Carbon Trading. Diederik Dian Schalk Nel. Christ Church University of Oxford

Transcription

1 Carbon Trading Diederik Dian Schalk Nel Chris Church Universiy of Oxford A hesis submied in parial fulfillmen for he MSc in Mahemaical inance April 13, 29

2 This hesis is dedicaed o my parens Nana and Schalk for all heir love and suppor.

3 Acknowledgemens I would like o hank my supervisor, Professor Sam Howison, for his suppor and guidance hroughou his hesis. A special hanks o my friend Eric Chin for all our mahs discussions we had. He made me hink of mahemaics in a differen way. I am graeful o he Isle of Man Governmen and REH plc for heir financial suppor. Many hanks o all my new friends I made a Oxford for making i such a wonderful experience.

4 Absrac Carbon rading is seen as a financial insrumen where markes can be used o lower greenhouse gas emissions such as carbon dioxide. Wih a good undersanding of he sochasic carbon price dynamics, companies can hedge heir posiions, aggregae risk exposure or speculae on underlying price movemens, and ulimaely have he abiliy o rade carbon derivaives. In his sudy we firs esablish a rading plaform for a wo-company scenario in a muli-period seing and derive an equilibrium carbon spo price. By using an Ornsein- Uhlenbeck process we can capure he mean-reversion characerisics associaed wih carbon emissions which in urn allows us o easily exend he equilibrium spo price model o a muli-company scenario. We hen propose a sochasic differenial equaion o capure he behaviour of he unique equilibrium spo price. In addiion by using a no-arbirage principle we can consruc a fuures sochasic differenial equaion and hence derive under risk-neural dynamics carbon opions on fuures. In paricular by incorporaing fuures prices of elecriciy and naural gas we exend our findings by analysing clean spark spread opions. inally we inerpre he above findings wih numerical resuls based on exensive Mone Carlo simulaions.

5 Conens 1 Inroducion 1 2 CO 2 Spo Price Dynamics Two-Company Scenario in Muli-Period Seing Modelling he CO 2 Emissions Esimaing he Equilibrium CO 2 Spo Price Muli-Company Scenario in Muli-Period Seing CO 2 Opion Pricing Modelling he CO 2 Spo Price Pricing CO 2 uures Conracs Pricing CO 2 Opions on uures Pricing Clean Spark Spread Opions Simulaion Resuls Opions on uures Clean Spark Spread and Spark Spread Opions Conclusions 43 A Equilibrium CO 2 Spo Price Esimaion of a Muli-Company Scenario in a Muli-Period Seing 45 B Clean Spark Spread PDE 49 C Theoreical Approximaion for a Clean Spark Spread Call Opion Price 51 Bibliography 57 i

6 Chaper 1 Inroducion Climae change is one of he greaes environmenal challenges of our generaion. Over he las decade numerous scienific sudies have proved ha greenhouse gases such as carbon dioxide CO 2 undoubedly conribues o climae change. The key concep underpinning carbon rading or CO 2 rading is he noion ha markes can be used o reduce greenhouse gas emissions a he lowes possible abaemen cos. A he Rio Earh Summi in 1992, where he idea of rading polluion among companies was firs suggesed, one of he speakers said: The righ wing objeced o our vision because hey hough we were environmenaliss. The lef wing objeced because hey hough we were capialiss [Sandor, 24]. Whichever way i is seen, he Kyoo Proocol and is carbon emissions rading scheme is here o esablish a marke and essenially reduce greenhouse gas emissions. More specifically carbon rading is seen as a useful vehicle o help member sae governmens of he Kyoo Proocol in achieving heir arges insead of using he old command-and-conrol approach such as environmenal axes. example [Muller and Meselman, 1994]. or more deailed economic comparisons see for The European Union Emission Trading Scheme EU ETS is formed under he Kyoo Proocol and also referred o as a cap-and-rade sysem. 1 The cap indicaes a arge or a limi which lowers over ime whereas he rade refers o radable credis, which in his case are CO 2 credis. 2 One credi gives he holder he righ o emi one onne of CO 2. Each member sae governmen submis a naional allocaion plan NAP o he EU before a rading phase sars, deermining he amoun of CO 2 credis each of heir indusry secors 1 The EU ETS is esablished in accordance wih Direcive 23/87/EC of he European Parliamen and Council 13 Ocober 23. Esablishmen of a Scheme for Greenhouse Gas Emission Allowance Trading 23: hp://ec.europa.eu/environmen/clima/emission/ 2 More accuraely referred o as European Union Allowances EUAs. Allowances, permis or credis are used synonymously. Noe ha anoher cap-and-rade sysem, mosly familiar in he Unied Saes, are sulphur rading wih sulphur credis. 1

7 and ulimaely companies will receive. 3 The cap on he oal number of credis graned or aucioned is wha creaes he scarciy in he marke as his is overall shor. 4 Companies ha paricipae in he EU ETS are mosly from he cemen, chemical and power generaion secors wih a fuure possibiliy o include he aviaion indusry. 5 By April 3 each year he operaor of each insallaion company surrenders a number of credis equal o he oal emissions of he preceding calendar year. 6 These emissions are verified by hird paries on and off-sie and he surrendered credis cancelled via a Communiy Independen Transacion Log CITL. Companies ha keep heir emissions below he level of heir credi allowances can sell heir excess credis. Those who have rouble keeping heir emissions in line wih heir credis have a choice beween aking measures o reduce heir own emissions, such as invesing in more efficien echnology, using less carbon-inensive energy sources like fuel swiching for power saions as seen in [Rong and Lahdelma, 27], or buying addiional credis on he marke. Unused credis can be banked save some of his year s credis o be used nex year bu no borrowed use some of nex year s credis now and no have hem available nex year. Any operaor who does no surrender sufficien credis o cover is excess emissions is liable for he paymen of an excess emissions penaly of e1 for each onne of CO 2 emied. 7 Paymen of he penaly does no release he operaor from he obligaion o surrender an amoun of credis equal o hose excess emissions when surrendering credis in relaion o he following calendar year. The key assumpion behind his ype of sysem is ha he rading of credis among he paries will allow a given reducion o be achieved a he lowes possible abaemen cos for he se of capped insallaions as a whole. Therefore he cap-and-rade sysem places a higher price on polluion as he demand rises over ime for he credis and evenually makes i cheaper o inves in cleaner echnologies. In addiion o he CO 2 credis raded wihin he EU ETS are he Kyoo s flexible projec-based mechanisms such as he Clean Developmen Mechanisms CDM and Join Implemenaion JI. Boh schemes allow governmens o inves in emissions-reducion projecs in developing for CDM and indusrialised for JI counries o achieve reducion agains heir se Kyoo arges. Companies can also paricipae and each of hese credis is equivalen o a CO 2 credi in EU 3 Phase I was from Phase II is from and Phase III will be from NAPs mus be approved by he EU before a rading phase begins. More deailed allocaion of credis secorspecific, for example he UK NAP submied in 26 for Phase II, can be found a 4 Member saes are allowed o aucion up o 1% of he oal credis and 9% of he oal credis are allocaed for free in Phase II. 5 Power generaion faciliies mus be greaer han 2 Megawa MW in generaion capaciy. Draf proposals have been drawn up o include aviaion emissions in he fuure. 6 See Aricle 123 of Direcive 23/87/EC. 7 See Aricle 163 of Direcive 23/87/EC. 2

8 ETS allowing for one onne of CO 2 o be emied. EU member sae governmens may se heir own CDM credi limis. In he UK for example companies may use CDM credis up o a limi of 8% of heir free carbon allocaion. In Ireland, by conras, a much higher proporion of CDM credis can be used - up o 22%. Carbon rading is sill a relaively new concep, however we foresee in he near fuure a porfolio of all he above CO 2 credis on for example a power saion s books forming par of overall abaemen cos sraegies. 8 Mos lieraure available on carbon rading consis of economeric analysis of CO 2 credis wih some marginal abaemen cos sudies and he effec he carbon price have on insallaions. A few papers however discuss carbon spo price modelling. In he paper by [Chesney and Taschini, 28], he auhors base heir argumens of heir CO 2 model on a wo-company scenario in a muli-period seing, which hey hen generalise ino a mulicompany scenario. The companies are characerised by exogenous emissions processes, which hey assume o follow a Geomeric Brownian Moion GBM wih no correlaion beween he companies emissions. More specifically hey inroduce he presence of asymmeric emissions informaion. or example in a wo-company scenario hey assume company 1 has complee knowledge of is own accumulaed emissions, bu only parial knowledge of company 2 s accumulaed emissions from company 1 s percepion and herefore imposes a lag-effec on he adjused expeced fuure emission levels of he oher company. This model works well since each company opimises is cos funcion by adjusing is CO 2 credi porfolio allocaions according o he expeced fuure CO 2 credi posiions. [Cein and Verschure, 28] discuss a model where banking of CO 2 credis is allowed beween phase I and phase II and derive a no-arbirage relaion o produce an explici semimaringale represenaion of he CO 2 spo price. urhermore he auhors derive an explici formula for hedging, using a local risk minimisaion approach, under he assumpion ha he marke s ne posiion is common knowledge among all marke paricipans. ollowing he same argumens as in [Chesney and Taschini, 28], in his sudy we esablish a rading plaform for a wo and muli-company scenario wihin a muli-period seing o derive an equilibrium CO 2 spo price. We also impose a lag-effec on he adjused expeced fuure emissions bu insead of modelling he emissions process of each insallaion wih a GBM, we propose a more realisic Ornsein-Uhlenbeck OU process. 9 urhermore we incorporae a correlaion beween he carbon emissions of companies. Given ha each of our emissions processes is normally disribued, he sum of all companies emissions 8 or simpliciy we will ignore he credis generaed from CDM and JI projecs in his sudy and is lef for fuure research. 9 CO 2 emissions daa was provided by The Naional Oceanic and Amospheric Adminisraion NOAA for various power saions in he USA. Saisical analysis shows CO 2 emissions for power saions end o have mean-revering characerisics. 3

9 processes is also normally disribued, and hence we can easily exend he opimisaion of CO 2 credis of he equilibrium model o a muli-company scenario. In his sudy we ignore annual banking of unused CO 2 credis which provides us wih a wo-sae soluion by he end of he rading year compliance ime as discussed in [Chesney and Taschini, 28]. Therefore under he assumpion of no marke-power if neiher company is in credi need, all lef-over credis have zero value. On he oher hand, if a leas one of he companies is in credi shorage, since by law all covered companies have o surrender sufficien credis a he compliance ime, he credi has a value equal o he penaly value. or he purpose of pricing carbon derivaive conracs we capure he characerisics of he bounded equilibrium spo price by proposing a sochasic differenial equaion SDE which models he equilibrium spo price. The carbon sochasic model is based on characerisics from boh GBM and Brownian bridge diffusion processes where he carbon spo price is eiher zero or a penaly value a he compliance ime in order o saisfy carbon rading rules. urhermore since CO 2 conracs are mosly raded as fuures we derive a fuures price for CO 2 using a no-arbirage principle and hence able o consruc a fuures SDE o price opions on fuures under risk-neural measure. In paricular we exend he opions on fuures o include a derivaive insrumen such as a clean spark spread opion. Concepually a clean spark spread opion is similar o a spark spread opion as seen in [Deng e al., 21] wih he only difference ha he price of CO 2 emissions is now included. or example one can analyse he cos of fuel swiching for power saions beween naural gas and coal from a real opions poin of view. Here coal migh be a cheaper fuel o generae one megawa hour MWh of elecriciy, bu i is more inensive in CO 2 emissions compared o naural gas, and if he price of CO 2 credis is high a he ime, i migh be cheaper o swich o naural gas. More informaion on comparisons beween fuel mix sraegies and how i can provide good radeoff beween profi-making and emissions reducion can be found in [Rong and Lahdelma, 27]. The hesis is arranged as follows: In Chaper 2 we derive he equilibrium spo price for wo and muli-company scenarios in a muli-period seing by combining he emissions of a company wih he rules of carbon rading. In Chaper 3 we discuss opion pricing by proposing a novel sochasic differenial equaion which allows us o model he equilibrium spo price. We hen derive heoreical approximaions o price CO 2 opions on fuures as well as clean spark spread opions for shor mauriies. In Chaper 4 we compare all he heoreical resuls wih Mone Carlo simulaions and finally in Chaper 5 we conclude our sudy. 4

10 Chaper 2 CO 2 Spo Price Dynamics In his chaper we derive he equilibrium CO 2 spo price of a wo and muli-company scenario in a muli-period seing. Here he acual emissions of companies are modelled wih an Ornsein-Uhlenbeck OU process where he emissions are assumed o be correlaed wih each oher. This is in conras wih he model proposed by [Chesney and Taschini, 28] where he emissions are modelled wih a Geomeric Brownian Moion GBM wih no correlaion. urhermore by applying he carbon rading rules he model is based on a wai-and-see scenario for each company and only acs if i needs o and herefore assume a risk-neural approach. Hence if a company requires more CO 2 credis a ime assuming fuure projecion of emissions a compliance ime T indicaes i will be shor hen i will buy more credis provided here are some o buy from he oher company in order o mee is arge a ime T. Similar argumens apply for he selling of CO 2 credis. 2.1 Two-Company Scenario in Muli-Period Seing Modelling he CO 2 Emissions In igure 2.1 we show he daily CO 2 emissions level over a year of a 312 Megawa MW coal power saion in Norh America. The emissions daa is provided by he Naional Oceanic and Amospheric Adminsraion NOAA and covers a period from 1 January 27 o 31 December 27. The CO 2 emissions are measured wih probes siuaed half-way up each of he four exhaus sacks. 1 We noe from igure 2.1 for he firs day in he year he mass of CO 2 recorded was 2 CO 2 where CO 2 represens one onne of carbon dioxide. The emissions level hen increases over he nex few days whereafer i flucuaes around he mean of 3 CO 2. Since he CO 2 emissions of a company, say for example a power saion, have mean-revering characerisics we herefore choose an OU process o model 1 or more informaion on coninuous sack monioring for power saions see for example hp:// 5

11 he emissions. urhermore an advanage of modelling he CO 2 emissions as an OU process is ha i is normally disribued and herefore we can easily esimae he parameers. In he following paragraph we describe how he CO 2 emissions are modelled. 6 Daily CO 2 emissions for a power saion CO 2 emissions level 5 Emissions Level x1 CO Days igure 2.1: Daily CO 2 emissions for a ypical coal power saion over a year. Le Ω,, { },P be he probabiliy space and = { } defined as he filraion where = σ i I Q i,s, s [,, and I = {1, 2} for a given OU process. We assume ha each company coninuously emis CO 2 over he period [, T ] according o an OU process of he form dq i, = θ i Q i, µ i d + σ i dw i, for i = 1, wih E [dw 1, dw 2, ] = ρ 12 d, ρ 12 [1, 1] and T is he compliance ime. The parameers θ i and µ i are he mean-reversion rae and long-erm mean respecively, σ i is he imeindependen volailiy and W i, is a sandard Brownian Moion. Inegraing from o we have Q i, = Q i, e θ i + µ i 1 e θ i + σ i e θ i or a small ime inerval he accumulaed emissions level is [ Q i,s ds = Q i, e θis + µ i 1 e θis + σ i e θ is s e θ is dw i,s. 2.2 e θ ik dw i,k ] ds. 2.3 or a simple analyical inerpreaion of accumulaed emissions level we can approximae he inegral of 2.3 as follows Q i,s ds = Q i, e θ i + µ i 1 e θ i + σ i e θ i e θ is dw i,s. 2.4 ollowing he same noaion as in [Chesney and Taschini, 28] each European Union EU member sae governmen under he Kyoo Proocol is assigned a oal number of 6

12 credis N according o he EU direcive. The iniial credi endowmen allocaed by he governmen o each company i I is given as N i, and he sum of all companies iniial credis is defined as N = N i,. i I The quaniy of credis ha each company i I buys a ime is given as X i, and he quaniy sold a ime is X i,. Therefore he ne amoun of credis ha any given company possesses a ime can be wrien as δ i, = N i, + s= X i,s = N i, + X i, + X i,1 + X i, X i,1 + X i, = N i,1 + X i, = 1, 2,..., T 1 and i = 1, 2 where N i,1 is he iniial credi endowmen plus he sum of he emission credis bough and sold by a company up o ime 1. Since N is fixed, he marke clearing condiion for I = {1, 2} is δ 1, + δ 2, = N or X 1, + X 2, = for all = 1, 2,..., T 1. A ime [, T ] company 1 has complee knowledge of is own accumulaed emissions and he amoun of credis cleared he previous rading day, 1, and expressed as Q 1,s ds δ 1,1. Based on company 1 s percepion, we assume i has parial knowledge abou he accumulaed emissions of company 2 and given as 1 Q 2,s ds δ 2,1. Therefore he presence of asymmeric informaion imposes a lag-effec on he adjused expeced fuure emission levels of he oher company. urhermore he paricipan compares he sum of is emissions and he amoun of CO 2 credis a hand before he end of he rading period. A he end of a rading period all covered companies have o surrender sufficien credis. If neiher company is in credi need, all remaining credis have zero value. Those who are shor of credis a compliance ime T, which is in effec paymen of excess emissions, have o buy addiional credis which is equal o he penaly price. We can herefore express he rading rules as follows: 7

13 1. A ime T, if neiher company is in credi need, all remaining credis have zero value. 2. If a company is in credi shorage, since by law all covered companies have o surrender sufficien credis a ime T he credis have a value equal o he penaly price P. 3. Therefore he bounded spo price value a ime T of he underlying CO 2 credis can be wrien as S T = { if i I, P if i I, T Q i,sds δ i,t 1 T Q i,sds > δ i,t 1 or where S T = P max [ 1{ T Q 1,sds>δ 1,T 1},1 { T Q 2,sds>δ 2,T 1} I = {1, 2} ] 2.5 and 1 {A>B} = { 1 ifa > B ifa B. 4. A ime T, if company 1 is in credi excess, i can sell o company 2 wha he laer wans o buy and we can express he amoun of credis o be sold o company 2 as { T + T + } Γ min δ 1,T 1 Q 1,s ds, Q 2,s ds δ 2,T On he oher hand, if company 1 is in credi shorage, i can buy from company 2 wha he laer wans o sell and we can wrie he amoun of credis o be bough from company 2 as { T Π min + T + } Q 1,s ds δ 1,T 1, δ 2,T 1 Q 2,s ds Therefore combining 2.6 and 2.7 we can wrie a boundary condiion for he credi quaniy a ime T as T + X i,t = Q i,s ds δ i,t 1 Γ i I 8

14 2.1.2 Esimaing he Equilibrium CO 2 Spo Price Under he physical measure P and given he iniial cash-flows a ime = T, where is a small ime inerval, plus he discouned expecaion of he poenial penalies of company 1, wih η he discoun rae, he oal cos a ime = T can be calculaed as H = S T X 1,T + e η E P [S T X 1,T T ]. 2.8 H To find S T, by firs order necessary condiion we se = and herefore have X 1,T H = S T + e η E P [ ] S T X 1,T X i,t + S T T X 1,T X 1,T X 1,T = or S T = e η E P [ ] X 1,T S T T X 1,T 2.9 since S T X 1,T =. urhermore, X 1,T X 1,T = T + Q 1,s ds δ 1,T X 1,T Γ X 1,T = 1 { T Q 1,sds>δ 1,T } 1 {δ1,t > T Q 1,sds} δ 1,T T Q 1,sds if 1 { T Q 2,sds>δ 2,T } if δ 1,T T Q 1,sds + T + Q 1,sds δ 1,T + T + > Q 2,sds δ 2,T and herefore S T X 1,T X 1,T = P1 { T Subsiuing 2.1 ino 2.9 we have Q 1,sds>δ 1,T } P1 {δ 1,T > T Q 1,sds} 1 { T Q 2,sds>δ 2,T }. 2.1 S T = e η PE P [ 1 { T Q 1,sds>δ 1,T } T + e η PE P [ 1 { T Q 1,sds<δ 1,T } 1 { T Q 2,sds>δ 2,T } T ] ]

15 I follows ha E P [ 1 { T Q 1,sds>δ 1,T } T ] = P T Q 1,s ds > δ 1,T T = P T T Q 1,s ds > N 1,T 2 + X 1,T T = P Q 1,T e θ 1 + µ 1 1 e θ 1 + σ 1 e θ 1 > N 1,T 2 + X 1,T = P σ 1 e θ 1 T Q 1,s ds T T Q 1,s ds T e θ 1s dw 1,s e θ 1s dw 1,s > Q 1,T e θ 1 µ 1 1 e θ 1 Q 1,s ds + N 1,T 2 + X 1,T T = P Z 1 < d 1,T T = Φ d 1,T 2.12 where Z 1 = eθ 1 e θ 1s dw 1,s 1 2θ 1 1 e 2θ 1 N, 1 and d 1,T = Q 1,T e θ1 + µ 1 1 e θ 1 T Q 1,s ds N 1,T 2 X 1,T + σ 1 1 2θ 1 1 e 2θ1 such ha Φ d 1,T = d1,t 1 2π e 1 2 z2 1 dz1. urhermore, E P [ 1 { T Q 1,sds<δ 1,T } 1 { T Q 2,sds>δ 2,T } T ] 1

16 = P T Q 1,s ds < δ 1,T, T Q 2,s ds > δ 2,T T = P T T 2 = P T T T T 2 T Q 1,s ds + Q 2,s ds + T T T 2 Q 1,s ds < N 1,T 2 + X 1,T Q 2,s ds > N 2,T 2 + X 2,T Q 1,s ds < N 1,T 2 + X 1,T, Q 2,s ds > N 2,T 2 + X 2,T T T T 2 Q 1,s ds, = P Q 1,T e θ 1 + µ 1 1 e θ 1 + σ 1 e θ 1 < N 1,T 2 + X 1,T Q 2,s ds T T e θ 1s dw 1,s Q 1,s ds, 2 Q 2,T 2 e 2θ µ 2 1 e 2θ 1 + 2σ 2 e 2θ 2 e θ2s dw 2,s T 2 > N 2,T 2 + X 2,T Q 2,s ds T = P Z 1 < d 1,T, Z 2 > d lag 2,T T = Φ d 1,T Φ d 1,T, d lag 2,T 2.13 where and d 1,T = Z 1 = eθ 1 e θ 1s dw 1,s 1 2θ 1 1 e 2θ 1 Z 2 = e2θ 2 2 e θ 2s dw 2,s 1 2θ 2 1 e 4θ 2 N, 1 N, 1 Q 1,T e θ1 + µ 1 1 e θ 1 T Q 1,s ds N 1,T 2 X 1,T + σ 1 1 2θ 1 1 e 2θ1 Q 2,T 2 e 2θ2 + µ 2 1 e 2θ 2 + d lag 2,T = σ 2 1 2θ 2 1 e 4θ 2 T 2 Q 2,s ds N 2,T 2 X 2,T 2 11

17 such ha Φ d 1,T, d lag 2,T = d lag 2,T d1,t 1 2π 1 ρ 2 12 e z 2 1 2ρ 12 z 1 z 2 +z2 2 21ρ 2 12 dz 1 dz 2. Subsiuing 2.12 and 2.13 ino 2.11 yields [ ] S T = e η P Φ d 1,T + e η P Φ d 1,T Φ d 1,T, d lag 2,T [ ] = e η P 1 Φ d 1,T, d lag 2,T 2.14 and similarly from company 2 s perspecive we have [ ] S T = e η P 1 Φ d 2,T, d lag 1,T By seing 2.14 equal o 2.15, assuming he same discoun facor η for each company, and knowing X 1,T = X 2,T, we can find he quaniy of credis, X 1,T, ha a company buys or sells a ime Φ d 1,T, d lag 2,T = Φ d 2,T, d lag 1,T. or general ime seps we can derive he following heorem. Theorem 2.1 In general for each ime sep k = 1, 2,..., T/ we can obain a pair I = {1, 2} of emission price equaions [ S T k = e ηk P 1 E P [ Φ d 1,T, d lag [ S T k = e ηk P 1 E P [ Φ 2,T d 2,T, d lag 1,T Wih he above equaions and he marke clearing condiion ]] T k ]] T k X 1, + X 2, = 2.17 a each ime sep k we can deermine he equilibrium credi price by numerically evaluaing he quaniy of credis X 1,T k and X 2,T k ha saisfies E P [ ] Φ d 1,T, d lag 2,T T k = E P [ ] Φ d 2,T, d lag 1,T T k.2.18 Proof We prove he above resul by means of mahemaical inducion. or he case of ime sep T we have for he spo price dynamics for company 1 and 2 as [ S T = e η P 1 E P [ ]] Φ d 1,T, d lag 2,T T [ ] = e η P 1 Φ d 1,T, d lag 2,T 12

18 and S T = [ e η P 1 E P [ Φ [ = e η P 1 Φ d 2,T, d lag d 2,T, d lag 1,T 1,T ]. ]] T Therefore X 1,T and X 2,T can be found by solving Φ d 1,T, d lag 2,T = Φ d 2,T, d lag 1,T. Hence he resul is rue for ime sep T. We assume he resul is rue for ime sep T k 1 such ha for companies 1 and 2 he emission price processes are S T k1 = [ e η P 1 E P [ Φ d 1,T, d lag S T k1 = [ e η P 1 E P [ Φ 2,T d 2,T, d lag 1,T ]] T k1 ]] T k1. or ime sep T k we can herefore wrie he cos funcion for company 1 as follows: [ k ] H = S T k X 1,T k + E P e ηh S T kh X 1,T kh T k. h=1 To find S T k by firs order necessary condiion we se Since H X 1,T k = S T k + E P [ k h=1 H X 1,T k +e ηh S T kh X X 1,T kh 1,T k = and herefore have e ηh S T kh X 1,T kh X 1,T k T k ]. X 1,T kh X 1,T k = { 1 for h = 1 for h = 2, 3,..., k S T kh X 1,T k = for h = 1, 2,..., k H = S T k E P [ e η ] S X T k1 T k 1,T k and herefore S T k = e η E P [ S T k1 T k ]. 13

19 Making use of he ower propery and knowing S T k1 = e ηk1 P we can wrie S T k = e η E P [ e ηk1 P and hus for company 1 we have S T k = e ηk P Similarly for company 2 we can wrie S T k = e ηk P 1 E P [ ] Φ d 1,T, d lag 2,T T k1 1 E P [ ] ] Φ d 1,T, d lag 2,T T k1 T k 1 E P [ ] Φ d 1,T, d lag 2,T T k E P [ ] Φ d 2,T, d lag 1,T T k. 2.2 Seing 2.19 equal o 2.2, X 1,T k and X 2,T k can be evaluaed by solving E P [ ] Φ d 1,T, d lag 2,T T k = E P [ ] Φ d 2,T, d lag 1,T T k and hence he resul is also rue for ime sep T k. Therefore using mahemaical inducion he proposiion is rue for all k = 1, 2,..., T/. q.e.d As a resul, from Theorem 2.1, he equilibrium spo price is derived using an OU process for he acual emissions insead of a GBM as seen in [Chesney and Taschini, 28]. urhermore in his sudy we incorporae a correlaion coefficien beween he companies emissions and do no assume he emissions of each company o be independen as discussed in [Chesney and Taschini, 28]. This in paricular produces a bivariae sandard normal cumulaive disribuion funcion in he final expression of 2.16 raher han he muliplicaion of wo independen sandard normal disribuion funcions. In he following discussions we simulae he acual emissions for wo companies ogeher wih illusraing he expeced CO 2 credis o be bough or sold by each company, and he equilibrium CO 2 spo price. In igure 2.2a, using 2.1, we simulae he CO 2 emissions Q of each company using an OU process over he period [, T ]. We assume T = 25 days wih ime sep = 1 day. By using he OU process, which consiss of a long-erm mean µ, mean-reversion rae θ, and volailiy σ we can capure he ypical CO 2 emissions for each company. Each company has an iniial emissions level of Q 1, = 5 CO 2 and Q 2, = 25 CO 2 respecively where CO 2 is one onne of carbon dioxide. urhermore we se he long-erm 14

20 8 7 a CO 2 Emissions Level for Company1 and Company2 Q 1, for Company1 Q 2, for Company2 5 4 b Expeced CO 2 Credis o be Bough + or Sold X 1, for Company1 X 2, for Company2 Emissions Level CO Expeced Credis Trading Days Trading Days 1 9 c Equilibrium CO 2 spo price for Company1 and Company2 S 8 Spo Price Euro/ CO Trading Days igure 2.2: a Acual emissions levels of company 1 and company 2 each following an OU process. b Expeced CO 2 credis of company 1 and company 2. c CO 2 spo price hiing penaly P. Parameers are Q 1, = 5 CO 2, Q 2, = 25 CO 2, N 1, = 125, N 2, = 125, µ 1 = 4 CO 2, µ 2 = 2 CO 2, θ 1 =.1, θ 2 =.2, σ 1 =.75, σ 2 =.5, ρ 12 =.5, η =.3, P = 1 e/co 2 and T = 25 days. means µ 1 = 4, µ 2 = 2, he mean-reversion raes θ 1 =.1, θ 2 =.2, he emissions volailiies σ 1 =.75, σ 2 =.5 and he correlaion coefficien ρ 12 =.5. Using he marke clearing condiion in 2.17 ogeher wih 2.18 where = T k, k = 1, 2,..., T and = 1 day, igure 2.2b shows he quaniy of expeced CO 2 credis o be bough or sold by each company a ime. Here each company is assigned an iniial endowmen of credis, N 1, = 125 and N 2, = 125, by is member sae governmen before rading sars. If a company requires more CO 2 credis a ime because fuure projecion up o ime T, indicaes i will be shor hen i will buy more credis provided here are some o buy in order o mee is arge a compliance ime T. Since we assume a risk-neural approach a company acs only if i needs o in order o mee is arge. In his wo-company simulaion we noe ha company 1 has a higher emissions level han company 2 and herefore expeced o buy more credis from company 2 during mos of he rading phase. urhermore we 15

21 can see ha company 2 is always in a posiion o sell is exra credis o company 1. This could be due o a number of reasons for example using a differen fuel ype or invesing in cleaner echnology. Taking noe ha since company 1 s acual emissions keep increasing from around ime = 18 days onwards, he equilibrium spo price of he CO 2 credis will evenually reach a penaly value whichever penaly is specified, due o company 1 s inabiliy o buy more credis from company 2. In igure 2.2c we show he equilibrium CO 2 spo price, S, of he wo companies marke where = T k, T = 25 days, k = 1, 2,..., T wih = 1 day and assuming a discoun rae η =.3 o be he same for boh companies. By seing he penaly P = 1 e/co 2 Euro per onne of CO 2 we noe he spo price is bounded beween zero and P for mos of he rading period. As he rading period draws o a close T and company 1 is in credi shorage and unable o purchase more credis from company 2 he spo price approaches he penaly value P. Therefore if here exiss a leas one company in credi shorage he spo price of he carbon credis will reach P. Given ha all companies have o surrender heir credis a ime T, as k 1, 2.16 approaches S T e η P since E P [ ] Φ d 1,T, d lag 2,T T. In order o illusrae he CO 2 spo price converging o zero, in he following igure 2.3a we simulae a new se of CO 2 emissions for each company. In his realisaion, whils keeping all oher parameer values he same, we only change he mean-reversion rae θ of company 1 from θ 1 =.1 o θ 1 =.5. This can ypically represen company 1 changing is fuel from coal o naural gas which will have a smaller CO 2 emissions inensiy and herefore emi less CO 2. igure 2.3b shows he quaniy of expeced CO 2 credis o be bough or sold a ime in order o be wihin is emissions limi a ime T. Using he same iniial credi allowance for each company N 1, = N 2, = 125 credis we can observe he effec he differen emissions realisaions have on he rading ha akes place. Here company 1 buys a large amoun from an early sage from company 2. A ime = 1 days company 1 s expecaion is ha i will make is arge a ime T which correlaes wih is emissions saring o decrease. A ime = 2 days company 1 has enough credis and do no need o buy more from company 2 and since here is no demand for any credis in he marke he value of he CO 2 credis will decrease o zero. igure 2.3c shows he CO 2 spo price, S, of he wo companies marke where = T k, T = 25 days, k = 1, 2,..., T wih = 1 day and η =.3 he same for boh companies. The spo price reaches a 16

22 8 a CO 2 Emissions Level for Company1 and Company2 Q 1, for Company1 2 b Expeced CO 2 Credis o be Bough + or Sold X 1, for Company1 7 Q 2, for Company2 15 X 2, for Company2 6 1 Emissions Level CO Expeced Credis Trading Days Trading Days 12 c Equilibrium CO 2 spo price for Company1 and Company2 S 1 Spo Price Euro/ CO Trading Days igure 2.3: a Acual emissions levels of company 1 and company 2 each following an OU process. b Expeced CO 2 credis of company 1 and company 2. c CO 2 spo price hiing zero. Parameers are Q 1, = 5 CO 2, Q 2, = 25 CO 2, N 1, = 125, N 2, = 125, µ 1 = 4 CO 2, µ 2 = 2 CO 2, θ 1 =.5, θ 2 =.2, σ 1 =.75, σ 2 =.5 and ρ 12 =.5, η =.3 and T = 25 days. high value during early rading indicaing a srong demand for he CO 2 credis. A ime = 2 days he CO 2 price sars o decrease rapidly, indicaing no buyers, and evenually reaches zero. This scenario emulaes wha happened during he rial phase in 26 when EU governmens announced hey assigned oo many CO 2 credis o heir indusrial secors for fear of paricipaion. This caused he carbon price o crash since here was no incenive o lower emissions, which in urn placed no value on he CO 2 credis. or furher informaion see [Laba, S. and Whie, R. 27, p149]. Given ha all companies have o surrender heir credis a ime T, as k 1, 2.16 approaches S T since E P [ ] Φ d 1,T, d lag 2,T T 1. 17

23 2.2 Muli-Company Scenario in Muli-Period Seing In his secion we exend our argumens used in he wo-company scenario o calculae he equilibrium CO 2 spo price for a muli-company scenario in muli-period seing. Given ha each company s emissions is normally disribued by using an OU process, he sum of he emissions processes for all companies is also normally disribued and herefore we can exend he model ino a muli-company scenario. In he following heorem we presen he equilibrium CO 2 spo price for a muli-company scenario in muli-period seing. Theorem 2.2 Le I = {1, 2,..., m}, where m is he oal number of companies rading CO 2 credis and define he se I l = I {l} as all he companies excluding company l. Given he emissions processes {Q l, } T = for company l = 1, 2..., m such ha dq l, = θ l Q l, µ l d + σ l dw l,, l = 1, 2,..., m 2.21 he price process S = {S } T = is called an equilibrium price process if here exiss {X l, } T = for company l = 1, 2..., m such ha for k = 1, 2,..., T/ E P [ ] Φ d 1,T, d lag 2,T T k, I1 = I {1} = E P [ ] Φ d 2,T, d lag I 1,T T k, I2 = I {2}. = E P [ ] Φ d m,t, d lag Im,T T k, Im = I {m} and he marke clearing condiion is saisfied X l, = l=1 =, 1,..., T. Therefore for k = 1, 2,..., T/, he CO 2 spo price is given as S T k = e ηk P 1 E P [ ] Φ d l,t, d lag I l,t T k

24 where d l,t = Q l,t e θ l + µ l 1 e θ l + σ l 1 2θ l 1 e 2θ l T Q l,s dsn 1,T 2 X l,t and Q I d lag = I l,t I l,t 2 e2θ l + µi l 1 e 2θ I l + σ I l 1 2θ I l 1 e 4θ I l T 2 Q I l,sdsn I l,t 2 X I l 2,T such ha σ I l µ I l = = σi 2 + 2ρ ij σ i σ j i=1,i l i=1,i l µ i i=1,i j,l j=1,j i,l where ρ li l θ I l ρ li l = = σ 2 i θ i + σ 2 I l i=1,i l i=1,i j,l j=1,j i,l ρ il σ θi i l θi i=1,i l σ I l 2ρ ij σ i σ j θi θ j is he Pearson correlaion coefficien of he process {Q l, } and Proof See Appendix A. { } Q I l,. In igure 2.4a and similar o a wo-company scenario we simulae he CO 2 emissions Q for hree companies using 2.21 over he period [, T ], where T = 25 days and ime sep = 1 day. Each company has an iniial emissions level of Q 1, = 5 CO 2, Q 2, = 3 CO 2 and Q 3, = 4 CO 2. We se he long-erm means µ 1 = 4 CO 2, µ 2 = 2 CO 2, µ 3 = 1 CO 2, he mean-reversion raes θ 1 =.1, θ 2 =.2, θ 3 =.3 and he emissions volailiies σ 1 =.75, σ 2 =.5, σ 3 =.3 and he correlaion coefficiens ρ 12 =.5, ρ 13 =.3, ρ 23 =.2. igure 2.4b shows he quaniy of expeced CO 2 credis o be bough or sold by each company a ime. As menioned before we assume a risk-neural approach in our model where a company only acs if i needs o in order o mee is arge a ime T. rom his figure we can see ha company 3 is buying credis during he enire rading year and ha he assigned iniial endowmen of N 3, = 6 credis is no enough and herefore expeced o buy more credis hroughou he year. On he oher hand company 1 and 2 19

25 7 a CO Emissions Level for Company1, 2 and 3 2 Q for Company1 1, 4 b Expeced CO Credis o be Bough + or Sold 2 X for Company1 1, 6 Q for Company2 2, Q for Company3 3, 3 X for Company2 2,1 X for Company3 3, Emissions Level CO Expeced Credis Trading Days Trading Days 1 c Equilibrium CO 2 spo price for Company1, 2 and Spo Price Euro/ CO S Trading Days igure 2.4: a Acual emissions levels of company 1, 2 and 3 following an OU process. b Expeced CO 2 credis of company 1, 2 and 3. c CO 2 spo price hiing penaly P. Parameers are Q 1, = 5 CO 2, Q 2, = 3 CO 2, Q 3, = 4 CO 2, N 1, = 125, N 2, = 8, N 3, = 6 µ 1 = 4 CO 2, µ 2 = 2 CO 2, µ 3 = 1 CO 2, θ 1 =.1, θ 2 =.2, θ 3 =.3. σ 1 =.75, σ 2 =.5, σ 3 =.3, ρ 12 =.5, ρ 13 =.3 and ρ 23 =.2, η =.3, P = 1 e/co 2 and T = 25 days. are in he posiion o sell heir excess credis during he enire rading year due o lower emission levels relaive o heir iniial assigned CO 2 credis. In igure 2.4c we show CO 2 spo price of he hree companies S where = T k, T = 25 days, k = 1, 2,..., T wih = 1 day and discoun he rae η =.3 o be he same for all hree companies. If here exiss a leas one company in credi shorage as T, he spo price approaches he penaly value P = 1 e/co 2. Therefore from Theorem 2.2, he marke clearing condiion for m = 3, X l, = l=1 =, 1,..., T is saisfied and we can opimise he quaniy of credis bough and sold by hree companies and hence derive an equilibrium spo price for hree companies. 2

26 In order o illusrae he equilibrium spo price converging o zero, in he following igure 2.5a we simulae a new se of CO 2 emissions for each company. In his realisaion, whils keeping mos parameer values he same, we only change he mean-reversion rae θ of company 1 from θ 1 =.1 o θ 1 =.6 and he volailiy σ of company 3 from σ 3 =.3 o σ 3 = a CO 2 Emissions Level for Company1, 2 and 3 Q 1, for Company1 Q 2, for Company2 Q 3, for Company3 4 3 b Expeced CO 2 Credis o be Bough + or Sold X 1, for Company1 X 2, for Company2 X 3, for Company3 Emissions Level CO Expeced Credis Trading Days Trading Days 8 c Equilibrium CO 2 spo price for Company1, 2 and 3 S 7 6 Spo Price Euro/ CO Trading Days igure 2.5: a Acual emissions levels of company 1, 2 and 3 following an OU process. b Expeced CO 2 credis of company 1, 2 and 3. c CO 2 spo price hiing zero. Parameers are Q 1, = 5 CO 2, Q 2, = 3 CO 2, Q 3, = 4 CO 2, N 1, = 125, N 2, = 8, N 3, = 6, µ 1 = 4 CO 2, µ 2 = 2 CO 2, µ 3 = 1 CO 2, θ 1 =.6, θ 2 =.2, θ 3 =.3, σ 1 =.75, σ 2 =.5, σ 3 =.4, ρ 12 =.5, ρ 13 =.3 and ρ 23 =.2, η =.3, P = 1 e/co 2 and T = 25 days. igure 2.5b shows he quaniy of expeced CO 2 credis o be bough or sold by each company a ime. rom his figure we noe ha ha company 3 is sill buying credis during mos of he rading period, however boh company 1 and company 2 sell heir excess credis during he enire rading period and never buy. Similar o he spo price for hree companies reaching penaly value P, from Theorem 2.2 he marke clearing condiion for 21

27 m = 3, X l, = l=1 =, 1,..., T is saisfied and we can opimise he quaniy of credis bough and sold by hree companies. In igure 2.5c we show he equilibrium CO 2 spo price of he hree companies S, where = T k, T = 25 days, k = 1, 2,..., T, = 1 day and η =.3. Therefore wih no demand for CO 2 credis he equilibrium CO 2 spo price for hree companies approaches zero as T. 22

28 Chaper 3 CO 2 Opion Pricing In his chaper we presen he mahemaical analysis for CO 2 opion pricing. irs we propose a sochasic differenial equaion SDE ha models he equilibrium spo price derived in Chaper 2, i.e. an SDE ha reaches eiher zero or penaly P a compliance ime T for CO 2 rading. By applying a no-arbirage principle we derive he fuures price and use i o obain a fuures SDE under risk-neural measure. We hen provide heoreical approximaions of European CO 2 call and pu opions on fuures prices. In addiion by adding elecriciy and naural gas fuures conracs o he CO 2 fuures conrac we can se up a risk-free porfolio and derive a PDE o price clean spark spread opions. 11 Since he clean spark spread call and pu opion prices have no closed-form soluion ha saisfies he PDE, we herefore derive heoreical approximaions by adaping Kirk s formula [Kirk, 1995] for hree commodiies. 3.1 Modelling he CO 2 Spo Price In order o model he equilibrium CO 2 spo price dynamics given in 2.22 which is graphically illusraed in boh igure 2.4c and igure 2.5c we se ou he following requiremens: A spo price, S, following a diffusion process wih independen incremens such ha S [, P for < T. A ime = T, S T is eiher zero or penaly P. Since we have a discoun facor in 2.22 and o avoid arbirage opporuniies we impose he condiion S P e rt [, T ]. 11 A clean spark spread opion is a relaively new concep/phrase used and is similar o a spark spread opion bu includes he CO 2 price. More informaion can be found a hp:// 23

29 Take noe ha if S > P e rt hen a rader can sell he carbon credis a he curren spo price S and hen inves he proceeds in a bank which will appreciae is value a a risk-free rae r. By compliance ime T, he invesmen will be higher han he penaly value P, and he difference beween he risk-free invesmen and S T is he arbirage profi. Hence he condiion S P e rt is needed o preven arbirage opporuniies. Before we presen our sochasic model we firs analyse wo diffusion processes namely he Geomeric Brownian Moion GBM and he Brownian bridge [Shreve, 24] ha will assis us in he consrucion of our diffusion model. By definiion, he GBM and Brownian bridge processes are given respecively as dx X = µ x d + σ x dw dy = P Y T d + dw where µ x is a consan drif, σ x is a consan volailiy and W is a sandard Brownian Moion BM. or he Brownian bridge he process is condiioned o be P, a fixed value, a a specified ime = T. In igure 3.1 we simulae he wo diffusion processes for [, 25] days. or he GBM we se X = 5 wih parameers µ x =.4 and σ x =.45. or he Brownian bridge we se Y = 5, P = 1 and T = 25 days. We noe from he lef-hand 13 A sample pah of a GBM 13 A sample pah of a Brownian bridge 12 X 12 Y Spo Price Spo Price Trading Days Trading Days igure 3.1: Illusraive sample pahs of he GBM and Brownian bridge processes wih X = 5, µ x =.4, σ x =.45, Y = 5, P = 1 and T = 25 days. side plo in igure 3.1 ha he GBM is no bounded above and also canno go below zero. or he righ-hand side plo of igure 3.1 we noe ha for he Brownian bridge process he spo price is no bounded above eiher, however he diffusion process is condiioned o be Y T = P a ime = T since he drif erm drives he price o P as T. In order o 24

30 model he CO 2 price we can use he characerisics of he above wo diffusion processes and herefore propose he following SDE: ds P e rt S = µ S, S T d + σ P e rt S dw 3.1 where µ S, = µ S 12 P ert 3.2 is he sae- and ime-dependen drif wih consan µ >, consan σ >, r is he risk-free rae, P and T are he penaly value and compliance ime for CO 2 rading respecively and W is a sandard BM. Noe ha he dimensions of µ and σ are differen compared o he GBM. urhermore by drawing our aenion o he drif erm and ignoring he noise erm we can analyse he sabiliy of he model in more deail. By focussing only on he deerminisic par of he diffusion process we consider he following coninuous dynamical sysem ds d = fs P e rt S = µs, S. T By assuming f is differeniable a S = S where fs =, we know ha: If f S > hen S is a repelling fixed poin. If f S < hen S is an aracing fixed poin. If we were o se µs, = µ, where µ is a consan, hen f µ ] S = [P e rt 2S. T Therefore we have S = as a repelling poin and S = P e rt as an aracing poin for µ > and vice versa for µ <. However our assumpions require boh of hese poins o be aracing poins. Therefore by seing µ S, = µ S 1 2 P ert such ha [ f µ S = P e rt S S 1 P ert T 2 S S 1 ] 2 P ert + S P e rt S and since µ >, we have S = 1 2 P ert as a repelling poin whils boh S = and S = P e rt become aracing poins. To illusrae how he proposed SDE can be used o model he original equilibrium CO 2 spo price, in igure 3.2 we simulae wo scenarios of 3.1 wih S = 4, µ =.6, 25

31 σ =.9, r =.45, P = 1 and T = 25 days. 12 Similar o he equilibrium spo price dynamics in 2.22 from Chaper 2, we noe ha he spo price based on he proposed SDE produce values S such ha S P e rt for < T. ollowing he equilibrium spo price erminaion crieria a compliance ime T as seen in Chaper 2, our spo price model also erminaes a S T = or S T = P a = T and herefore represens a ypical cap-and-rade sysem. 1 CO 2 Sochasic Model Hiing Penaly P 1 CO 2 Sochasic Model Hiing 9 S 9 S Spo Price Spo Price Trading Days Trading Days igure 3.2: Illusraive sample pahs of he SDE for CO 2 approaching penaly P and zero respecively wih parameers S = 4, µ =.6, σ =.9, r =.45, P = 1 and T = 25 days. inally i is worh poining ou ha for he underlying CO 2 spo price a ime = T, where S T is eiher zero or P we have a one-sep binomial model for European claims. By resricing our aenion o European call ζ = 1 and pu ζ = 1 opion prices a ime, V S, = e rτ E Q [max {ζ S τ K, } ] wih exercise price K such ha < K < P and mauriy ime τ, < τ T, we can wrie he call opion price a ime for mauriy τ = T as CS, = e rt P K P S T = P and he pu opion price a ime for mauriy τ = T as P S, = e rt K P S T =. In he following igures , for various µ and σ values, we plo a selecion of hreedimensional hisograms o illusrae he behaviour of he proposed CO 2 SDE over ime. 12 Due o he annualised drif µs 1 2 P ert P e rt S T < 1 and he annualised volailiy < σ P e rt S < 1, he µ and σ parameer values are relaively small. 26

32 Using Mone Carlo simulaion wih he Euler discreisaion mehod and by concenraing on hree differen saring spo price values S = 4, 5, 6 wih P = 1, r =.45 and T = 25 days, we show he disribuion of spo prices. We noe from igure 3.3a ha for a relaively large drif erm in 3.2 seing µ =.8 and relaively large volailiy a b µ =.8 σ =.9 µ =.8 σ = requency % T requency % T 5 Spo Price P 5 1 Trading Days 5 Spo Price P 5 1 Trading Days c d µ =.3 σ =.9 µ =.3 σ = requency % T requency % T 5 Spo Price P 5 1 Trading Days 5 Spo Price P 5 1 Trading Days igure 3.3: Three-dimensional hisograms of he proposed CO 2 SDE wih S = 4, P = 1, r =.45 and T = 25 days. seing σ =.9, he CO 2 spo price approaches eiher zero or he penaly value P from > 15 days wih a larger percenage approaching zero han P. Conversely using he same µ =.8 coupled wih a smaller σ =.4, igure 3.3b shows he spo price ending o zero for mos of he Mone Carlo simulaed realisaions wih only a very small percenage approaching P. This is due o he fac ha by using a saring value S = 4, we have a negaive drif erm in 3.2, and since he volailiy is relaively small he spo price will mainly approach zero. rom igure 3.3c we noe by using a smaller µ =.3 and keeping he volailiy relaively large wih σ =.9, he price approaches eiher zero or P, bu a a slower rae as compared o igure 3.3a. inally in igure 3.3d wih a relaively small 27

33 µ =.3 and relaively small σ =.4 he CO 2 spo price mainly approaches zero wih a small percenage ending o P. This is because for a smaller µ, he diffusion process will end o have a slower deerminisic decay rae and hence allow he spo price o approach P for some Mone Carlo simulaions. a b µ =.8 σ =.9 µ =.8 σ = requency % T requency % T 5 Spo Price P 5 1 Trading Days 5 Spo Price P 5 1 Trading Days c d µ =.3 σ =.9 µ =.3 σ = requency % T requency % T 5 Spo Price P 5 1 Trading Days 5 Spo Price P 5 1 Trading Days igure 3.4: Three-dimensional hisograms of he proposed CO 2 SDE wih S = 5, P = 1, r =.45 and T = 25 days. By using a saring value of S = 5, igure 3.4a illusraes ha for a relaively large drif erm in 3.2 wih µ =.8 and relaively large volailiy wih σ =.9, he CO 2 spo price approaches eiher zero or he penaly value P wih approximaely equal percenages from > 2 days. Using he same µ =.8 ogeher wih a smaller σ =.4, igure 3.4b shows he spo price ending o P for a larger percenage of he realisaions compared o zero. This is due o he fac ha unlike igure 3.3b, a saring value of S = 5 corresponds o a posiive drif erm in 3.2 and wih a relaively small volailiy he spo price will mainly approach P. rom igure 3.4c we noe by using a smaller µ =.3 and keeping he volailiy relaively large wih σ =.9, he price approaches eiher zero or P 28

34 similar o igure 3.4a, bu wih a slower rae. inally in igure 3.4d wih a relaively small µ =.3 and relaively small σ =.4 he CO 2 spo price approaches o zero or P wih a greaer spread as compared o igures 3.4a - 3.4c. a b µ =.8 σ =.9 µ =.8 σ = requency % T requency % T 5 Spo Price P 5 1 Trading Days 5 Spo Price P 5 1 Trading Days c d µ =.3 σ =.9 µ =.3 σ = requency % T requency % T 5 Spo Price P 5 1 Trading Days 5 Spo Price P 5 1 Trading Days igure 3.5: Three-dimensional hisograms of he proposed CO 2 SDE wih S = 6, P = 1, r =.45 and T = 25 days. By using a higher saring value of S = 6, igure 3.5a illusraes ha for a relaively large drif erm in 3.2 seing µ =.8 and relaively large volailiy seing σ =.9, he CO 2 spo price approaches eiher zero or he penaly value P wih a larger percenage approaching P han zero. On he oher hand by using he same µ =.8 coupled wih a smaller σ =.4, igure 3.5b shows he spo price ending o P for all of he realisaions. This is due o he fac ha a higher saring value of S = 6 ogeher wih a relaively large µ corresponds o a large growh rae in 3.2, leading o he spo price only reaching P. rom igure 3.5c we noe by using a smaller µ =.3 and keeping he volailiy relaively large wih σ =.9, he price approaches o mainly P similar o igure 3.5a, bu wih a slower growh rae. inally in igure 3.5d wih a relaively small µ =.3 and 29

35 relaively small σ =.4 he CO 2 spo price mainly approaches o P wih a very small percenage of spo prices ending o zero. This is because, unlike igure 3.5b, a smaller µ corresponds o a smaller growh rae and hence allow he spo price o approach zero for a very small percenage of realisaions. Based on he proposed spo price dynamics wih is unique properies, in he nex secion we focus on developing a sochasic differenial equaion for fuures conracs under risk neural measure and derive heoreical approximaions for opions on fuures as well as clean spark spread opions. 3.2 Pricing CO 2 uures Conracs As discussed earlier, under curren rading condiions, he price of CO 2 credis is normally quoed in erms of fuures conracs. In his secion we derive he fuures price in order o consruc he fuures SDE for CO 2 and hence price opions on fuures as well as clean spark spread opions. To begin wih we consider he diffusion process for he CO 2 spo price as ds P e rt S = µ S, d + σ P e rt S dw 3.3 S T where µ S, = µ S 12 P ert. Under risk-neural measure Q we have ds S = [ P e rt S µ S, T λσ where λ is he marke price of risk such ha ] d + σ P e rt S dw Q is a Brownian moion. W Q = W + λudu Given ha he CO 2 credis are in consan supply unil compliance ime T, he fuures price can be fixed by a no-arbirage principle. Hence he relaionship beween he spo price S and he fuures price, τ for delivery ime τ, τ T is, τ = e rτ S. 3.4 Since S P e rt 3.5 3

36 and by subsiuing 3.5 ino 3.4 we herefore have, τ P e rt τ 3.6 which shows he fuures price is bounded above by he penaly value P in order o avoid arbirage opporuniies. By differeniaing 3.4 we can obain he following fuures SDE d, τ = re rτ S d + e rτ ds = re rτ S d + P e rt µ S, S T λσ e rτ S d +σ P e rt S e rτ S dw Q [ ] = r + P e rt µ S, S T λσ e rτ S d +σ P e rt S e rτ S dw Q and since he fuures price is a Maringale wih respec o he risk-neural measure Q and subsiuing S =, τe rτ we have d, τ = σ,, τ, T, τdw Q 3.7 where σ,, τ, T = σ P e rt, τe rτ. 3.8 Take noe ha under he risk neural measure Q, he fuures diffusion process 3.7 is able o ake values beween zero and P for all T. On he oher hand, under he physical measure, due o he drif erm in he spo price dynamics 3.3 he fuures price is consrained eiher a zero or P a compliance ime T. Hence he risk neural densiy for he diffusion process 3.7 is only valid for < T whils for = T, he corresponding risk neural densiy is binomial Pricing CO 2 Opions on uures Under he risk-neural measure Q, adaped o he filraion, for srike value K > and mauriy τ, where τ T, we can express he CO 2 call ζ = 1 and pu ζ = 1 opions on fuures prices a ime as V, τ, = e rτ E Q [max {ζ τ, τ K, } ]. or he case when τ = T, due o he binomial densiy, he CO 2 call opion on fuures price a ime for mauriy τ = T is given as C, T, = e rt P K P S T = P 31