ln r h h S1 (5.9) Where S is he asse price a ime, r is he consan one period risk-free rae, is he consan uni risk premium, and I N(0, h ) and i1 1 1 follows a GARCH(p,q) process where h h (5.10) I is he informaion se, p 0, q 0, a0 0, a1 0, i1,..., q; 0, i 1,..., q. The opion pricing resuls rely on condiional normaliy in order o develop an opion pricing model. Furhermore, i is assumed ha he GARCH(p,q) process is less han 1 o ensure covariance saionariy. If p=0 and q=0 hen he GARCH process specified above reduces o he sandard homoscedasic lognormal process in he Black-Scholes-Meron model, hence he Black-Scholes-Meron model is a special case of he GARCH opion pricing model. i 5

53 As previously menioned, he risk-neural valuaion relaion has been a problem in previous sudies of he GARCH opion pricing model. Duan (1995) generalizes he convenional risk-neural valuaion relaionship o accommodae for heeroskedasiciy of he asse reurn process. Duan (1995) provides he firs risk-neural valuaion relaionship (RNVR) analysis of GARCH processes. The analysis is buil upon he works of Rubinsein (1976) and Brennan (1979), and characerize sufficien condiions in a represenaive agen economy which ensures he exisence of RNVR, which is referred o as a local RNVR. Duan s analysis is based on he exisence of a represenaive agen wih consan relaive or absolue relaive risk aversion. Duan (1995) analyses he case of a condiional normally disribued variable and exends his framework o include non-normal innovaions such as he inverse normal and GED disribuions, in Duan (1999). The presen research will only cover he case of condiional normaliy, because of he problems wih nonsaionariy and he insignificance of a second regime when using suden- and GED disribuions. Insead of characerizing a represenaive agen economy o obain opion prices as done in Duan (1995). Chrisoffersen e al. (01) specify a class of Radon- Nikodym (RN) derivaives o derive resricions ha ensure he exisence of an equivalen maringale measure (EMM) which makes he discouned sock price process a maringale. Chrisoffersen e al. (01) arrive a he same resul as Duan (1995), bu i is required o be familiar wih Radon-Nikodym derivaives which is why he presen research prove he risk-neural GARCH process as done in Duan (1995). Since S / S 1 I 1 as S ln v S1 disribues log-normally under measure Q, equaion (5.9) is wrien (5.11) Where v is he condiional mean and is a Q-normal random variable. has zero condiional mean and an undeermined variance. Firs, is i necessary o prove ha 1 v r h Q S Q v E I 1 E ( e I 1) e S1 h v (5.1) Where h by he local RVNR is P S Q S h Var ln I 1 Var ln I 1 S 1 S 1 (5.13) 53

54 Q S r 1 The local RVNR saes ha E I 1 e and hen i follows ha v r h. Now S1 he only hing needed o be proved in-order o obain he risk-neural GARCH process, is o prove ha h under local RVNR is equal o h ( h ) h (5.14) When seing he previous resul equal o equaion (5.9) o obain he following expression 1 1 r h h r h (5.15) This implies ha h and when subsiuing his expression ino he sandard GARCH(1,1) in equaion (5.10), he following expression is found for condiional variance under local RNVR h 01( h ) 1h 1 (5.16) The same procedure is used for he GJR and he IGARCH models presened in chaper 4, hen h becomes h ( h ) I ( h ) (5.17) 0 i 1 1 i i h (1 )( h ) h (5.18) 1 1 For he GJR GARCH and IGARCH respecively To price coningen payoffs, i is required o emporal aggregae one-period reurns o arrive a he random erminal price a some fuure poin in ime. I follows from equaion (5.17) ha he erminal price S T a mauriy T is T T 1 ST S exp (T ) r hs s s1 s1 (5.19) Then European call and pu opion prices are obained hrough he following expressions r(t) * C e E [max(s T K,0)] (5.0) P E K S T r(t) * e [max(,0)] (5.1) 54

55 The condiional disribuion given oday s price S T canno be deermined analyically, bu has o be approximaed using Mone Carlo simulaion. A Mone Carlo framework for pricing opions is developed and discussed in secion : MRS-GARCH opion pricing The increased amoun of research ino GARCH opion pricing models lays he foundaion for research for MRS-GARCH opion pricing models. Duan e al. (00) develop a framework for pricing opions using MRS-GARCH models. The framework of Duan e al. (00) expands he findings of Duan (1995) o include a regime swiching effec. However, Duan e al. (00) does no allow for he GARCH process o swich beween regimes, bu only allow for volailiy levels o changes. Chen & Hung (010) develop a framework for pricing opions using MRS GARCH wih a laice algorihm, and he GARCH process o swich beween regimes. Ellio, Siu & Chan (005) develop an opion pricing formula using a MRS version of he Heson-Nandi GARCH opion pricing model, bu Duan s (1995) framework has proven o be superior compared o Heson and Nandi s (010) model. Therefore, he MRS-GARCH opion-pricing framework developed in his research is a MRS version of Duan s (1995) opion pricing model, discussed in secion 5.1., developed by Duan e al. (00). For pricing opions in a MRS GARCH framework, Duan s (1995) LRNVR is exended o allow for all of he GARCH parameers o depend on he regime, hence a 0, a and 1 1 changes depending on he regime. The MRS-GARCH allows differen regimes o have differen GARCH behaviors and herefore differen volailiy srucures. As menioned in chaper 4, only a model wih wo regimes is implemened in his research, bu he model is easily exended o include more regimes. When he number of regimes is se o one, he model is idenical o he above sudied GARCH opion pricing. Following he mehodology provided by Chen & Hung (010) and he framework developed in he previous secion, hen he wo regime swiching risk-neural GARCH process is expressed as S 1 () i ln r h S1 (5.) Where S is he asse price a ime, r is he one period risk-free ineres rae, is he consan uni risk premium, i 1, is he regime, and follows a GARCH(1,1) process where I 1 N(0, h ) and h ( i) ( i) ( i) ( i) ( i) 0 1 ( 1 h ) i h 1 (5.3) 55

56 As wih he single regime GARCH models, i is necessary o emporal aggregae oneperiod reurns as done in equaion (5.19) o obain a erminal price o obain payoffs. However, i is imporan o esablish how he variable h is calculaed. h is, as when forecasing volailiy, calculaed as a weighed sum of he volailiy of each regime, hen he volailiy of each sep is calculaed as p1 h (1 p1 ) h (5.4) 1,, Where p 1 is calculaed as in equaion (4.5). Call and pu opion prices are obained he same way as in equaion (5.0) and (5.1). 5.: Mone Carlo simulaion Mone Carlo mehods are used exensively in finance for approximae prices and hedge derivaive insrumens when closed form soluions are no available. Mone Carlo simulaion is a procedure for sampling random oucomes of a process o obain numerical resuls. Mone Carlo simulaion akes advanage of he law of large numbers, which was presened in chaper one, by simulaing a sochasic process a large amoun of imes and compuing he average of he oucomes of he sochasic process. In is basic form, Mone Carlo simulaion is flexible, easy o implemen and work wih, while yielding an inuiive resul. However, when applied in is basic form i is compuaional heavy and compuaions ake a lo of ime which makes i an undesirable mehod for eniies who need quick resuls. However, numerical mehods for decreasing he number of ieraions and increasing he compuaion ime exis. Since ime is no a consrain for he presen research, and analyical soluions for he GARCH and MRS GARCH opion pricing models do no exis, Mone Carlo is an excellen ool for obaining numerical resuls from he GARCH and MRS GARCH models. Mone Carlo simulaion is especially usefull when pricing pah-dependen asse prices or opion-payoffs as wih he GARCH or MRS GARCH. Furhermore, Chrisoffersen e al. (01) finds ha Mone Carlo mehods for GARCH opion pricing models are reliable. Therefore, Mone Carlo simulaion is implemened for pricing opions hrough GARCH and MRS GARCH models. Opions is priced wih Mone Carlo simulaion by simulaing he risk-neural process a large amoun imes o obain a large amoun of payoffs, hen discouning he mean payoff wih he risk-free ineres rae o obain he opion price. Through Mone Carlo simulaion, he opion price for call and pu opion is expressed as follows 1 C e [max(s exp( K),0)] (5.5) T r(t) R * i, MC 1 1 Pe [max(k S exp( ),0)] (5.6) T r(t) * Ri, MC 1 56

57 Where r is he risk-free ineres rae, MC is he number of simulaions, K is he srike price, and R * i, denoes he fuure daily log-reurn under he risk-neural measure. The approximaion of he rue opion prices, hrough Mone Carlo simulaion, is a good one as long as he number of simulaions is large enough (Hull, 01; Wilmo. 007; Chrisoffersen e al., 01). 5.3: Empirical resuls To es he empirical validiy of he hree proposed opion pricing models, opion prices are obained hrough he hree opion pricing models, and are compared o acual raded opions on EUA fuures. The mean absolue percenage error (MAPE) is calculaed o assess he performance of he five models. MAPE is compued as follows 1 n Ai Pi MAPE n (5.7) A i1 Where n is he number of opions, price. i A i is he acual price of he opion, and P i is he heoreical Due o daa limiaions, an analysis across moneyness levels is no performed. Furhermore, i is imporan o noe ha a large par of he opions in he daase is ouof-he-money opions, and only a very limied amoun of opions can be declared as inhe-money. The opion daa sample consiss of prices on 140 call and 77 pu opions oaling of 17 which were raded on he ICE beween and The mauriy of he opion varies beween 0, 0,6 years, which is a fairly long mauriy. The long mauriies are due o he fac ha EUA fuures opions are only acively raded on he December fuures. This means ha he opions expire on The parameers of each model is re-esimaed for each opion, using daa unil he dae of he opion and simulaions are used for he Mone Carlo simulaion. Table 5.1 presens he resuls of he empirical sudy. Table 5.1: Opion pricing accuracy All of he single regime GARCH and he MRS GARCH models ouperform Black s model. Of he single regime GARCH ype models, he sandard GARCH model is performing bes. However, he bes model for pricing opions on EUA fuures is by far he MRS GARCH 57