analysis of carbon market price behaviour · by hailong bao this thesis focuses on the european...

Universiteit van Amsterdam

Master Thesis

Analysis of Carbon Market PriceBehaviour

Author:Hailong [email protected]

Supervisors:dr.P.J.C. Spreij

Universiteit van Amsterdamdr.M.F.M. Nuijens

Statkraft Markets B.V.

A thesis submitted in fulfillment of the requirementsfor the degree of Master of Science

August 2013

“If you look for truth, you may find comfort in the end; if you look for comfort you willnot get either comfort or truth only soft soap and wishful thinking to begin, and in theend, despair.”

- C. S. Lewis

UNIVERSITEIT VAN AMSTERDAM

AbstractFaculteit der Natuurwetenschappen, Wiskunde en Informatica

Korteweg de-Vries Instituut voor Wiskunde

Master of Science

Analysis of Carbon Market Price Behaviour

by Hailong Bao

This thesis focuses on the European emissions market, the EU ETS, and performs acomprehensive analysis to the EUA prices. First, the structure of the EUA market isintroduced; then we analyze the EUA end-of-day spot price and real-time price usingsome quantitative methods. A distribution analysis is done, and we use a changepointanalysis to explore any distributional heterogeneity in the price curve. A simple jumpdetection procedure follows. A key result is that the EUA spot price can be decomposedinto two parts: a diffusion part which resembles a white noise, and a jump part whichcan be linked to influential political news in the market.

http://www.uva.nl/

Faculty Web Site URL Here (include http://)

http://www.science.uva.nl/math/

Acknowledgements

Until the beginning of 2013, I was still a “pure” student. Every day I would go backand forth between my home and the faculty building in Science Park Amsterdam, andI didn’t have a clear idea what I would be doing for my master thesis. Nor did I knowexactly what I would do after graduation. One day Peter Spreij told me that I mighthave a chance to work as an intern in Statkraft Markets B.V., and I would then neverimagine what I could have learned and achieved in the 19th-floor office. Without theopportunity I might probably not have heard of the emissions trading market; I evendidn’t know about Statkraft then!The first time I met Alexis Gillett and Misja Nuyens was in late January, at Dicky’sCafe. I was quite nervous at that time; I was even more nervous during the followingformal interview! Anyway, I didn’t let the chance slip through my fingers. My firstthanks thus go to Alexis and Misja, who have been really patient and tolerant to me,and are always there to give me guidance. They have taught me far more than merelyhow they do business.I would also express my gratitude to the whole Global Carbon team in the StatkraftAmsterdam office. They have treated me as a full member of the team, and we havespent quite some unforgetable moments together which I would definitely cherish in mymind. In particular I thank Arjan Karreman, Eric Boonman, James Liu, Koen De-jonghe, Mike Bluett, Peter Sprengers, Scott Zuercher, Stef Peters and Wim Luyckx.They never hesitated to offer help when asked.My thanks also go to the rest of the Statkraft Amsterdam office. Special thanks go toHao Wen, who helpt me so much both in and out of the office; to Vinh Thinh Tran,with whom I go back home by train together. Also, I would thank Marieke Sniedersand Peter Verver, who have spent some effort sorting out a solution to my travel costreimbursement; Sylvia Lof, who amongst other things prepares lunch for us day afterday; Lodewijk Antonides, with whom around we never feel unsafe with our IT facilities(the feeling of insecurity about the IT system can never be totally eliminated, though).I would, of course, sincerely say thanks to Peter Spreij, my supervisor at the Universityof Amsterdam. I thank him for giving us splendid and extremely beneficial lectures aswell as the lecture notes, although I didn’t quite enjoy the time when I spent dozensof hours in the library squeezing my mind just for a solution to his homework. I alsoappreciate his recommendations for me, without which it could be much harder for meto find a good PhD position.Finally, thanks to my parents. They always give me support for any decisions I’ve made,and they never shed a tear in front of me when we say farewell at the train station inmy hometown. I hope they could know how much I love them.

I love the days and nights in Amsterdam,

Hailong Bao

iv

Contents

Abstract iii

Acknowledgements iv

List of Figures ix

Abbreviations xi

1 EU ETS: An Overview 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Emissions Trading: An Economic Incentive . . . . . . . . . . . . . . . . . 2

1.3 The Linking Directive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4 Emissions Trading Market . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.5 Trading Periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.6 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 The EUA Secondary Market 7

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 EUA Futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 EUA Forward Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.1 Implied Interest By EUA Forward Curve . . . . . . . . . . . . . . 8

2.3.2 Comparison To The Market Interest Rates . . . . . . . . . . . . . 10

2.4 Price Drivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5 EUA European Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5.1 Exchange-traded Options . . . . . . . . . . . . . . . . . . . . . . . 12

2.5.2 Black-Scholes Formula For ETOs . . . . . . . . . . . . . . . . . . . 12

2.5.3 Options Pricing: A Mathematical View . . . . . . . . . . . . . . . 14

2.5.4 Implied Volatility Surface . . . . . . . . . . . . . . . . . . . . . . . 15

2.5.5 ATM Implied Volatility . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5.6 Is Options Liquidity A Problem? . . . . . . . . . . . . . . . . . . 16

2.6 Other EUA Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.6.1 Forwards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.6.2 Time Spreads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.7 Comparison To Other Financial Markets . . . . . . . . . . . . . . . . . . . 18

v

Contents vi

3 The EUA Auction Market 21

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.1 Auction Venues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.2 Auction Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.1.3 Published Information . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.1.4 A Generic Bidding Strategy . . . . . . . . . . . . . . . . . . . . . . 23

3.2 Market Participation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3 A Nonparametric Analysis: Exploiting The Published Information . . . . 24

3.3.1 Statistics Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3.2 Relationship Among PA, Pmed, Pµ . . . . . . . . . . . . . . . . . . 27

3.3.3 Market Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3.4 A Way Of Illustration . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3.5 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4 Beyond The Nonparametric Analysis . . . . . . . . . . . . . . . . . . . . . 31

4 Carbon Price Data: A Statistical Analysis 33

4.1 Unevenly-Spaced Time Series . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.1.1 A Smoothing Procedure . . . . . . . . . . . . . . . . . . . . . . . . 35

4.1.2 Implementation And Examples . . . . . . . . . . . . . . . . . . . . 36

4.2 Distribution Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2.1 Kolmogorov-Smirnov Test And Jarque-Bera Test . . . . . . . . . . 39

4.2.1.1 Kolmogorov-Smirnov Test . . . . . . . . . . . . . . . . . . 39

4.2.1.2 Jarque-Bera Test . . . . . . . . . . . . . . . . . . . . . . . 41

4.2.2 Tick Data: Deals Arrival Time . . . . . . . . . . . . . . . . . . . . 42

4.2.2.1 Maximum Likelihood Fitting: Results . . . . . . . . . . . 44

4.2.3 Tick Data: Price Changes . . . . . . . . . . . . . . . . . . . . . . . 45

4.2.3.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2.4 EOD Settlement Prices: Price Changes . . . . . . . . . . . . . . . 46

4.2.5 Auction Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.3 Changepoint Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.3.1 A Simple Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.3.2 Single-Changepoint Case . . . . . . . . . . . . . . . . . . . . . . . 52

4.3.2.1 Parametric Methods: Changes In Mean/Variance . . . . 53

4.3.2.2 Parametric Methods: Examples . . . . . . . . . . . . . . 55

4.3.2.3 Non-Parametric Methods: Examples . . . . . . . . . . . . 57

4.3.3 Multi-Changepoint Case . . . . . . . . . . . . . . . . . . . . . . . . 59

4.4 Jump Detection: A Non-Parametric Methodology . . . . . . . . . . . . . . 59

4.4.1 Static vs. Dynamic Filtering . . . . . . . . . . . . . . . . . . . . . 61

4.4.2 Implementation And Results . . . . . . . . . . . . . . . . . . . . . 62

4.4.2.1 Jump Filtering: Tick Data . . . . . . . . . . . . . . . . . 62

4.4.2.2 Jump Filtering: EOD Price . . . . . . . . . . . . . . . . . 63

4.4.3 The Structure Of The AR Series . . . . . . . . . . . . . . . . . . . 66

4.4.3.1 Detected Jumps In AR Series . . . . . . . . . . . . . . . . 66

4.4.3.2 The Filtered AR Series . . . . . . . . . . . . . . . . . . . 68

4.4.3.3 Comments On Modelling The EUA Spot Price . . . . . . 69

Contents vii

A Sample Prices In Other Markets 71

A.1 Stock Price: ING Groep N.V. . . . . . . . . . . . . . . . . . . . . . . . . 72

A.2 Stock Index: S&P 500 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

A.3 Energy Price: Brent Crude Oil . . . . . . . . . . . . . . . . . . . . . . . . 74

Bibliography 75

List of Figures

1.1 Micro-Mechanism Of Emissions Trading . . . . . . . . . . . . . . . . . . . 2

1.2 Phase 3 Market Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 EUA Spot Price, 04/2009 - 06/2013 . . . . . . . . . . . . . . . . . . . . . 4

2.1 EUA Forward Curve, 28/03/2013 . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 EUA Implied Rate vs. Risk-free Rate, 28/03/2013 . . . . . . . . . . . . . 10

2.3 Daily EUA Call Option Data from ICE, 07/05/2013 . . . . . . . . . . . . 14

2.4 Dec-13 EUA Call, Price & Volatility, 07/05/2013 . . . . . . . . . . . . . . 15

2.5 Implied Volatility Surface, Dec-13 EUA Call, 07/05/2013 . . . . . . . . . 16

2.6 ATM Implied Volatility, 07/05/2013 . . . . . . . . . . . . . . . . . . . . . 17

3.1 The Mechanism To Decide The Auction Price . . . . . . . . . . . . . . . . 22

3.2 Number of Bidders, 01/2013 - 06/2013 . . . . . . . . . . . . . . . . . . . . 24

3.3 Auction Volume, 01/2013 - 06/2013 . . . . . . . . . . . . . . . . . . . . . 24

3.4 Daily Changes: Pmin/Pmax vs. PF . . . . . . . . . . . . . . . . . . . . . . 26

3.5 Daily Changes: Pmed/Pµ vs. PF . . . . . . . . . . . . . . . . . . . . . . . 26

3.6 Pmed & Pµ vs. PF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.7 Chart Template . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.8 Auction On 18/04/2013 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.1 ICE Dec-13 EUA Tick Data, 05/04/2013 . . . . . . . . . . . . . . . . . . . 33

4.2 ICE Spot EUA EOD Prices, 01/01/2013 - 21/06/2013 . . . . . . . . . . . 34

4.3 Candlestick Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.4 Dec-13 EUA Intraday Averaged Price, s=30 min, 05/04/2013 . . . . . . . 37

4.5 Dec-13 EUA Intraday Averaged Price, s=15 min, 05/04/2013 . . . . . . . 37

4.6 Dec-13 EUA Intraday Averaged Price, s=5 min, 05/04/2013 . . . . . . . . 37

4.7 Dec-13 EUA Intraday Total Volume, s=30 min, 05/04/2013 . . . . . . . . 38

4.8 Dec-13 EUA Intraday Total Volume, s=15 min, 05/04/2013 . . . . . . . . 38

4.9 Dec-13 Intraday Total Volume, s=5 min, 05/04/2013 . . . . . . . . . . . . 38

4.10 Deals Arrival Time Histogram . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.11 Deals Arrival Time Histogram In log10 Scale, Partial . . . . . . . . . . . . 42

4.12 QQ-Plot, Deals Arrival Time vs. A Sample From The Estimated Distri-bution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.13 Tick Data Price Changes, in log10 scale . . . . . . . . . . . . . . . . . . . 45

4.14 Abs. Price Changes, 04/2009 - 06/2013 . . . . . . . . . . . . . . . . . . . 47

4.15 Log Returns, 04/2009 - 06/2013 . . . . . . . . . . . . . . . . . . . . . . . . 47

4.16 Abs. Price Changes QQ-Plot, 04/2009 - 06/2013 . . . . . . . . . . . . . . 47

4.17 Log Returns QQ-Plot, 04/2009 - 06/2013 . . . . . . . . . . . . . . . . . . 48

ix

List of Figures x

4.18 Auction Deviation QQ-Plot . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.19 QQ-Plot, AD vs. Rounded Normal Random Series . . . . . . . . . . . . . 50

4.20 Log Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.21 Change-In-Variance Detection, AR . . . . . . . . . . . . . . . . . . . . . . 56

4.22 Change-In-Variance Detection, LR . . . . . . . . . . . . . . . . . . . . . . 56

4.23 Changepoint Detection Based On K-S Test, AR . . . . . . . . . . . . . . . 58

4.24 Changepoint Detection Based On K-S Test, LR . . . . . . . . . . . . . . . 58

4.25 Multi-Change-In-Mean Detection On EOD Price, BinSeg Method . . . . . 60

4.26 Multi-Change-In-Var Detection On Log-Return, PELT Method . . . . . . 60

4.27 Tick Data, Fixed-Threshold Filtering . . . . . . . . . . . . . . . . . . . . . 62

4.28 Abs. Return, 3σ Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.29 Log-Return, 3σ Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.30 Jumps In The Price Curve, 3σ Filtering On AR . . . . . . . . . . . . . . . 64

4.31 Jumps In The Price Curve, 3σ Filtering On LR . . . . . . . . . . . . . . . 64

4.32 QQ-Plot, Abs. Return After Filtered . . . . . . . . . . . . . . . . . . . . . 65

4.33 QQ-Plot, Log-Return After Filtered . . . . . . . . . . . . . . . . . . . . . 65

4.34 ACF Function, Filtered AR . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.35 PACF Function, Filtered AR . . . . . . . . . . . . . . . . . . . . . . . . . 69

A.1 ING N.V.: Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

A.2 ING N.V.: Abs Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

A.3 ING N.V.: Log Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

A.4 S&P 500 Index: Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

A.5 S&P 500 Index: Abs Return . . . . . . . . . . . . . . . . . . . . . . . . . . 73

A.6 S&P 500 Index: Log Return . . . . . . . . . . . . . . . . . . . . . . . . . . 73

A.7 Brent Crude Oil: Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

A.8 Brent Crude Oil: Abs Return . . . . . . . . . . . . . . . . . . . . . . . . . 74

A.9 Brent Crude Oil: Log Return . . . . . . . . . . . . . . . . . . . . . . . . . 74

Abbreviations

ATM At The Money

CDF Cumulative Distribution Function

CDM Clean Development Mechanism

CER Certified Emission Reduction

ECDF Empirical Cumulative Distribution Function

EEX European Energy eXchange

EOD End-Of-Day

ERU Emission Reduction Unit

ETO Exchange-Taded Option

ETS Emissions Tading Scheme

EUA European Union Allowance

ICE InterContinental Exchange

IVS Implied Volatility Surface

JI Joint Implementation

xi

To my grandmother.

xiii

Chapter 1

EU ETS: An Overview

1.1 Introduction

The European Union Emission Trading Scheme, known as the EU ETS, was the world’sfirst emissions trading system and has been the biggest since it was born. It was launchedby the European Commission in 2005 as part of the EU climate policy to fight againstglobal warming.The EU ETS aims to reduce greenhouse gas (GHG) emissions, including Carbon dioxide(CO2), Nitrous oxide (N2O) and Perfluorocarbons (PFCs), in line with the requirementsof the Kyoto Protocol which came into force in 2005. It is a cap-and-trade system: atotal amount of emission rights is released to the market, and the market allocates thoserights by way of trading. The market then enables polluters to find the cheapest wayof emission reduction: either they reduce emissions by technological innovation, or theybuy the emission rights from the market.Lots of sectors and installations are included in the ETS Directive, for example theoil refineries and coal industries, iron and steel industries, paper and pulp industries,cement, glass and ceramics industries, and aviation from 2012. According to EuropeanComission, the EU ETS covers more than 11,000 power stations and industrial plantsin 31 countries, and covers around 45% of the total GHG emissions from the 27 EUcountries [8].Within the EU ETS, there are 3 emission rights that can be submitted for compliance:the EU Allowances, known as EUAs, the Emission Reduction Units, known as ERUs,and the Certified Emission Reductions, known as CERs. This will be further introducedin section 1.3.As ETS is a trading system, after the emission rights are created or distributed, theycan be traded by companies in the secondary market. Section 1.4 gives some basicintroduction.Currently the EU ETS has 3 phases: the first was from January 2005 to December 2007,the second from January 2008 to December 2012, and currently it’s in the third phasewhich begins in 2013 and will span till the end of 2020. More phases will follow afterthe end of phase 3. All of these will be introduced in section 1.5.

1

Chapter 1. EU ETS: an Overview 2

1.2 Emissions Trading: An Economic Incentive

Emissions trading aims to achieve the goal of emissions reduction by an economic ap-proach. For instance we consider a company which owns a power plant. It generateselectricity by burning coal and thus inevitably emits CO2. By the end of each year thecompany is obliged to submit a sufficient amount of emission permits, corresponding tothe amount of CO2 it has released. The company has two ways of dealing with thiscompliance issue: either it buys emission permits in the market from another companywho may have a surplus of permits, or it turns to technological innovation to reduce itsemissions. Which way to choose depends on the cost. Similarly, a company who hasdone well in emissions reduction earns a bonus by selling its surplus permits to others.The figure below shows this simple microscopic mechanism.

Figure 1.1: Micro-Mechanism Of Emissions Trading

Macroscopically, a cap limit is set to the whole emissions trading market, and the marketis responsible for re-allocating the permits and finding an equilibrium.

1.3 The Linking Directive

The final purpose of the Emissions Trading Scheme is to limit the total amount ofemissions in order to comply with the requirements set by the Kyoto Protocol, and letthe polluters adjust the emission themselves via a liquid market. Since there are otheremission permits created within the United Nations Framework Convention on ClimateChange (UNFCCC), the ETS has set up a series of rules, known as the Linking Directive,to link the use of different emission permits.The EU ETS fits into the Kyoto Protocol and corresponding to the Protocol the ETSestablishes 3 mechanisms.

• Emissions Trading: emission rights can be traded. The main emission permitintroduced in ETS is EUA, on which we will mainly focus in the following chapters.The unit of EUA is ton(s) CO2 equivalent (tCO2e), which means if 1 ton of CO2

is emitted, correspondingly 1 EUA is supposed to be submitted for compliance.CERs and ERUs, as are introduced below, can also be traded.


• Clean Development Mechanism (CDM): companies can invest in emission re-duction projects in developing countries, which are registered under the UNFCCC,and earn reduction credits, the CERs. The idea behind this is that developed coun-tries can help reducing emission in developing countries, and they are given someemission permits in return, and altogether the total emissions are reduced. Likethe EUAs, each CER represents 1 tCO2e.

• Joint Implementation (JI): Just as the CDM, within the JI framework com-panies can invest in emission reduction projects in developed countries and earnreduction credits, the ERUs. Like the EUAs, each ERU represents 1 tCO2e.

There are restrictions when polluters surrender the permits for compliance. On average,CERs and ERUs altogether cannot comprise more than 10 - 20% of all the submittedpermits, which means EUAs should be the major emission rights in the complianceaccount. The regulations vary from country to country, and different installations maybe subjected to different rules. Since the use of CERs and ERUs is so much restrictedand the supply is abundant, the CERs and ERUs are thus over-supplied in the marketnowadays and the CER and ERU prices are significantly lower than the EUA prices.

1.4 Emissions Trading Market

The EU ETS is a large market; every day millions of EUAs are auctioned and traded.Usually emissions trading has two steps:

• Primary allocation: for EUAs there is an auction market where EUAs areconstantly distributed by different EU countries; for CERs and ERUs, companieswill get these emission rights after they have participated in certain projects, asmentioned above.The EUA auction market will be discussed in Chapter 3.

• Trading in the secondary market: emission rights are traded both over thecounter (OTC) and in the exchanges. Concepts of spots, forwards and futuresare introduced into the market, and plain-vanilla derivatives such as Europeanoptions are also available. The market is liquid and lots of market participants areinvolved.Chapter 2 will mainly focus on the EUA secondary market and will discuss someof the quantitative characteristics.

There are 3 kinds of players in the game:

• Permit originators. They are typically companies/organizations who have par-ticipated in CDM or JI projects and have earned the corresponding emission rights,which they may not need for compliance themselves and thus want to sell formoney. The participating countries in the scheme can also fall in this category asthey constantly release EUAs to the market via auctions.

• End-users. They are companies/organizations who have installations in the tar-geted industries and have to surrender the emission permits for compliance.


Figure 1.2: Phase 3 Market Structure

• Intermediaries. They are companies who trade emission permits in the market,but they may not need the permits themselves. For example, there are brokersmaking the market, and there are financial institutions that aim at building thebridges between the originators and end-users.

The prices in the market has been volatile since the market was born, especially theEUA prices: the EUA spot price has been decreasing for years, from 20 - 30 EURs in2005 to around 3 EURs in the first quarter of 2013, having dropped sharply due to somenegative market information. The factors that can have an influence on the EUA priceswill be dealt with in Chapter 2.

Figure 1.3: EUA Spot Price, 04/2009 - 06/2013


1.5 Trading Periods

The ETS is an artificial market and is really young compared to some rather maturemarkets like the equity or the fixed income markets. However, the market has beengrowing very fast and there have been many stories to tell. This section gives a briefintroduction to the history of the trading system.Up to now there are 3 trading periods, the first of which was from 2005 to 2007, thesecond from 2008 to 2012, and the third from 2013 to 2020. The trading periods areusually referred to as “phases”. In the future there may be more phases coming up;phase 4 has already been planned.

• Phase 1 was a learning-by-doing process, during which the system was built upand experiences were accumulated for further improvement of the market. EUAswere allocated freely by the participating countries.

• Phase 2 expanded the scope of the scheme significantly. Norway, Iceland andLiechtenstein joined the scheme as non-EU members, and CDM and JI credits wereintroduced into the trading system and were formalized by the Linking Directive.

• Phase 3 saw some further changes e.g. auctioning became the major way to releaseEUAs to the market.

1.6 Thesis Structure

The EU ETS aims to reduce the GHG emissions by means of a market, letting themarket allocate the emission permits itself. However, for the companies involved thevolatile price also means an exposure to market risk. Quantitative methods are thusneeded for both risk management and a better understanding of the market. This thesisaims to perform an analysis to the quantitative aspects of the EUA market. This willbe achieved in several steps.In Chapter 2, we give an introduction to the EUA secondary market. Key productsare introduced and the forward curve will be analyzed. Also, we give some qualitativeanalysis, including price drivers and comparison with other markets.Chapter 3 is about the EUA auction market. In this chapter, auction rules are describedand we give an quantitative approach on how to understand the market.Chapter 4 is a statistical analysis to the carbon prices, both to tick data and to end-of-day (EOD) settlement prices. This include a smoothing procedure, distribution analysis,checkpoint analysis and in the end we propose a simple jump detection approach.

Chapter 2

The EUA Secondary Market

2.1 Introduction

As we have mentioned in Chapter 1, EUAs trading makes up the major part of the entireETS market. This chapter aims to give an extensive analysis to the EUA secondarymarket; in particular, we will look for answers to the following questions:

• What is the structure of the market?

• What should the EUA price term structure look like?

• What determines the price?

• Is the EUA market similar to other financial markets?

2.2 EUA Futures

The word “EUA price” can be ambiguous. Indeed, there are EUA futures of differentexpiry dates, and only some of them are actively traded at the exchange. This sectiongives an introduction to the EUA futures market, and the term structure is explored.Futures are standardized contracts at the exchange. This is the same in the emissionstrading market. An EUA futures contract is a standardized contract between one partyand the exchange to trade 1,000 EUAs at a price agreed upon today (the futures price orstrike price) with delivery and payment occurring at a specified future date, the deliverydate.We denote F (t, T ) as the price at time t of the futures with expiry time T , witht, T ∈ [0,∞).There are already some exchanges that serve as venues for EUA trading within the EUETS: two of them are the EEX (European Energy Exchange) in Leipzig, Germany andthe ICE (InterContinental Exchange) in London, the UK.There are EUA futures with different expiries traded at the exchange. Expiry datesinclude the next business day (the daily future, which is usually regarded as the spot),

7

Chapter 2. The EUA Secondary Market 8

the last business-day Monday in March, June, September and December of each year1.For example, a Dec-13 EUA futures contract at the ICE exchange means that the trad-ing parties agree to deliver 1,000 EUAs on the last Monday of December 2013. Theexchange serves as the intermediary: it buys the futures from the seller and then sellsit to the buyer. The exchange also serves as the clearing house and daily margins arerequired for all the trading parties.Currently the ICE serves as the main venue for the EUA secondary market, provid-ing liquidity for different EUA futures. Now the most actively traded futures is theaforementioned Dec-13 futures, which makes up usually more than 80% of all the EUAfutures daily trade volumes. There is also an EUA spot product, namely the EUA dailyfutures, whose delivery time is the next business day. The daily futures is not reallyactively traded. Altogether, every day there are 15 - 20 million EUAs traded at the ICEon average.It is argued in [13] that the futures price is the unbiased expectation of the correspond-ing spot price. Denote S(t) as the EUA spot price, F (t, T ) as the price at time t, ofthe futures with expiry time T , t, T ∈ [0,∞). Also, denote {Ft}t∈[0,∞) the filtrationgenerated by {S(s)}s∈[0,t]. Then we have

F (t, T ) = E[S(T )|Ft].

Another important result in [13] is that under the risk-neutral measure Q, the futuresprice follows the following relationship

EQ[F (T, T )|Ft] = F (t, T ),

which is the initial fair price of the futures contract.

2.3 EUA Forward Curve

Emission rights are rather simple commodities in the energy market: the products arealmost homogeneous, easy to store (one just need an electronic account), and there isliquidity in the market. All of these characterize the EUA forward curve: theoreticallyit has to be identical with the interest rate term structure from the money market.

2.3.1 Implied Interest By EUA Forward Curve

There are futures with different expiries. We can make an investment by simultaneouslybuying futures of some expiry and selling the same amount of longer-expiry futures.We would have some risk-free profit. Denote the corresponding implied interest rate byrE(t, T ), and we take it as the simple linear rate. Note that by the definition we haveF (t, t) = S(t), where F (t, t) denotes the price of a daily future, whose delivery is thenext trading day.By definition we have

1 + rE(t, T )(T − t) =F (t, T )

F (t, t)

1The exact expiry date is always the last Monday of the month. However, if the last Monday isa Non-Business Day or there is a Non-Business Day in the 4 days following the last Monday, the lastday of trading will be the penultimate Monday of the delivery month. More information is available onhttps://www.theice.com/productguide/ProductSpec.shtml?specId=197.


Assume the time now is at t, we consider some future time T . We have money F (t, t).There are two ways we can invest the money:

1. Buy 1 EUA daily futures (equivalent to the spot), and sell 1 EUA futures of expirytime T , with the price F (t, T ). The annual return rate, calculated as the annualprofit divided by the money invested, is

F (t, T )− F (t, t)

F (t, t)(T − t)= rE(t, T ).

2. Deposit the money in a bank account with the money market interest rate rM (t, T ).The annual return rate is

F (t, t)(1 + rM (t, T )(T − t))− F (t, t)

F (t, t)(T − t)= rM (t, T ).

The EUA spot and futures markets in reality are liquid enough. Then, based on a non-arbitrage statement we must have that the two ways of investment yield the same profit,i.e.

rE(t, T ) = rM (t, T ),

which means the implied interest rate by EUA forward curve is the same as the moneymarket interest rate.The approach we adopted above is essentially the interest rate parity in the foreignexchange market theory. In other words, to some extent the EUA can be viewed as acurrency which can exchange freely with the Euros.

Figure 2.1: EUA Forward Curve, 28/03/2013

The EUA forward curve thus will have less flexibility. As we have discussed, for theforward curve {F (t, T ), T ∈ [t, T ′]}, if some F (t, T0) is determined and the money marketinterest rates are known, then the structure of the whole curve will be clear. In the EUAcase, the T0 will usually be the coming December, as the corresponding futures is themost liquid at the exchange, and currently it’s the Dec-13 EUA futures. Figure 2.1shows what a typical EUA forward curve may look like.


2.3.2 Comparison To The Market Interest Rates

We may want to see whether the EUA implied interest curve is really the same asthe interest rates from the money market, as the theory indicates. Figure 2.2 showsthe comparison. The risk-free rate here is derived from Euribor R© rates (for short-termrates) and interest rate swaps from Bloomberg R© (for long-term rates).

Figure 2.2: EUA Implied Rate vs. Risk-free Rate, 28/03/2013

The figure reveals the fact that the implied rate is significantly higher than the risk-free rate from the money market. The reasons may be that the financing costs of thecompanies in the EU ETS are generally higher than the risk-free rate. Taking intoconsideration the financial constraints of the trading desks in the market as well as thetrading costs, the implied rate curve has been exploited of any arbitrage opportunities.We can then decompose rE as

rE = rRF + rP ,

where

• rRF stands for the risk-free rate;

• rP stands for the average premium that companies in the market have to paybesides rRF . Also, it incorporates implicitly the financial constraints of the marketplayers, e.g. the amount of available capitals by the trading desk and the liquidityconditions of the company.

2.4 Price Drivers

The EU ETS is an artificial market. Theoretically the price should be determined bythe relationship between supply and demand. Since the ETS is a cap-and-trade system,the supply is pre-determined; however, the demand is rather flexible and can hardlybe predicted. Thus any price drivers should either influence the EUA supply, or havean impact on the EUA demand. If the market will adjust its expectation of the futuresupply-demand ratio, the EUA price will change accordingly.Generally speaking, the factors that can influence the EUA prices are:


• Changes in regulations. This turns out to be very influential, as changes inregulations can essentially change the supply of the EUAs. For instance, on theaforementioned day when the veto of the Backloading Plan took place, the EUAprice jumped down and the market lost 1/3 of its total value, as the veto meantthat the over-supply situation would continue at least for the near future. TheEU ETS still faces regulatory uncertainty in the coming years, as the current lowprice is blamed to hinder the intended functioning of the system and the EuropeanCommittee is expected to take actions on this point.

• The influence of other energy commodity prices. Theoretically speaking,carbon prices should be strongly correlated to the energy markets. For example, ifthere is strong demand for energy perhaps because of a colder winter, the emissionprice will also increase since there are expected to be more emissions. There havebeen research on the interactions between the carbon prices and the electricity, oiland gas prices (e.g. [3]). However, it has been observed that there is no strongconsistent connection between emissions prices and other energy prices; sometimesthere are short-term strong correlations and sometimes the prices go in totallydifferent direction.

• Macroeconomic and financial markets indicators. Most installations sub-jected to the emission compliance are in the industry or energy sectors. Thus, theiractivities will be heavily influenced by the macroeconomic prosperity. As for now,the Eurozone economy is far from optimistic, and that leads to a weak demand forEUAs.The financial markets also have a huge impact. It has been observed, that duringthe credit crisis in 2012, the EUA spot price dropped sharply in months. Thefinancial markets may have at least two ways of influencing the emissions market.Once there is a negative shock in the financial markets, the financial situations ofindividual firms will be affected; meanwhile, the economy may deteriorate, whichin turn shrinks the emissions demand.

2.5 EUA European Options

As we have discussed, as the carbon price has been volatile, portfolio managers are facingthe market risk all the time. Some derivatives like options may be needed for dealingwith such risk. However, on the other hand, the market is rather reluctant to provideliquidity in the European options market. This may be partly due to the immaturityof the carbon market; also, the EUA prices are so much volatile that it may be difficultfor the derivative sellers to deal with the risk they have to take. They may ask for highprices in return, which ruin the attractiveness of such derivatives.European options are among the most fundamental derivatives due to their simplicity.Options price is a barometer of the market expectation on future prices, since the futuredistributions of the underlying are implicitly incorporated into the pricing procedure.The implied volatility surface, referred to as IVS hereafter, obtained by inverting theBlack-Scholes formula given the market prices of options, can thus reflect the marketexpectation and is regarded to be an important market feature.


2.5.1 Exchange-traded Options

European options are the very basic derivatives. A European option gives the buyer theright, not the obligation, to buy a certain amount of underlyings at a pre-specified price,the strike price K, at a certain time, the expiry time T .With regard to the emissions trading, standardized European options contracts can betraded via the ICE exchange, and will be referred to as exchange-traded options (ETOs).Strictly speaking, such EUA options are indeed a kind of Futures-style European optionon futures, which means the following:

• Options on futures: If in the money at the expiry time, one lot of EUA optionwill exercise into one lot of the underlying EUA futures, which is the Decemberfuture of the corresponding year. The expiries of the EUA options are usuallyMarch, June, September and December of each year.

• Futures-style options: Like the futures, if one wish to trade such exchange-traded options, he has to put collaterals, i.e. he has to deposit money in themargin account. Thus the quoted prices from the exchange are undiscounted.

2.5.2 Black-Scholes Formula For ETOs

The Black-Scholes (BS) model was first proposed by Fischer Black and Myron Scholesin 1973. It has been so popular that it is now the important cornerstone of the modernmathematical finance. This is also true in the EUA market, for example, in the optionprice quotes, implied volatilities and delta values are included according to the Black-Scholes formula.The Black-Scholes model assumes the following infinitesimal time-scale dynamics to theunderlying spot price, under the risk-neutral measure Q:

dS(t) = rS(t)dt+ σS(t)dWt,

S(0) = s.

The parameters are

• r: the market interest rate which is regarded as a constant in this setting;

• σ: the volatility parameter;

• s: the initial value.

Two immediate but important results following are

1. The distribution of any future spot price is log-normal;


2. The Black-Scholes formula for a ordinary European call option on spots with strikeK and expiry T is given by

c(t;T,K) = S(t)N(d1)− e−r(T−t)KN(d2),

d1 =ln(S(t)/K) + (r + σ2/2)(T − t)

σ√

(T − t),

d2 =ln(S(t)/K) + (r − σ2/2)(T − t)

σ√

(T − t),

where N(·) denotes the standard normal cumulative distribution function (CDF).Options on futures is slightly different from ordinary options whose underlyings areusually future spots. By noticing that

F (t, T ) = S(t)er(T−t),

the risk-neutral dynamics of F (t, T ) becomes, by Ito’s formula,

dF (t, T ) = σF (t, T )dWt.

thus a modified version of Black-Scholes formula for call options on futures is given in[13]:

c(t;T,K) = e−r(T−t)[F (t, T )N(d1)−KN(d2)],

d1 =ln(F (t, T )/K) + σ2(T − t)/2

σ√

(T − t),

d2 =ln(F (t, T )/K)− σ2(T − t)/2

σ√

(T − t).

Then, for the futures-style feature, as there should be no discount, the final Black-Scholesformula for such ETO becomes

c(t;T,K) = F (t, T )N(d1)−KN(d2),

d1 =ln(F (t, T )/K) + σ2(T − t)/2

σ√

(T − t),

d2 =ln(F (t, T )/K)− σ2(T − t)/2

σ√

(T − t).

This formula is very useful in that it sets up a one-to-one correspondence between thecall option price and the volatility parameter σ, in other words, the formula defines apricing function BSF that is monotone w.r.t. σ:

c(t;T,K) = BSF (σ, F (t, T ), T − t,K).

The so-called implied volatility is calculated using the inverse formula of BSF . If wesee σ as a function of T,K, then we have

σI(T,K; t) = BSF−1(c(t;T,K), F (t, T ), T − t,K). (2.1)


Figure 2.3 below show a typical end-of-day report available on the ICE website.2 Settle-ment prices and the changes against the previous trading day are published. The impliedvolatility and delta values are calculated according to the aforementioned Black-Scholesformula for ETOs. We will also be interested in the “Vol”, which means the daily tradevolume, and the “Open Int”, which means the number of outstanding open contractsin total. It is worth mentioning that for different expiries, the total number of strikeprices can be quite different. The Dec-13 EUA option enjoys the most strikes, showingits relevant liquidity. This will be further discussed in Section 2.5.6.

Figure 2.3: Daily EUA Call Option Data from ICE, 07/05/2013

2.5.3 Options Pricing: A Mathematical View

Recall that the ETO price can be written as

c(t;T,K) = EQ(F (t, T )−K)+.

The price can be determined once we get the marginal distribution of F (t, T ) under Q.Denote the probability distribution function (PDF) of that distribution as pQ(x; t, T ).

2For more information see https://www.theice.com/productguide/ProductSpec.shtml?specId=196.


Then we have

c(t;T,K) =

∫(x−K)+pQ(x; t, T )dx.

Note that we can regard k(x;K) = (x − K)+ as a kernel function and the pricing isnothing more than doing an integral transform, i.e.

c(t;T,K) =

∫k(x;K)pQ(x; t, T )dx. (2.2)

One question would be how we can recover pQ(x; t, T ) once we get the call prices data.There are several ways to deal with this:

• A common way is to parametrize it “properly”, in which case we can rewrite p aspQ(x; t, T, θ) with θ ∈ Θ the extra parameters. The famous way to do this is toassume a model for the dynamics of the underlying spot price, which gives what pQmay look like. For example, the Black-Scholes model assumes that F (t, T ) followsa log-normal distribution, and θ = (r, σ). For some other models the PDF maynot have a closed form; sophisticated methods would have to be used to calculatethe call options prices in Formula 2.2.

• Alternatively, we can adopt non-parametric methods. However, we may thenencounter a difficult problem when trying to find out how the distribution shouldevolve w.r.t. T , which is more important when dealing with exotic derivatives.

2.5.4 Implied Volatility Surface

Recall that in Formula 2.1, we know that we can get σ from Black-Scholes formula oncewe know the call option price. It has been observed that σ is different if T,K change.Thus we can construct an implied volatility surface (IVS) σI(T,K; t). Note that thissurface contains exactly the same amount of information as the call option prices.We can construct an IVS using the options price data from ICE, as is shown in Figure2.5. From Figure 2.5 we observe that shorter-expiry curves have a more clear volatilitysmile structure, and deep-in-the-money options have a higher implied volatility. Also,short-term volatilities are expected to be higher. For every strike, the implied volatilitydeclines over the expiry date. This may be an indication that the market is expectingbig fluctuations in the short term. This can also mean that the market expects the priceto stablize in the future: there may be mean-reversion in the price in the long run.

Figure 2.4: Dec-13 EUA Call, Price & Volatility, 07/05/2013


Figure 2.5: Implied Volatility Surface, Dec-13 EUA Call, 07/05/2013

2.5.5 ATM Implied Volatility

From the IVS we can get the at-the-money (ATM) implied volatility, which is the impliedvolatility of the ATM call option on the same-expiry futures. Under the classical Black-Scholes setting, this ATM implied volatility can be seen as the standard deviation of thecorresponding forward price, whose distribution is assumed to be log-normal.From the Black-Scholes model we know the distribution at time t of the future spotprice S(T ) follows

S(T ) ∼ LN(F (t, T ), σ√T − t),

which meansln(S(T )) ∼ N(F (t, T ), σ

√T − t).

Thus, the standard deviations of this distribution family should increase as a functionof T by the rate

√T − t if σ is really a constant.

Take t = 07/05/2013. Figure 2.6 shows that the standard deviation grows much slowerthan the

√T − t rate. More precisely, this figure reveals that the short-term variance is

relatively higher. According to the market, the near-future uncertainty is no less thanthe uncertainty in the long run. Predicting even the near future is no easier than makingpredictions for 3 or 4 years from now.

2.5.6 Is Options Liquidity A Problem?

Options prices contain market opinions on future spot prices. However, this is true onlywhen the prices are a fair reflection of the market expectation. One necessary conditionis that there is “enough” liquidity in the market. One should be very careful whenliquidity comes in. For example, a hedging strategy may involve incessantly buyingand selling some security, where it is implicitly assumed that the security doesn’t lackliquidity. If this is not the case, then one may suffer from high transaction costs or eventhe impossibility to realize the strategy which would be theoretically wonderful.Whether the option is liquid or not can be reflected in the three following aspects:


Figure 2.6: ATM Implied Volatility, 07/05/2013

• Daily trading volume and open interest. A liquid option should be closelyrelated to high daily volume and high open interest. As we can see from the end-of-day report, the EUA options are not quite actively traded; most of the time,especially for options with longer expiries, the daily volumes are zero most of thetime, which means the options with those expires are rather illiquid. The Dec-13option is the most liquid one, and it also suffers from low daily volume, althoughthere are some outstanding open interests.

• The number of strikes. As we have discussed, since the Dec-13 option is themost liquid, it has more strikes than options with other expiries. For more liquidoptions, market-makers may increase the strikes in order to satisfy the marketdemand. Some options with longer expiries suffer from too few strikes.

• Bid-ask spread. More illiquid options lead to a wider bid-ask spread. Thismeans if one wants to trade such options, he will find himself starting from adisadvantagous stage: he has to pay more (get less) for the illiquidness whenbuying (selling) the option.

We should be rather careful when using the data. The prices listed for illiquid optionsmay be calculated by the brokers, in a sense this may be a distortion of the market: theprice may be quite different if the options were actively traded.

2.6 Other EUA Products

Besides futures at the exchange, there are also other carbon products that are extensivelytraded everyday. In this section we give some brief introduction to some of the commonproducts.

2.6.1 Forwards

An EUA forward contract is a non-standardized contract between two parties to buy orsell an specified amount of EUAs at a specified future time at a price agreed upon today.This is in contrast with futures contract which is standard and is extensively traded at


the exchange. Forwards are usually traded OTC, possibly via a broker. Also, in theemissions market, in order to avoid credit risk trading parties are now required to putcollaterals according to the contract.

2.6.2 Time Spreads

At the exchange, EUA time spreads are also frequently traded. an EUA time spread con-tract allows the buyers to simultaneously purchase an EUA futures expiring at particulardate and sell the same instrument expiring another date. For example, a Dec-13/Dec-14 EUA spread means the buyer buys a Dec-13 futures and meanwhile sells a Dec-14futures.The EUA time spreads are no more than a composition of two futures deals. However,the structure allows traders to realize the spread value without being affected by anyshort-time price volatility. Also, traders don’t have to pay twice the bid-ask spread.

2.7 Comparison To Other Financial Markets

Within the category of financial market, we have several well-established markets. Toname some, we have

• Equity markets

• Fixed income markets

• Foreign exchange markets

• Money markets

• Commodity markets

Emissions trading falls in the category of energy commodities trading. However, evenwithin the broad concept of energy commodities market, the emissions market can beso different in some aspects; meanwhile, it shares some features with other markets likethe equity markets. The following are some characterizations:

• There is only one underlying asset trading in the market. Thus we don’thave to consider correlations among different assets within the market. This issimilar to the crude oil market. The electricity market provides heterogeneousproducts like “base-load” and “peak-load”. Outside the energy market, thereare thousands of stocks in the equity market, dozens of currencies in the foreignexchange market, etc.

• The storage cost of emission rights is almost negligible. All one needsis an electronic account. Also the actual delivery is easy: just electronic transferbetween accounts. This is an important reason that makes the forward curve ofEUA really inflexible. Other energy-related products are different, crude oil forexample; most traders would close out their futures position prior to the expiryand no delivery will happen.


• Variable demand, fixed supply. This is the key fact that enables the emissionsmarket to identify itself. As we have discussed, in a sense the emission rightsfeel like currencies which can freely be exchanged with Euros. However, the EUAprice is much more volatile since the supply is pre-determined and the demand ischanging.

• There is not an active options market. The ICE exchange does provide stan-dardized European options products, but it’s not quite liquid. Even some ETOs ofspecific expiries will be regarded as exotic. For other more complicated derivatives,it’s only possible to trade OTC; generally it’s not easy to find a counterparty totrade these derivatives. This is quite different from other major markets, whereboth plain-vanilla and sophisticated derivatives are extensively traded.

Chapter 3

The EUA Auction Market

3.1 Introduction

The EU ETS is a cap-and-trade system. Within the system, auctioning is regarded asan efficient way to primarily allocate the EUAs: by way of bidding, the allowances go tothose participants that value them most. In the phases 1 and 2, most of the allowanceswere allocated freely to the polluters: according to the EU, only about 1% and 4% ofthe total EUA cap amount was auctioned respectively. This has changed in phase 3,when auctioning will serve as a major approach to distribute the allowances.As we have stepped into phase 3 in year 2013, it has turned out that the auction markethas become a successful and active market: every day 3 to 5 million EUAs are auctioned,and the market interacts closely with the secondary market. In order to get a betterunderstanding of the auction market, it deserves further scrutiny and analysis of thepublished information.

3.1.1 Auction Venues

The product auctioned is mainly spot EUAs: they are delivered on the day after theauction. The EEX is the major platform since most of the participating countries willauction their allowances in this exchange. The ICE serves as the platform for the UKauction. This setting may change in the future.Generally, the auctions at the EEX are held every day except on Wednesdays. Theauction usually starts at 9:00 and is cleared at 11:00 CET. The EUAs auctioned aremainly of phase 3 issued by the EU1 as well as by Germany, with the total auctionvolume ranging from 3 to 5 million.2 An auction report is usually published after everyauction, which will be analyzed in the following sections.In contrast, auctions at the ICE are much less frequent: only once on Wednesday inevery two weeks. The EUAs auctioned are issued by the UK, with the volume beingaround 4 million per auction.

1which means the EUAs auctioned are a proportional combination from all the participating countries.2In the first several months in 2013, occasionally there will be 2 auctions in a day, one in the morning,

usually the aforementioned phase 3 auction, and the other one in the afternoon, usually phase 2 auctionsfrom countries/districts other than Germany, e.g. Lithuania, Czech Republic or Flanders.

21

Chapter 3. The EUA Auction Market 22

3.1.2 Auction Rules

Traders are able to give bids in the auction after they meet all the relevant requirements.Every single bid can be characterized by the pair (p, v), where p is the bidding price andv is the corresponding bidding volume. Bidders can give multiple bids, possibly withdifferent volumes in different bidding prices. The minimal volume of a single bid is 500EUAs.The auction is designed as a single-round uniform-pricing system: the auction will becleared at such a price that the total bidding volume at and above this price is equal orjust above the aimed auction volume. This means all the bidders will pay at this pricefor their bidding volumes whose bidding prices are no less than this price. 3

Denote Vaim to be the total volume to be auctioned. We denote p1, p2, . . . , pN to beall the bidding prices with p1 ≤ p2 ≤ · · · ≤ pN . In this way of notation a largerindex corresponds to a higher bidding price. The corresponding bidding volumes arev1, v2, . . . , vN . According to the auction rule we have that the auction price is determinedby

PA = pm,where

m := max{j = 1, 2, . . . , N − 1 :N∑i=j

vi ≥ Vaim}

Figure 3.1: The Mechanism To Decide The Auction Price

3.1.3 Published Information

An Emission Spot Primary Market Auction Report is published on the EEX website andis updated immediately after every auction. Generally, the report contains the following:

• the auction names and types: the state that issues the EUAs and the eligible phaseof those EUAs;

3The bidding volumes whose bidding prices are just equal to the clearing price will be cleared ran-domly, if the auctioned volume remained for this price is less than the total bidding volume of thisprice.


• the auction price PA: at which the auction is cleared;

• the minimal price Pmin: the minimal bidding price;

• the maximal price Pmax: the maximal bidding price;

• the mean Pµ: the average of all the bidding prices;

• the median Pmed: the median of all the bidding prices;

• the cover ratio CR: defined as total volume of bids divided by the auction volume;

• the auction volume Vaim: the total volume to be auctioned;

• total volume of bids VT : the total volume from the bidders;

• total number of bidders;

• the number of successful bidders.

The symbols assigned to some of the terms are prepared for later use.Note that Pµ and Pmed are unweighted : they are totally determined by the biddingprices and are not influenced by the bidding volumes.In contrast, the ICE discloses less information: only auction price, bidding volumes,auction volumes, number of bidders and successful bidders are available after each auc-tion. Hence we will mainly focus on the EEX auctions.In the content we will also consider the auction deviation

AD = PF − PA,

where PF is defined as the fair price of the spot EUA at the exchange, at the time theauction is cleared. It is subject to the judgement of the trader, since the price is changingall the time at the exchange and it’s up to the trader what is the at-the-moment spotprice. AD tells how much the auction is cleared below the exchange-traded price.Note that the PF is just the discounted price at which the EUA Dec-13 futures are tradedat the exchange, at the time when the auction is cleared. However, this doesn’t meanthat one can actually buy EUAs at this price at the exchange. The auction enables oneto buy a relatively large amount of spot EUAs (e.g. 200,000 EUAs) without seriouslyinfluencing the market price. If one would buy the same amount of EUA Dec-13 futuresfrom the exchange in a short time, the average price at which he pays for those EUAswill significantly increase, based on the trading mechanism at the exchange.

3.1.4 A Generic Bidding Strategy

Different bidders may have their own bidding strategies. The auction rules give thebidders enough flexibility to customize their bidding strategies. For example, some willbid at a relatively high price to guarantee they get some EUAs from the auction; somewould bid at relatively low prices so that they can possibly profit if the auction is clearedreally low. More bidders would give a stratified bidding-price structure so that they getmore if the auction price hits lower. Bidders pay nothing for the non-cleared bids.By a non-arbitrage statement, we may give a generic assumption that the bidders willgive their bidding prices according to the EUA fair value right before the auction iscleared. The majority of the bidding prices should not deviate too much from this fairvalue.


3.2 Market Participation

The number of bidders involved has been stable in year 2013: most of the time it fluc-tuates between 15 and 20, as is shown in the following chart. Usually the number ofsuccessful bidders is 3 to 8 less than the total number. The unsuccessful bidders fail togive a sufficiently high bidding price and thus don’t get any EUAs from the auction.

Figure 3.2: Number of Bidders, 01/2013 - 06/2013

In contrast, the total bidding volume can be rather volatile. With the auction volumebeing stable, this means that the cover ratio can change dramatically even in the nextday.

Figure 3.3: Auction Volume, 01/2013 - 06/2013

3.3 A Nonparametric Analysis: Exploiting The PublishedInformation

As we have seen, we can access only very limited data about the auction. Only knowingsome statistics such as the mean and median of the bidding prices, it’s rather difficult toassume any underlying probability distribution to the bidding behavior of the market.


However, we can still say something once we know the published information.In this section we will give a simple but useful analysis approach. The auction data weuse are from the EEX website.As before we denote p1, p2, . . . , pN to be all the bidding prices with Pmin = p1 ≤ p2 ≤· · · ≤ pN = Pmax, and the corresponding bidding volumes are v1, v2, . . . , vN . Note thatin order to be consistent with the statistics, different bids with the same prices are notaggregated and will be treated individually. For simplicity we always assume N to beodd, thus we have

Pmed = pN+12.

In fact if N is even, we put a “dummy” bid pd = 12(pN

2+ pN

2+1) with vd = 1(1 lot of

EUA would be negligible in reality!) and consider the new situation.According to the auction rule we have the following

PA = pm,where

m := max{j = 1, 2, . . . , N − 1 :N∑i=j

vi ≥ Vaim}

Section 3.3.2 will discuss the relationship among PA, Pµ and Pmed.Next to this are the concepts of the upper/lower interval length, which are defined by

LU := Pmax − Pmed, LL := Pmed − Pmin.

The interpretation is that the upper half of the bidding prices are in the upper intervalwith length LU , while the other half lie in the lower interval with length LL.In the content the auction deviation

AD = PF − PA

will also be an important figure.

3.3.1 Statistics Stability

We have some variables that partly characterize the bidders’ behavior in the auction.Some of them are purely statistics, including Pmin, Pmax, Pmed, and Pµ. Whether theyare robust or not is important, since we will have to decide how much valid informationthese statistics can convey.As we have discussed in Section 3.1.4, the spot EUA fair price at around 11:00 is a goodbenchmark. Figure 3.4 and 3.5 shows the daily change of each statistic against the dailychange of the spot fair price. The daily change is calculated as today’s price subtractedby the corresponding price of the last trading day that had a phase 3 auction at theEEX. The data are from Bloomberg R© and the EEX website, from 01/2013 to 06/2013.

We would have the following comments:

• Pmax: this is not stable as it can be easily influenced by any individual behavior.

• Pmin: also not stable; its stability is even slightly worse than Pmax, as bidding lowand high are not treated symmetrically according to the auction rule: bidders willhave to pay for the volumes whose bidding prices are higher than the final auction


Figure 3.4: Daily Changes: Pmin/Pmax vs. PF

Figure 3.5: Daily Changes: Pmed/Pµ vs. PF

price, but won’t pay for the un-cleared low-price bids. Bidders are implicitlyencouraged to be more aggressive when giving low-price bids.

• Pmed: Figure 3.5 as well as 3.6 shows that the median of the bidding prices seemsto be the most stable one. The daily change of the median is almost linear againstthe change of PF . Also, generally the median will be around 7 cents lower than thefair price. This may be explained by our assumption in Section 3.1.4: as bidderswould take the fair price as an important reference when giving bids, there can beso many bids whose associated prices are several cents lower than the fair price.This stablizes the Pmed−PF spread and to a large extent immunizes it from largechanges that are caused by individual bidding behavior.

• Pµ: the mean is not very stable; it can also be easily influenced by individualbehavior. For instance, one bidder can give 10 bids whose bidding prices are onaverage 2 Euros lower than the fair price. Suppose altogether there are 100 bids.Then because of this Pµ decreases by 20 cents.


Figure 3.6: Pmed & Pµ vs. PF

3.3.2 Relationship Among PA, Pmed, Pµ

PA, Pmed, Pµ can give some enlightment on what has happened in an auction. Notethat among the three, PA is the only variable that is linked to the bidding volumes.

If Pmed > PA:a naive conclusion would be that the auction price lies in the lower half of the orderedsequence of all the bidding prices. In this case we must have N+1

2 > m. Then by thedefinition of m, immediately

N∑i=N+1

2

vi < Vaim,

thus ∑Ni=N+1

2vi

VT<

1

CR=VaimVT

.

This already tells something about the bidders’ behavior: the total volumes of theupper half in price is no more than VT

CR . For example, on the day 04/03/2013, we havePmed = 4.55 > PA = 4.54, and CR = 2.47. Then∑N

i=N+12vi

VT<

1

2.47= 0.405,

which means the bidders tended to put more of their bidding volumes in the lower halfof the price on that day.

If Pmed ≤ PA:analogously this tells the auction price lies in the upper half of the ordered sequence ofall the bidding prices and ∑N

i=N+12vi

VT≥ 1

CR=VaimVT

.


If Pmed > Pµ:by definition

pN+12

>

∑Ni=1 piN

,

which can be rewriten as

N · pN+12

>N∑i=1

pi,

N+12∑i=1

(pN+12− pi) >

N∑i=N+1

2

(pi − pN+12

).

This means the lower half of the prices are farther away from the median price than theupper half, i.o.w. the relatively higher prices are more concentrated. The bidders aremore aggressive when bidding lower prices.

If Pmed ≤ Pµ:this serves as the corresponding converse situation: the upper half of the prices are moresparsed from the median than the lower half.

If PA > Pµ:recall that

PA = pm,where

m := max{j = 1, 2, . . . , N − 1 :N∑

i=j+1

vi ≥ Vaim}

we thus have

pm >

∑Ni=1 piN

,

m∑i=1

(pm − pi) >N∑i=m

(pi − pm).

Note that this cannot happen if Pmed > PA and Pmed ≤ Pµ.

If PA ≤ Pµ:analogously this implies

m∑i=1

(pm − pi) ≤N∑i=m

(pi − pm).

3.3.3 Market Special Cases

As we have discussed, the published information usually give some characterizationsabout what happened in the auction market. Below we list some cases when we canhave a clear picture of what the bidders are doing.


• Case I+: Pmed > PA while CR > 2We have shown that when Pmed > PA,∑N

i=N+12vi

VT<

1

CR< 0.5

since CR > 2. This shows that the bidders generally put more of their biddingvolumes in the lower half of the bidding prices. The auction usually closes with apositive discount.

• Case I−: Pmed ≤ PA while CR < 2 This is the converse situation, when Pmed ≤PA with CR < 2, in which case∑N

i=N+12vi

VT>

1

CR> 0.5.

This can mean that not so many bidders would give low-price bids.Case I+ and I− perceives the auction from the angle of bidding volumes.

• Case II+: Pmed > Pµ while LL < LUPmed > Pµ tells that the lower half of the prices are on average farther away fromthe median. However, its span interval is smaller, which indicates that either Pmaxis too high and is really an individual behavior, or the lower half of the prices areon average really low.

• Case II−: Pmed ≤ Pµ while LL ≥ LUAnalogy can apply to the converse case of II+, when either there is an extremelylow price which widens the interval, or there are a lot of high-price bids.Case II+ and II− perceives the auction by way of extreme values.

• Case III+: Pmed > PA > PµNow naturally it follows that Pmed > Pµ. The situation happens when the auctionprice lies in the lower half of the bidding prices (N+1

2 > m), and the lower pricesare on average more close to the minimum price while the higher prices are moreclose to the median. Furthermore, PA > Pµ, which means

m∑i=1

(pm − pi) >N∑i=m

(pi − pm),

and the left hand side has even fewer terms. This means either the lower prices arereally low in general, or the higher prices are relatively not quite high and moreconcerntrated, or both. This case can also signify a market atmosphere when nottoo many bidders would bid high in the auction.

• Case III−: Pmed ≤ PA ≤ PµThis is the corresponding converse case of III+.Case III+ and III− perceives the auction from the angle of bidding prices.

3.3.4 A Way Of Illustration

A neat and clear way to visualize the published information facilitates any further anal-ysis and interpretation. Here we give a simple approach to make such charts.The chart template is shown in Figure 3.7. In the chart, key statistics like median,


mean, auction price as well as cover ratio are included. The chart helps to perceiveintuitively how participants were bidding. The “100/CR %” means the percentage ofthe total volumes that are above the auction price.

Figure 3.7: Chart Template

3.3.5 Case Study

In the year 2013 we have seen quite a few EUA auctions. Among these auctions, themost common case is that PA ≥ Pmed ≥ Pµ and 1.80 ≤ CR ≤ 4.50. However, specialcases do occur sometimes, and we give a brief summary of the highlights of the EEXPhase 3 auctions from 01/2013 to 07/2013. Altogether there are 112 auctions.

• 9 auctions have been identified as case I+, which indicates there are not so manyvolumes in the upper half of the bidding prices. Most of these auctions also fall inthe case III+, which gives us a hint that the higher prices are more concentratedthan the lower ones. All of the auctions have an auction deviation of at least 5cents.

• 13 auctions fall in the case II+. The statistics show that some bidders give reallylow bidding prices but the average of all the bidding prices is not seriously affected.This may indicate that relatively speaking there were not many low bidding prices.In contrast, we also observe that in May and June the minimal bidding price wasalways low and the mean was significantly lower than the median as well as theat-the-moment fair value. The mean is sensitive to the low-bidding behavior.

• Only 1 auction, the auction on the day 18/04/2013, is flagged as case II−, whichmeans in that auction there were generally more higher bidding prices. The auctionwas the first auction after the aforementioned backloading veto. Figure 3.8 showsthat lots of bidding volumes were in the upper side, and the auction was cleared 7cents above the fair value at the exchange.

• Only 1 auction is flagged as case III−, which means that in that auction the higherbidding prices were really high. The auction was on the day 22/04/2013, the firstauction in the week after the backloading veto. However, the auction deviation was3 cents. This may indicate that several bidders tried to get some volume from theauction by giving very high-price bids, but the volume was not very substantial.


Figure 3.8: Auction On 18/04/2013

3.4 Beyond The Nonparametric Analysis

The nonparametric analysis on the published statistics is a useful tool for one to under-stand the auction market. However, one should be aware that such an analysis is notfully satisfactory, due to the following problems with the auction market.

1. Problems with the published data. Sometimes the data are insufficient. Inthe previous section, we do not fully utilize the magnitude of the numbers. Inan extreme example, consider the cases when 1.PA = 5.00, Pmed = 4.00 and 2.PA = 4.01, Pmed = 4.00. Under both cases we have PA > Pmed, but the twocases are essentially different if they really happen in the market. However, tofix this problem we may need the total number of bidding prices N , which isnot available. Another example of this would be that we cannot incorporate thenumber of bidders into the analysis, since we need further information about thebidders’ bidding strategies.Another problem with the data is that the definition of mean and median aredefined in such a way that every bid has the same weight regardless of its volume.Thus we cannot get too much information of the bidding volume, which is as animportant aspect of a bid as the bidding price.Lastly, the minimal and maximal bidding price can be misleading, as they are notrobust at all and can be changed simply by an individual behavior. As we haveobserved, in mid-March some bidder(s) would always give a constant bidding priceof 6.00 that reached the maximum all the time, regardless of the fair value of spotEUAs during the auction time. However, the associated bidding volume could besmall, and the second highest price would be just 50 cents above the fair value,which was then between 3.50 and 4.50. Thus one should be really careful whendealing with extreme prices.

2. The bidding behaviors of the bidders change all the time. With the lackof information, it’s almost impossible and can be dangerous to assume any time-homogeneous underlying probability distributions to the daily bidding behavior.Parametric analysis is not a good choice, while the nonparametric approaches wehave done are not powerful enough. To conclude, qualitative methods should beadopted along with the quantitative analysis.

Chapter 4

Carbon Price Data: A StatisticalAnalysis

It has been 8 years since the emissions trading system started; during these years we haveseen lots of carbon price movements, and we are sometimes interested in the statisticalaspects of the prices. In this chapter, we talk about some statistical methods that canbe applied to process the price data.We will use two series of EUA price data from the ICE exchange:

• Tick data: the EUA Dec-13 futures price.1 Tick data refers to the real-timemarket data which shows the price and volume of every deal that is done from08:00 to 18:00 CET at the exchange. Figure 4.1 shows a typical tick data curvestructure.

• End-of-day settlement prices: the EUA spot price. These prices are calcu-lated and published by the exchange and are also referred to as the closing prices.Open, intraday high and low prices are also published along with the closing prices.All these prices are often integrated into one chart, the candlestick chart, which isbriefly illustrated in Figure 4.3.

Figure 4.1: ICE Dec-13 EUA Tick Data, 05/04/2013

1The tick data are from ReutersR©.

33

Chapter 4. Carbon Price Data: A Statistical Analysis 34

Figure 4.2: ICE Spot EUA EOD Prices, 01/01/2013 - 21/06/2013

Figure 4.3: Candlestick Chart

These two kinds of data are closely related to each other: the closing prices containonly part of the information from the tick data. On the other hand, they demonstratedifferent statistical features and thus should be treated differently.In particular, we are interested in the following questions:

• Empirical distributions. What can we say about the distributions of the dailyreturns of the prices as well as the Auction deviations? Are the deals at theexchange coming according to the famous Poisson pattern, i.e. in a fixed intervalof time the number of deals is Poisson distributed?

• Possible changes in the market. Statistically speaking, are there any signs whichmay indicate a significant change of the market conditions?

• The carbon price has been volatile and some jumps are observed from time totime. Are there any simple approaches to filter out the jumps?

In this section some programming will be involved. We will mainly work in the environ-ment R [24] and VBA, depending on the situation.


4.1 Unevenly-Spaced Time Series

The real-time tick data are indeed an unevenly-spaced time series: deals are done andrecorded one by one according to the time they happen, so the time differences amongthese data points are not equal.The tick data series can be fully characterized by the 4-tuple series:

{Dn = (n, tn, pn, vn), n ∈ N} ,

where tn is the time, pn the settlement price of the deal, and vn the volume of the deal.Note that it can definitely happen that for some m 6= n we have tm = tn, pm = pn,and vm = vn. Thus the cardinal number is important in that different deals can bedistinguished from each other.

4.1.1 A Smoothing Procedure

Sometimes equispaced time series are more convenient. To convert the tick data seriesinto an equispaced time series, we first need to divide the timeline into equispacedintervals, and choose a price and a volume to represent the deals that are done withineach time interval. Essentially we are doing smoothing with regard to some specifiedweight series {ωi}i∈I : the price is then determined by

PI =∑i∈I

ωipi

For instance, for the deals {D′i}i=1,...,N with N > 0 within the target time interval[T1, T2), consider the following methods:

• Take the average prices: the weights are

ωi =1

N, i = 1, . . . , N ;

• Take the volume-weighed average prices: the weights are

ωi =v′i∑Nj=1 v

′j

, i = 1, . . . , N ;

• Take the last deal price: the weights are

ωi = 1N (i), i = 1, . . . , N.

The EOD settlement price is calculated as the volume-weighted average price in thelast 10 minutes of the trading period, i.e. 17:50 - 18:00 CET. Thus we can formulate itsweight function by the following:consider

{D′i}i=1,...,N , [T1, T2] = [08 : 00, 18 : 00],

define the set of index

J = {i = 1, . . . , , N : ti ∈ [17 : 50, 18 : 00].}


Thenωi =

vi∑j∈J vj

1J (i), i = 1, . . . , N.

This sets up the link between the tick data and the EOD settlement price series.We can then formulate a way to transform the tick data into a equispaced time series:

1. Choose a time window span s min.

2. Construct time intervals by dividing the whole trading time into intervals withlength s min and divide the series in groups according to the time intervals. Notethat it can happen that for some time interval there are no data that fall withinthat interval.

3. Choose a weight function.

4. Do the smoothing w.r.t. the chosen weight function.

5. For those intervals which don’t have a price because there are no deals that hap-pened in that interval, we give a price according to the nearby prices. Possiblesolutions are

• linear interpolation;

• take the last interval price.

Volume-wise, the question becomes quite simple, as we only have to sum up the volumeswithin each time interval because the total trade volume in each time interval is the onlything that we are interested in most cases. We continue with the example {D′i}i=1,...,N

with N > 0 within the target time interval [T1, T2). Then the weight function would be

ωi = 1, i = 1, . . . , N.

The smoothing procedure is also valid for the volumes.

4.1.2 Implementation And Examples

To implement the algorithm we use Microsoft R© Excel combined with VBA as theprogramming environment. To show the results, we take the tick data on the day05/04/2013. We choose the weight function

ωi =1

N, i = 1, . . . , N,

i.e. we will take the average prices. Respectively we take s = 30 min, 15 min and 5 min.The results will be shown below.

Figure 4.7 - 4.9 show the intraday total trade volumes. The unit of the vertical axis is1,000 EUAs.

From the curves we can see clearly the intraday price trend on that day; also, tradingactiveness can be reflected partly by the volume traded.


Figure 4.4: Dec-13 EUA Intraday Averaged Price, s=30 min, 05/04/2013




Figure 4.7: Dec-13 EUA Intraday Total Volume, s=30 min, 05/04/2013

Figure 4.8: Dec-13 EUA Intraday Total Volume, s=15 min, 05/04/2013

Figure 4.9: Dec-13 Intraday Total Volume, s=5 min, 05/04/2013


4.2 Distribution Analysis

We have two sets of data: the intraday tick data of the EUA Dec-13 futures

{Dn = (n, tn, pn, vn), n = 0, . . . , N0} ,

and the EOD settlement prices of spot EUAs

{Pn = (n, τn, ρn), n = 0, . . . , N1}

where τn stands for the date and ρn is the EOD EUA spot price.In this section we will mainly deal with the distributions of

• the arrival time among each individual deals from the tick data, which is definedby

∆tn = tn − tn−1, n = 1, . . . , N0;

• the tick data price change

∆pn = pn − pn−1, n = 1, . . . , N0;

• price change of the EOD prices. We will consider

∆ρn = ρn − ρn−1, n = 1, . . . , N1,

the absolute return series and

δρn = log

(ρnρn−1

), n = 1, . . . , N1,

the log-return series2;

• the Auction deviation series.

We will mainly use the software R.

4.2.1 Kolmogorov-Smirnov Test And Jarque-Bera Test

From time to time we may want to test whether a sample comes from a certain dis-tribution, or whether two samples share the same distribution or not. In this sectionwe introduce two famous tests: the non-parametric Kolmogorov-Smirnov test, and anormality test, the Jarque-Bera test.

4.2.1.1 Kolmogorov-Smirnov Test

The Kolmogorov-Smirnov test (K-S test) is a nonparametric test for the equality ofcontinuous, one-dimensional probability distributions that can be used either to compare

2The term “absolute return” here is corresponding to the “relative return”, which is the return in aproportional basis and can be approximated by the log-return.


a sample with a reference probability distribution (one-sample K-S test), or to comparetwo samples (two-sample K-S test).To adopt the K-S test we shall first introduce the concept of the Kolmogorov-Smirnovstatistic (K-S statistic):

Definition 4.1. (Kolmogorov-Smirnov statistic.) For any given data {xi : i = 1, . . . n},the Kolmogorov-Smirnov statistic for a given CDF F (x) is defined as

KSn = supt|FXn (t)− F (t)|,

where the FXn is the Empirical CDF of {xi}.

The Kolmogorov-Smirnov statistic quantifies a distance between the ECDF of the sampleand the CDF of the reference distribution. By a small modification it can also measurethe distance between the ECDFs of two samples.We are interested in that if {xi} are drawn according to F (x), what distribution theK-S statistic would follow.

Definition 4.2. (Brownian bridge.) A Brownian bridge is a continuous-time stochasticprocess B(t) whose probability distribution is the conditional probability distribution ofa Wiener process W (t) given the condition that B(1) = 0, i.e.

B(t) := (W (t)|W (1) = 0), t ∈ [0, 1].

Proposition 4.3. (A characterization of Brownian bridge.) We have

B(t) := W (t)− tW (1), t ∈ [0, 1]

is a Brownian bridge given that W (t) is a standard Wiener process.

Proof. See [25].

Then we can define

Definition 4.4. (Kolmogorov distribution.) The Kolmogorov distribution is the distri-bution of the random variable

K := supx∈[0,1]

|B(t)|,

where B(t) is a Brownian bridge. The CDF of K is given by

P(K ≤ x) = 1− 2∞∑k=1

(−1)k−1e−2k2x2 .

With all these definitions we can then proceed to the one-sample K-S test. The nullhypothesis is that the samples {xi} are drawn from the distribution F (x), under whichwe have the following theorem:

Theorem 4.5. If F (x) is continuous, then under the null hypothesis above,√nKSn

converges to the Kolmogorov distribution, where KSn is the K-S statistic.

Proof. See [15].


Based on theorem 4.5 we can have the following procedure for a K-S test.

1. Get the target data {xi, i = 1, . . . , N} and compute the corresponding ECDFFXN (x).

2. Choose the target distribution F (x) for which we would like to test.

3. Choose a significance level α.

4. Compute the K-S statistic KSX,FN .

5. Hypothesis test: the null hypothesis is rejected if

√NKSX,FN > Kα,

where Kα is the α-th quantile from the Kolmogorov distribution.

This procedure can also be used to test whether two empirical distributions differ fromeach other. An advantage of this test is that there is no need to presume any parametricdistributions.

Theorem 4.6. (Two-sample Kolmogorov-Smirnov test. ) Let {Xi, i = 1, . . . , n} and{Yi, i = 1, . . . ,m} be two iid series from two independent distributions. Denote FX , FYas their ECDFs based on a realization. Then under the null hypothesis that the twodistributions are the same, we have the modified K-S statistic√

nm

n+mKSn,m =

√nm

n+msupx|FX(x)− FY (x)|

converges to the Kolmogorov distribution.

Proof. See [15].

4.2.1.2 Jarque-Bera Test

The Jarque-Bera test is a famous way to test whether the sample data have the skewnessand kurtosis matching a normal distribution.

Proposition 4.7. (Jarque-Bera test.) Let {Xi, i = 1, . . . , , N} be a series of iid variableswith a realization {xi, i = 1, . . . , , N}. Then, under the null hypothesis that {Xi} are froma distribution with skewness being 0 and excess kurtosis being 0, asymptotically we havethat

JB =N

6

(S2 +

1

4(K − 3)2

)∼ χ2(2),

where S is the sample skewness and K is the sample kurtosis.

Proof. See [14].

Note that the J-B test only tests whether the sample data have the right skewness andkurtosis as a normal distribution; some other tests, the aforementioned K-S test forinstance, relies more on the relevant CDF features. In the content we will apply boththe J-B test and the K-S test to increase the credibility of our results.


4.2.2 Tick Data: Deals Arrival Time

The first interesting question is whether the counting process of the number of deals is aPoisson process, under which the deals inter-arrival time, defined as the time differencebetween every two consecutive deals, should be exponentially distributed.We use the EUA Dec-13 real-time price data from 22/02/2013 to 20/06/2013. As before,mathematically we denote them as

{Dn = (n, tn, pn, vn), n = 0, . . . , N0} ,

N0 = 165339. We will mainly focus on the deals arrival time series, which is denoted as{∆tn = tn − tn−1, n = 1, . . . , N0

},

and the unit is second(s). Figure 4.10 shows the histogram of the deals arrival time. Asthe data has a long fat tail, a zoom-in picture with a logarithm scale in the vertical axismay be desired to show the low-value part of the histogram, as in Figure 4.11.The histograms show two things. First, the data has a really long fat tail; second, there

Figure 4.10: Deals Arrival Time Histogram

Figure 4.11: Deals Arrival Time Histogram In log10 Scale, Partial

are so many observations that have small values. These two features also coincide with


Table 4.1: Deals Arrival Time Stats

Statistics ∆tn Exp(λ)

Mean 17.82 λ−1

Std Dev 66.82 λ−1

Skewness 9.26 2Ex. Kurtosis 144.63 6

Range [0, 2836] N/A

Table 4.1, the descriptive statistics of the data, which shows that the deals arrival timeis much more dispersed than a general exponentially distributed series.Before we do any testing on the series, we should notice the following:

• the deals arrival time are all integers if we use the second as its unit. Thus wesee that many times the deals arrival time is zero. This means that two or moredeals happen at the same time, which is in contradiction with the definition of aPoisson process: the probability of such an incident is zero.

• It is partly due to the rules of the trading system at the exchange that there are somany zeros in the sequence. According to the system, bids/offers with the samebid/ask price will be aggregated together. This means it can be quite often that abuyer buys EUA futures from several sellers simultaneously, just by one click onthe screen; however, this will be recorded as several separate deals as the sellersare different. This feature partly explains the peak at 0 in the histogram of thedeals arrival time series.

Based on these two concerns, we propose a modified Poisson process, whose inter-arrivaltime

• is from a discretized exponential distribution with parameter λ, with probability(1− a), 0 < a < 1. The discretization is done by rounding, i.e.

P(T = k) =

{Fexp(k + 0.5)− Fexp(k), k = 0;

Fexp(k + 0.5)− Fexp(k − 0.5), k = 1, 2, . . .

where Fexp is the CDF of the exponential distribution.

• is 0 with an extra probability of a.

Thus, by noting thatFexp(x|λ) = 1− e−λx,

the distribution will be

FGE(k|a, λ) = P(T = k) =

{a+ (1− a)(1− e−0.5λ), k = 0;

(1− a)(e−λ(k−0.5) − e−λ(k+0.5)

), k = 1, 2, . . .

where “GE” means “generalized exponential”. The fitting can be done by a maximum-likelihood approach.


4.2.2.1 Maximum Likelihood Fitting: Results

In our content, we choose {∆tn = tn − tn−1, n = 1, . . . , N0} as the target data. To

adopt the maximum likelihood approach, we try to maximize the log-likelihood i.e. tofind

arg maxa∈(0,1),λ∈(0,+∞)

ML({∆tn}|a, λ) = arg max

a∈(0,1),λ∈(0,+∞)

N0∑i=1

lnFGE(∆ti|a, λ).

This is a two-dimensional nonlinear optimization problem. We try to solve this in Excelusing the Generalized Reduced Gradient Algorithm (GRG) 3 [21]. The result is

a = 0.506257, λ = 0.027707.

We may want to see the goodness of fit. We first simulate a sample from the estimateddistribution with the length equal to the deals arrival time series, and then compare itto the deals arrival time series by a QQ-plot, as is shown in Figure 4.12.

Figure 4.12: QQ-Plot, Deals Arrival Time vs. A Sample From The Estimated Dis-tribution

We may conclude that the model we proposed failed to explain the deals arrival timeseries. As we can see from the statistics, the deals arrival time series is much moredispersed in value and has larger chances of extreme values.The reason for this may be that the series is not homogeneous, i.e. on different days thedeals arrival time may have totally different distributions. On some days the market ismuch more active than the others.

3We refer interested readers to [19].


Table 4.2: Deals Arrival Time Stats

Statistics ∆tn Simulated Sample

Mean 17.82 17.84Std Dev 66.82 31.05Skewness 9.26 2.69

Ex. Kurtosis 144.63 12.96Range [0, 2836] [0, 437]

4.2.3 Tick Data: Price Changes

The tick data records every single deal that were made at the exchange. We are interestedin the price change series. Consider our setting

{Dn = (n, tn, pn, vn), n = 0, . . . , N0} ,

and the price change series are defined by

∆pn = pn − pn−1, n = 1, . . . , N0.

Figure 4.13 shows the histogram in a log10 scale. The minimal value of change is 1 cent,thus the series ∆p

n indeed follows a discrete distribution.As we can see also from Table 4.3 as well as Figure 4.134,

• the empirical distribution is very peaked around 0. This is because most of thetime the price will change by taking small steps, usually 1 or 2 cents.

• the empirical distribution has a significantly fat tail, which means extreme pricechanges happen from time to time.

Figure 4.13: Tick Data Price Changes, in log10 scale

4.2.3.1 Results

We do the J-B test on the tick data price change series. The results are

4In Figure 4.13 we let log10 0 = 0.


Table 4.3: Tick Data Price Change Stats

Statistics ∆pn

Mean 0.00Std Dev 0.009Skewness -9.50Kurtosis 788.80Range [-0.79, 0.26]

> library(tseries) # load the tseries package

> jarque.bera.test(TDP) # TDP is the tick data price change series

Jarque Bera Test

data: TDP

X-squared = 624121538, df = 2, p-value < 2.2e-16

As expected, with such a small p-value we should reject the null hypothesis and weconclude that the tick data price change series is not normal distributed. From Table4.3 we already see that the data have a negative skewness and a strikingly high kurtosis.

4.2.4 EOD Settlement Prices: Price Changes

Sometimes the changes in the EOD EUA price is more interesting since it shows moregenerally the carbon price trend. Also, for accounting reasons daily settlement pricesare an important reference to mark to market.To distinguish from the previous symbols we denote the data set of EOD prices as

{Pn = (n, τn, ρn), n = 0, . . . , N1} ,

where τn stands for the date and ρn is the EOD EUA spot price. N1 = 1086.There are two ways to show the price changes. One way is the absolute return:

∆ρn = ρn − ρn−1;

the other way is the so-called log-return which measures the price change approximatelyin a proportional way:

δρn = log

(ρnρn−1

).

Figure 4.14 and 4.15 show the curves of the two respectively.Figure 4.16 and 4.17 are the QQ-plots of the two series against normal distribution.

The figures show that compared to a normal distribution, the price changes series mayhave too many extreme values. These extreme values, also shown in Figure 4.14 and4.15, are corresponding to the price jumps, and in section 4.4 we will introduce a jumpfiltering procedure.A further J-B test confirms our intuition.


Figure 4.14: Abs. Price Changes, 04/2009 - 06/2013

Figure 4.15: Log Returns, 04/2009 - 06/2013

Figure 4.16: Abs. Price Changes QQ-Plot, 04/2009 - 06/2013


Figure 4.17: Log Returns QQ-Plot, 04/2009 - 06/2013

> library(tseries) # load the tseries package

> jarque.bera.test(AR) # AR is the absolute return series

Jarque Bera Test

data: AR

X-squared = 510.6052, df = 2, p-value < 2.2e-16

> jarque.bera.test(LR) # LR is the log-return series

Jarque Bera Test

data: LR

X-squared = 2189.61, df = 2, p-value < 2.2e-16

4.2.5 Auction Deviation

Another series we are interested in is the Auction deviation series. Take the Auctiondeviation data from 01/2013 to 06/2013. Figure 4.18 shows the QQ-plot of the seriesagainst a normal distribution. A J-B test gives us the following result, from which wecannot reject the null hypothesis.

> jarque.bera.test(AD) # AD is the Auction deviation series

Jarque Bera Test


Figure 4.18: Auction Deviation QQ-Plot

data: AD

X-squared = 3.6313, df = 2, p-value = 0.1627

The K-S test also tells us that the data series are from a normal distribution.

> ks.test(AD,’pnorm’) # AD is the Auction deviation series

One-sample Kolmogorov-Smirnov test

data: AD

D = 0.0881, p-value =0.4068

alternative hypothesis: two-sided

From Figure 4.18, the distribution is indeed discrete; the minimal difference in the datais 1 cent. For this sake we use a two-sample K-S test to compare the AD series to arandom number series which

• is from a normal distribution with both mean and variance equal to those of theAD series;

• is rounded to the second decimal place.

The comparison is done 10 times. The p-values are

0.9997 0.8221 0.5934 0.9123 0.1625

0.4804 0.4804 0.711 0.8221 0.9699

We may conclude that statistically speaking the AD series cannot be distinguished froma rounded series from a normal distribution. This can give some theoretical support


for doing hypothesis testing or giving confidence intervals based on AD. For example,consider the 95% confidence interval

> qnorm(0.025,mean(AD),sd(AD)) # quantiles from a normal distr.

[1] -0.06890511

> qnorm(0.975,mean(AD),sd(AD))

[1] 0.1214541

> quantile(AD,0.025) # quantiles from AD sample

2.5%

-0.05

> quantile(AD,0.975)

97.5%

0.13475

The theoretical and empirical quantiles are different, but the difference is not big. Wemay conclude that in most cases the Auction deviation will lie between around −0.06cent and around 0.13 cent.

Figure 4.19: QQ-Plot, AD vs. Rounded Normal Random Series

4.3 Changepoint Analysis

The emissions market is changing all the time. It could be useful to judge whether thereis something that has changed in the market. Statistically, this means that we have tojudge whether some of the statistical features of a series of data have changed or not.This is called the changepoint problem.Consider a series of independent random variables {Xi, i = 1, . . . , N}, from whose dis-tributions a sequence of observations {xi, i = 1, . . . , N} are drawn accordingly. We


have to judge, according to the observation series, that whether there exists a parti-tion of {1, . . . , N}, such that within each subset of {1, . . . , N} the Xis are identicallydistributed, while the distributions across the subsets are different? If so, what is thepartition?In our content, we call the index right at the beginning of each subset in the partitiona changepoint. We will have to give answers to the following questions:

• Are there any changepoints?

• How many changepoints are there?

• Where are the changepoints?

There has been vast research in dealing with this problem. To this end we will use R forthe analysis; for the general methodology, we will also follow the approaches cited anddescribed in the documentations of the R packages cpm and changepoint.

4.3.1 A Simple Example

We will start by solving, both theoretically and numerically, a simple problem abstractedfrom the practical situation.Consider the Auction deviation series from 01/2013 to 06/2013 {ADi, i = 1, . . . , N2},N2 = 102. We may first assume that

• the series are independent, and

• whether the Auction deviation is positive or not follows a Bernoulli distributionwith probability p.

We can first binarize the AD series and consider the new series {bi, i = 1, . . . , N2}. Wethen assume that there is exactly one change point, c, such that

Sub-series 1 : b1, . . . , bc−1 ∼ Ber(p1),Sub-series 2 : bc, . . . , bN2 ∼ Ber(p2),

while p1 6= p2 ∈ [0, 1].We are now interested in what c is. We will use a maximum-likelihood procedure, i.e.we will try to find

c = arg maxk=1,...,N2

{logPB(b1, . . . , bk−1|p1) + logPB(bk, . . . , bN2 |p2)} ,

where PB(b1, . . . , bk−1|p1) is the probability that the series {b1, . . . , bk−1} happen underthe unknown parameter p1. We will use an estimate pi for the calculation, which isestimated from the binary series by

pi =ni,k(1)

ni,k(0) + ni,k(1), i = 1, 2,


where ni,k(j) is the number of js in sub-series i, when the whole series is devided intotwo at the kth point. Finally we have the analytical expression of c

c = arg maxk=1,...,N2

{n1,k(1) log

(n1,k(1)

n1,k(1) + n1,k(0)

)+ n1,k(0) log

(n1,k(0)

n1,k(1) + n1,k(0)

)+ n2,k(1) log

(n2,k(1)

n1,k(1) + n2,k(0)

)+ n2,k(0) log

(n1,k(0)

n2,k(1) + n2,k(0)

)}

Figure 4.20 shows the log-likelihood as a function of k. Statistically, we would say thatunder the assumption that there is one changepoint, the point 77 is the most likelyto be the changepoint (while the likelihood of point 62 is only slightly smaller). Theprobabilities of the 2 subsets are

p1 = 0.684, p2 = 0.485.

Following the log-likelihood argument we can easily generalize the approach to multi-

Figure 4.20: Log Likelihood

changepoint situations. However, too many changepoints in the assumption would turnthe approach into a purely numerical game: statistically the best solution would be thatevery point follows a distribution itself. Thus, to avoid the risk of over-fitting we shouldintroduce a penalty function.A more serious problem is that we directly assume there is a changepoint, thus runthe risk that we mistakenly judge that there is a changepoint when this is not the case.A hypothesis-testing procedure is ideal, but in this case it’s not easy to find a properstatistic that follows a common parametric distribution. This problem will also bediscussed in the following sub-sections.

4.3.2 Single-Changepoint Case

A changepoint indicates a statistically significant change in the underlying distribution.Under the assumption that there can be at most 1 changepoint, detecting such a point isequivalent to find a point that the two subsets before and after this point show differentstatistical features. Thus, the problem becomes to compare two sample distributions.


A famous way is the aforementioned likelihood-based approach, which is to go throughevery possible partition and choose the point that both has statistical significance andmaximizes the associated likelihood. The entire procedure can be characterized as fol-lows:

• Construct the hypotheses. Usually

– H0: there is no changepoint;

– H1: there is a single changepoint.

• Construct a statistic S that is related to the likelihood, and choose a threshold κ,such that we reject the null hypothesis if S > κ.

• Go through every possible partition and if H0 can be rejected, choose the pointwhose associated S is maximized.

A direct approach is to construct S according to the likelihood function; however, themost challenging part then is to find a “proper” threshold.Another way is that for each partition we compare the two sub-series by a statistical test.We use the p-value of the associated statistical test instead of the likelihood function:the changepoint is the point which gives the smallest p-value (thus the most significantresult).The procedure is

• Construct the hypotheses. Usually

– H0: there is no changepoint;

– H1: there is a single changepoint.

• Choose a statistical test and a significance level α.

• For any partition, do the test based on the two sub-series and compute the p-value.Reject H0 if the p-value is smaller than α.

• Go through every possible partition and if H0 can be rejected, choose the pointwhose p-value is minimized.

Note that for some tests, minimizing the p-value is equivalent to maximize the associatedtest statistic; which one to choose depends on the test we use.In the following content we will follow the latter approach. Both parametric tests andnon-parametric tests will be used; R will be the main implementation tool.

4.3.2.1 Parametric Methods: Changes In Mean/Variance

Consider the series {Xi, i = 1, . . . , N} with realizations {xi, i = 1, . . . , N}. In theparametric setting, we will assume that all Xis follow a distribution with PDF p(x; θ)and the parameters θ ∈ Θ. We further assume that the changepoint means a changeonly in parameter values.If we assume the underlying distribution to be normal, then a change in mean can betested by the paired t-test and a change in variance by the paired F-test. For the pairedt-test, if we don’t assume the equality of sample sizes as well as the sample variancesbetween the two samples, then the Welch’s t-test would be appropriate.


Proposition 4.8. (Welch’s t-test for two samples with unequal variance.) Let {Xi, i =1, . . . , n} and {Yi, i = 1, . . . ,m} be two iid series from two normal distributions. Thenunder the null hypothesis that the two distributions have the same mean, we have

tW :=X − Y√S2Xn +

S2Ym

∼ t(df),

and the degree of freedom df can be approximated by

df ≈

(S2Xn +

S2Ym

)2(S2Xn

)21

n−1 +(S2Ym

)21

m−1

where S2X and S2

Y are sample variances of the two series.

Proof. See [31].

Also, as we have to compare the variances from two samples whose sizes are different,the Bartlett’s test may satisfy our need.

Proposition 4.9. (Bartlett’s test for two samples.) Let {Xi, i = 1, . . . , n} and {Yi, i =1, . . . ,m} be two iid series from two normal distributions. Then under the null hypothesisthat the two distributions have the same variance, we have

SB :=(m+ n− 2) ln(S2

p)− (n− 1) ln(S2X)− (m− 1) ln(S2

Y )

1 + 13

(1

n−1 + 1m−1 −

2m+n−2

) ∼ χ2(1),

where

S2p =

1

m+ n− 2

[(n− 1)S2

X + (m− 1)S2Y

].

Proof. See [29].

With regard to judging the changepoint, we do the following:

1. For every k = 1, . . . N , split the series into two at the point k (the point k belongsto the second series).

2. Compute the associated statistics tW (k) and SB(k) for each k.

3. If for all k no significant cases are detected, conclude that there’s no changepoint.Otherwise, denote K = {k = 1, . . . N |the associated tests give significant results}and go to step 4.

4. For the detection of change in mean, let the changepoint be

arg maxi∈K

{|tW (i)|}.

For the detection of change in variance, let the changepoint be

arg maxj∈K

{SB(j)}.


4.3.2.2 Parametric Methods: Examples

The package cpm in R gives the implementations of the change-in-mean/variance testsmentioned above. We will apply change-in-mean test to the Auction deviation seriesand change-in-variance test to the EOD price change series.We first consider the Auction deviation series from 01/2013 to 06/2013, {ADi, i =1, . . . , N2}. As is discussed in Sub-section 4.2.5, from a J-B test we may say that thedata are from a normal distribution. We perform the change-in-mean test based onProposition 4.8: results are

> library(cpm) # load the cpm package

> AD.cptm <- detectChangePointBatch(AD, cpmType = ’Student’, +

+ alpha = 0.05) # AD is the Auction deviation series

> AD.cptm$changeDetected

[1] FALSE

This means that we cannot refuse the null hypothesis, and no changepoints are detected.The test shows that statistically there’s no evidence that shows the mean of the Auctiondeviation has changed.To give an example on the change-in-variance test, let us consider the EOD price changeseries from 04/2009 to 06/2013: the absolute return series AR = {∆ρ

i , i = 1, . . . , N1},and the log-return series LR = {δρi , i = 1, . . . , N1}. We would like to see which one ismore stable. As is discussed in Section 4.2, we have rejected the hypothesis that thetwo series may come from a normal distribution. However, here we may assume firstthat the underlying distributions are indeed normal distributions; some later results inSection 4.4 would tell us the AR series may be seen as a composition of a normal seriesand some extreme values.Figure 4.21 and 4.22 show the change-in-variance test based on Proposition 4.9. Thevertical line shows where the changepoint lies.


> AR.cptm <- detectChangePointBatch(AR, cpmType = ’Bartlett’, +

+ alpha = 0.05) # AR is the absolute return series

> AR.cptm$changeDetected

[1] TRUE

> AR.cptm$changePoint

[1] 1052

> LR.cptm <- detectChangePointBatch(LR, cpmType = ’Bartlett’, +

+ alpha = 0.05) # LR is the log-return series

> LR.cptm$changeDetected

[1] TRUE


[1] 567

From the figures we see that the variance of the AR series almost stays the same, whilethe variance of the LR series rises significantly in the middle. We would conclude thatthe absolute return series shows more stability than the log-return series: the EODabsolute returns are not proportional to the price, as revealed by the data we use. As


Figure 4.21: Change-In-Variance Detection, AR

Figure 4.22: Change-In-Variance Detection, LR


the EUA price gets lower, the log-return becomes larger in value and more volatile aswell.

4.3.2.3 Non-Parametric Methods: Examples

Theorem 4.6 tells us that the K-S test can be applied to test any distributional changesbetween two series. The package cpm in R gives the implementations of the two-sampleK-S tests mentioned above.As described in [28], if we use the K-S statistic as our target, it’s easier to computethe associated p-value under the null hypothesis rather than using directly the statisticitself. Once detected, the changepoint is

arg maxi∈K

{1− pKS(i)}.

We first consider the Auction deviation series. We perform the two-sample K-S testbased on Theorem 4.6: results are


> AD.cptm <- detectChangePointBatch(AD, cpmType = ’Kolmogorov- +

+ Smirnov’, alpha = 0.05) # AD is the Auction deviation series

> AD.cptm$changeDetected

[1] FALSE

Still no changepoints are detected even if we change to a non-parametric method.Next we perform the change-in-variance test to the AR and LR series respectively. Theresults are


> AR.cptm <- detectChangePointBatch(AR, cpmType = ’Kolmogorov- +

+ Smirnov’, alpha = 0.05) # AR is the absolute return series

> AR.cptm$changeDetected

[1] TRUE


[1] 30

> LR.cptm <- detectChangePointBatch(LR, cpmType = ’Kolmogorov- +

+ Smirnov’, alpha = 0.05) # LR is the log-return series

> LR.cptm$changeDetected

[1] TRUE


[1] 563

Compared to the results in sub-section 4.3.2.2,

• Still Figure 4.23 and 4.24 show similar results as in sub-section 4.3.2.2: absolutereturn series shows a more stable variance structure, i.e. the magnitude of changeis not seriously influenced by the magnitude of the price.

• The changepoint in AR series is 1052 by Bartlett’s test and is 30 by the K-Stest. This means that the majority of the AR series has a more stable volatility


Figure 4.23: Changepoint Detection Based On K-S Test, AR

Figure 4.24: Changepoint Detection Based On K-S Test, LR


structure. Note that when the size of one sample is too small, the test can be verysensitive to the values in that sample.

• The changepoint in LR series is 567 by paired Bartlett’s test and is 563 by theK-S test: they are almost equal. This means that the change is significant enoughand different methods can successfully give similar results.

4.3.3 Multi-Changepoint Case

In the previous several sub-sections we presume that there can only be at most ONEchangepoint, if there is any. This, on one hand, simplifies the detection procedure,while on the other hand makes the test too restrictive and inflexible. If one wants togeneralize the methodology to multi-changepoint case, then the two important questionsto be solved are

• How many changepoints are there? How to decide, statically or dynamically?

• Where are the changepoints? What is the most efficient way of searching?

A popular way is to minimize the score function

κ(x,Σ, θ) + π(θ),

where κ is a cost function e.g. negative log-likelihood, which measures the goodness offit; π is a penalty function which encourages parsimony. x is the observed data, θ is theset of changepoints and Σ is the associated partition.There have been some problems in the area of multi-changepoint detection which arestill left open, and there have been some papers published trying to solve them, as arebriefly summarized in [17].We won’t go further in theory on this point; one thing worth mentioning is that theR package changepoint enables us to implement some approaches. Figure 4.25 isthe change-in-mean detection result when we apply the Binary Segmentation (BinSeg)method, and Figure 4.26 is the change-in-variance detection result when we apply thePruned Exact Linear Time (PELT) method. For BinSeg method, we have to decidethe maximal number of changepoints. The PELT method will do that itself; however,the result on LR series shows that it may suffer the over-fitting problem. As we havediscussed, it’s not easy to appropriately select a good cost function and a good penaltyfunction; by doing this the method becomes too much artificial and may not deserve toomuch belief in it.

4.4 Jump Detection: A Non-Parametric Methodology

As we have discussed in Chapter 2, the EUA price has been rather volatile. From timeto time there are jumps in the spot price curve. Strictly speaking, we haven’t given anyformal definition of what jumps are; most of the time people judge a jump by intuitivelylooking at the price charts and pointing to the time when the price suddenly “rose” or“dropped” with a great magnitude.If we turn the price series into a first-order difference series, we can then characterizethe jumps as any outliers w.r.t. a appropriately pre-chosen threshold. Then we turn the


Figure 4.25: Multi-Change-In-Mean Detection On EOD Price, BinSeg Method

Figure 4.26: Multi-Change-In-Var Detection On Log-Return, PELT Method


jump-detection problem into a statistical puzzle: how should we choose the thresholdso that we can properly filter out the jumps according to the statistical features of theseries?In this section we will give two simple but efficient ways of jump detection tailoredrespectively to the tick data and to the EOD price series. The final purpose is to choosea threshold easily but reasonably. It can also be possible to construct more complicatedstrategies; however, we should be aware of the over-fitting pitfalls.

4.4.1 Static vs. Dynamic Filtering

Basically there are two ways of choosing a threshold:

• The threshold is chosen BEFORE the filtering procedure. In the content we callthis a static filtering.

• The threshold is chosen DURING the filtering procedure. In the content we callthis a dynamic filtering.

With regard to the tick data price change series {∆pn}, as we have discussed, it has really

a high kurtosis, which means it has a thin peak and/or a fat tail. Most of the time thechanges are 1 or 2 cents, however occasionally large changes do occur. Based on theseconsiderations we may propose the following prefixed-threshold filtering :

1. Choose a two-sided quantile level α;

2. Get the α2 - and (1 − α

2 )-quantiles from the ECDF of the series {∆pn}. The two

quantiles are the chosen upper and lower thresholds.

3. Filtering: any outliers, i.e. points that lie on or outside the threshold lines areregarded as jumps.

This is indeed a static filtering procedure: thresholds are chosen before the procedurestarts.With regard to the EOD prices change series: the absolute return {∆ρ

n} and the log-return series {δρn}, we see from Figure 4.16 and 4.17, that except for a few points, themajority of the points lie almost in a straight line, which means that the two sets mayshare some distributional features with the normal distribution. We then propose thefollowing 3σ filtering, which originates from the fact that in a sample from a normaldistribution 99.74% of the data should lie within the interval [µ− 3σ, µ+ 3σ].

1. Compute the mean µ and the sample standard deviation σ of the series.

2. Let µ± 3σ as the upper and lower thresholds.

3. Do the filtering: any outliers w.r.t. the thresholds are regarded as jumps, whichwill be removed from the series. We then get a new series.

4. Do iterations: repeat steps 1-3 w.r.t. the new series, until no more jumps can bedetected.

This is indeed a dynamic filtering procedure: thresholds are chosen during the procedure.


4.4.2 Implementation And Results

In this section we will implement the filtering strategies to the corresponding data series.Both VBA and R will be used.

4.4.2.1 Jump Filtering: Tick Data

We choose 0.1% to be the two-sided quantile level. Then the corresponding thresholdsare

> quantile(TDP, 0.0005) # TDP is the tick-data price change series

0.05%

-0.07

> quantile(TDP, 0.9995) # TDP is the tick-data price change series

99.95%

0.07

Figure 4.27: Tick Data, Fixed-Threshold Filtering

The results are shown in Figure 4.27; the two blue horizontal lines are the thresholds.Altogether, 163 points are detected as jumps out of 165339 points altogether.This filtering approach is relatively simple. This is because that the tick data pricechanges show simple distributional features: most of the time the price changes by


several cents. Another comment would be that there have been many jumps detected(also strikingly shown in Figure 4.27) in the middle of the series. These jumps mainlyhappened on the day 16/04/2013, when the aforementioned backloading vote took place.The market was struggling to find a new price after the regulator refused to reduce futuresupply.

4.4.2.2 Jump Filtering: EOD Price

We implement the 3σ filtering in the environment of VBA. The filtering procedure aredone to both the absolute return series and the log-return series.Figure 4.28 and 4.29 show the locations of the jumps. The dark blue lines denote thepoints that are detected as jumps.

Figure 4.28: Abs. Return, 3σ Filtering

Figure 4.29: Log-Return, 3σ Filtering

Altogether, for the AR series 12 points are detected as jumps out of 1086 points after 4iterations, while for the LR series the number of jumps detected is 40 after 5 iterations.Figure 4.30 and 4.31 shows the jumps in the EUA spot price curve.As is shown, the 3σ method does not produce quite desirable results when it is appliedto the log-return series. The method then becomes sensitive to the price changes whenthe price is low. This is because the EUA price change is not proportional to the price


Figure 4.30: Jumps In The Price Curve, 3σ Filtering On AR

Figure 4.31: Jumps In The Price Curve, 3σ Filtering On LR

itself, thus the log-return is not a good measure. This has been revealed already inSection 4.3, when the AR series show more stability in variance along time.We will conclude the implementation by a normality test to the filtered AR and LRseries. Figure 4.32 and 4.33 are the QQ-plots of the two filtered series against normaldistribution. J-B and K-S tests give the following results:

> jarque.bera.test(ARF) # ARF is the filtered AR series

Jarque Bera Test

data: ARF


>

> ks.test(scale(ARF), "pnorm") # ARF is first normalized


data: scale(ARF)

D = 0.0364, p-value = 0.1158


Figure 4.32: QQ-Plot, Abs. Return After Filtered

Figure 4.33: QQ-Plot, Log-Return After Filtered



>

> jarque.bera.test(LRF) # LRF is the filtered AR series

Jarque Bera Test

data: LRF


>

> ks.test(scale(LRF), "pnorm")


data: scale(LRF)

D = 0.0491, p-value = 0.01293


With a 5% significance level, both the J-B test and the K-S test tell us the filtered ARseries almost follows a normal distribution. On the other hand, both the J-B test andthe K-S test reject the normality assumption on the filtered LR series.

4.4.3 The Structure Of The AR Series

As we have seen, with regard to the EOD price changes, the absolute return seriesshows a more stable volatility structure. As we have seen, the 3σ filtering proceduredecomposes the absolute return series into a normal distributed series with a relativelystable variance, and a jump process.The AR series has a relatively stable volatility structure, while the associated LR seriesis not. This means that the changes in the EUA spot price does not depend on themagnitude of the price. For prices in the stock market or energy market this is not thecase; the changes of some price are roughly proportional to the magnitude of the priceitself. 5

4.4.3.1 Detected Jumps In AR Series

Most of the jumps we have detected from the AR series by the 3σ filtering can be foundto have a clear connection with significant market news, as we can see below.

• 14/04/2009: N/A.

• 08/06/2009: N/A.

• 21/12/2009: This is the first trading day after the failure of the 2009 UN ClimateChange Conference (the “Copenhagen summit”). The conference ended only witha weak deal to combat with the global warming, far from the market expectation.Carbon prices went down significantly as a response to the disappointing result.

5As for comparison, Appendix A gives 3 price curves: the S&P 500 index, the price of the ING Groupstock in the New York Exchange, and the Brent Crude Oil price.


Table 4.4: Jumps Arrival Time Stats (In Years)

Statistics ∆tn Exp(λ)

Mean 0.3643 λ−1

Std Dev 0.3507 λ−1

Skewness 1.390 2Ex. Kurtosis 4.4259 6

• 16/03/2011: After the Fukushima nuclear plant disaster, the German governmentannounced that it would temporarily shut down 7 of its 17 reactors, i.e. all reactorsthat went online before 1981. The carbon prices went up as traders saw futuredemands on coal-generated electricity.

• 23/06/2011: The European Commission made its proposal for an Energy EfficiencyDirective, which brings forward measures to step up Member States efforts to useenergy more efficiently. Carbon prices went down sharply as a response to thepotential cutting-off of GHG emissions in the future.

• 24/06/2011: The second day in a row when the emissions market reacted on thepublication of the EC Energy Efficiency Directive.

• 20/12/2011: The EU Environment Committee voted to approve a backloadingproposal to boost carbon price, which would allow the European Commission towithhold up to 1.4 billion EUAs during the phase 3 of the ETS.

• 02/04/2012: Publication of 2011 verified emissions data. The data showed thatthe real emissions were below the EU ETS cap and evidence was made that thetrading scheme was oversupplied for a sixth consecutive year since the launch ofthe European cap and trade system in 2005. This triggered a strong downwardmove on the carbon prices.

• 15/11/2012: The European Commission announced a market reform plan. Themarket was disappointed by the plan since the market was expecting more seriousreforms.

• 23/01/2013: The European Parliament Committee on Industry, Research andEnergy (ITRE) rejected the backloading plan, leading the EUA Dec-13 prices tobelow 3 Euros, a record-low level.

• 01/02/2013: The German Chancellor Angela Merkel supported plans of the en-vironment minister, which was seen as indication of support for the backloadingproposal. The market responded positively and the prices jumped above 4 Euros.

• 16/04/2013: The backloading proposal, which proposed deferring the auctioning of900 million EUAs in future years, was narrowly rejected by a vote in the Europeanparliament. The EUA price dropped sharply from 4 Euros to below 3 in minutes.

If the price jumps in the real-time price data are signs that the market takes pains to finda new equilibrium in turbulence, then the price jumps in the EOD prices are caused byan imbalance between the long force and the short force. Such an imbalance is usuallycaused by the market news, like those we have mentioned above.Another result w.r.t. the jumps is that statistically speaking, we cannot reject the as-sumption that the inter-arrival time of the jumps are exponentially distributed, i.e. thenumber of jumps can be modelled by a Poisson process.


> library(exptest) # load the exptest package

> ks.exp.test(JT) # JT is the jumps arrival time set

Kolmogorov-Smirnov test for exponentiality

data:

KSn = 0.1569, p-value = 0.8045

This means that we cannot refuse the null hypothesis that the jumps arrival time seriesis from an exponential distribution. We should be aware of the sample size we use here,though; with only 12 detected jumps we should be careful with the results we get fromthem. We can be more confident with the result if we get more detected jumps.

4.4.3.2 The Filtered AR Series

The filtered AR process is almost homogeneous in volatility and both J-B and K-Stest tell us that the data in the series may come from a normal distribution. Under arelatively moderate assumption that the filtered series is stationary, the ACF and PACFfunctions show that statistically the series resembles a (shifted) white noise series; seeFigure 4.34 and 4.35.

Figure 4.34: ACF Function, Filtered AR

However, we should be aware that this may not be valid when the EUA price is reallylow, e.g. below 1 Euros when there’s little space for the price to further go down. Westill haven’t seen the EUA price to fall below 1 Euro (nor do we hope).


Figure 4.35: PACF Function, Filtered AR

4.4.3.3 Comments On Modelling The EUA Spot Price

We have shown that w.r.t. the variance, the AR series show more stability than LRseries. Thus it may not be appropriate to model the diffusion part of the EUA spotprice using the classic Black-Scholes model. Recall that the B-S dynamics implies underthe objective measure P

d lnSt = qdt+ σdWt,

which can be discretized as

ln

(StSt−1

)= q + σ(Wt −Wt−1),

which indicates that the LR series should be stable in variance along time.Our analysis above gives motivation to model the price based on the AR series. Aswe have discussed, the AR series can be decomposed into two parts: the diffusion partwhich resembles a (shifted) white noise, and a jump part with the associated countingprocess being the Poisson process.However, when one tries to apply any quantitative models, he should be aware that

• It’s important to avoid negative prices. Merely a superposition of the aforemen-tioned diffusion part and jump part cannot guarantee the positiveness of the price;thus, to build a model we need more assumptions and then we can add moreconstraints into the model to avoid negativeness.

• When it comes to calibration, the historical data is more reliable than the marketoptions data, as the options market has been suffering from liquidity problems.


Calibrating the model to the historical data is an optimization problem based ona maximum likelihood approach.

Appendix A

Sample Prices In Other Markets

In Chapter 4 we showed that the absolute return of the EUA spot price has a relativelystable volatility structure. The absolute return is not dependent on the magnitude ofthe price itself.However, this is not always the case. In this appendix we choose 3 price curves forcomparison: the stock price of ING Groep N.V. in the New York Stock Exchange, theS&P 500 Index, and the Brent Crude Oil spot price.A short comment: for all these 3 prices, neither the absolute return nor the log returnshows the stability like the absolute return of the EUA spot prices. For these prices it’smore complicated to find a good quantitative model candidate.

71

Appendix A. Sample Prices In Other Markets 72

A.1 Stock Price: ING Groep N.V.

Data source: Yahoo! R© FinanceDate range: 01/2000 - 08/2013Prices are in dollars and are adjusted for dividend.

Figure A.1: ING N.V.: Price

Figure A.2: ING N.V.: Abs Return

Figure A.3: ING N.V.: Log Return


A.2 Stock Index: S&P 500

Data source: Yahoo! R© FinanceDate range: 01/2000 - 08/2013

Figure A.4: S&P 500 Index: Price

Figure A.5: S&P 500 Index: Abs Return

Figure A.6: S&P 500 Index: Log Return


A.3 Energy Price: Brent Crude Oil

Data source: US Energy Information Administration (EIA) websiteDate range: 01/2000 - 08/2013The unit is dollars per barrel.

Figure A.7: Brent Crude Oil: Price

Figure A.8: Brent Crude Oil: Abs Return

Figure A.9: Brent Crude Oil: Log Return

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