A New N-factor Affine Term Structure Model of Futures Price for CO 2 Emissions Allowances: Empirical Evidence from the EU ETS

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1 WSEAS RASACIOS on BUSIESS and ECOOMICS Ka Chang, Su-Sheng Wang, Je-Mn Huang A ew -facor Affne erm Srucure Model of Fuures Prce for CO Emssons Allowances: Emprcal Evdence from he EU ES KAI CHAG, SU-SHEG WAG, JIE-MI HUAG Shenzhen Graduae School Harbn Insue of echnology Shenzhen Unversy own n Shenzhen cy HE PEOPLE S REPUBLIC OF CHIA, ,.chang6@yahoo.cn, wangsusheng@hmal.com Absrac: - In recen years, carbon emsson mares have become lqud and promsng mares whn he European Unon emssons radng scheme (EU ES. In order o f and forecas fuures prce for CO emssons allowances, we propose a new -facor affne erm srucure model for CO fuures prce and esmae parameers n he new affne model by usng he Kalman fler echnque. Our emprcal resuls show ha CO fuures prce follow sgnfcan mean-reverson process, and he esmaed coeffcens of meanreverson speed, mare rs premum, volaly and correlaon among sae varables are almos sgnfcan. Compared wh one-facor model, mean absolue errors (MAE and roo mean square errors (RMSE n predcon errors from wo-facor and hree-facor model are lower, accordngly wo-facor and hree-facor model can accuraely descrbe he erm srucure of CO fuures prce. Key-Words: CO emssons allowances; fuures prces; new affne model; erm srucure; Kalman fler Inroducon In recen years, CO emsson allowances mares have become lqud and promsng mares whn he European Unon emssons radng scheme (EU ES. he EU ES domnaes he global carbon mares, a varey of specalzed fnancal producs such as spo, fuures and opons are raded. he radng scale of global emssons mares has acheved 9 bllon euros or abou 0 bllon dollars unl 00. he emsson rghs of greenhouse gas, called EU allowances (EUA, allow for he rgh o em one on of CO n he EU ES. herefore carbon emssons rgh has gven specfc ownershp, and s sgnfcanly valuable cred asses for he nvesors, hedgers, oher mare praconers. Benz and ruc (006 propose emsson allowances prces are drecly deermned by he expeced mare scarcy and her emprcal resuls show fuures prces have srongly mevaryng rend []. Correspondng auhor. el: , Fax: ,emal:.chang6@yahoo.cn(Ka Chang; wangsusheng@gmal.com(su-sheng Wang he auhors are graeful for research suppor from aonal aural Scence Foundaon of Chna (70050; Guangdong aural Scence Foundaon ( and Guangdong Sof Scence Foundaon (0B Sochasc models of commody prces have receved much aenon among he scholars, hedgers, fnancal praconers, and sochasc models are manly abou he prcng and hedgng of commodes asses. Early sudes n he feld ypcally assume ha sorable commodes prces follow Brownan moon process. Gbson and Schwarz (990 develop wo-facor model of commody prcng, where he spo prce follows a geomerc Brownan moon and he convenence yelds follow mean-reverng Ornsen-Uhlenbec (O-U process []. Schwarz (997, Mlersen and Schwarz (998 add he hrd sochasc neres raes by assumng mean reverson process, and hey propose he hree-facor model of commody fuures prcng [-4]. herefore he commody spo prce, he nsananeous convenence yelds, and he nsananeous neres rae are of mporan sae varables for commody fuures prce. Lauer (005 ndcaes ha he erm srucure s defned as he relaonshp beween spo prce and fuures prces wh dfferen maures [5]. In many commodes mares, he concep of erm srucure becomes sgnfcan because provdes useful nformaon for he hedgng or nvesmen decson of dfferen asses. Many scholars consder ha he erm srucure of commody prce s affeced by many sae varables. Schwarz and Smh (000 develop a wo-facor model of commody prce E-ISS: Issue, Volume 9, Aprl 0

2 WSEAS RASACIOS on BUSIESS and ECOOMICS Ka Chang, Su-Sheng Wang, Je-Mn Huang whch allows mean-reverson n shor-erm prces and uncerany n he equlbrum level [6]. Manolu and ompads (00 presen mul-facor sochasc model of erm srucure for energy fuures prce [7]. Corazar and arano (006 propose an -facor Gaussan model o explan he sochasc behavour of ol fuures prces [8]. Wang e al (00 pu forward an -facor affne erm srucure model n erms of behavoural characerscs of copper fuures prces n Shangha Fuures Exchange (SHFE [9]. hey propose ha spo prces are composed of mul arbrary sae varables, and hey esmae unobserved sae varables and calbraed model parameers by usng Kalman fler mehod. he above affne models can well smulae and forecas erm srucure of commodes fuures prces, and hey have become he sgnfcanly hedgng and rs-managng ools for all mare parcpans. CO emsson allowances prces are drecly deermned by he expeced mare scarcy whch s nduced by he change of emssons regulaon polcy, exreme weaher, energy prces, abaemen echnology progress ec. Benz and ruc (009 analyze he shor-erm dynamcs behavour of spo prce for carbon doxde (CO emsson allowances [0]. Wagner and Homburg (009, Dasalas and Psychoyos (009 propose he dynamcs behavour of fuures prce for CO emsson allowances. Benz and ruc (009 reveal ha hese fuures conracs for CO emsson allowances lead he prce dscovery process n he EU ES [-]. Dasalas and Psychoyos (009 develop he emprcally and heorecally vald framewor for he prcng and hedgng of nra-phase and nerphase fuures and opons on emsson fuures conracs. Chevaller (00 analyzes me-varyng rs premum and posve relaonshp beween rs premum and me-o-maury n CO spo and fuures prces []. herefore fuures prces for CO emssons allowances are affeced by many sae varables, especally non-observable sae varables. Undersandng erm srucure of fuures prces and accuraely forecasng fuures prces for emsson allowances are of crucal mporance for all mare parcpans. he paper has hree maor obecves. Frsly on bass of undersandng he behavour feaure of CO fuures prce, we propose a new mul-facor affne erm srucure model of fuures prce for CO emssons allowances. Secondly we also show how o esmae model parameers of unobservable sae varables by usng he Kalman fler and maxmum lelhood mehods n he whole perod. hrdly we compare wh measuremens errors n observable fuures prce by mang an opmal use of all mare prces avalable. he remander of hs paper s organzed as follows. he second secon descrbes he daa samples and he sascal analyss resuls of CO fuures prces. Secon proposes a new -facor affne erm srucure model of fuures prce for emssons allowances. Secon 4 presens he Kalman fler echnque. Secon 5 shows esmaed model parameers by usng he Kalman fler and maxmum lelhood mehods. Secon 6 compares he evaluaon of model Robusness. Secon 7 concludes he paper. Daa descrpon. Daa descrpon he EU ES s he larges greenhouse gas (GHG emssons radng sysem n he world. European Clmae Exchange (ECX, now s merged by ICE s he mos lqud plaform for CO fuures mare n European. he EU ES has he exsng wo phrases: he plo phase ( and he Kyoo phase ( Monagnol and Vres (00 ndcae ha mare effcency was neffcen n he ral and learnng perod, hen shows resorng sgns n he Phase II [4]. Snce EU governmen banned ou-of-phrase banng and borrowng, hen he spo prce for CO emssons allowances fell down o zero from Ocober 006 unl December 007 (see Chevaller, 00. herefore emssons allowances prces had los her real value. he mnmum radng volumes for each fuures conrac are,000 ons of CO equvalens. We selec ha dae samples are me-varyng daly selemen prces for EUA fuures conracs wh varyng maures gong from December 00 o December 04. he radng of fuures conracs wh vnages December 0 and 04 were nroduced on Aprl 8, 008.Consderng he avalably and connuy of EUA fuures prce, we choose ha dae samples cover he perod gong from Aprl 8, 008 o December 0, 00 n he ECX mare.. Descrpve of sascal evdence for emssons fuures prce In he followng fgure, F,F,F,F 4,F 5 denoe he raded EUA fuures conracs wh varyng maures gong from December 00 o December 04.Among hem, F s he closes o maury for E-ISS: Issue, Volume 9, Aprl 0

3 WSEAS RASACIOS on BUSIESS and ECOOMICS Ka Chang, Su-Sheng Wang, Je-Mn Huang EUA fuures conrac, F s he second closes o maury for EUA fuures conrac, and so on. In he fgure, we see he prces of EUA fuures conracs wh varyng delvery daes have srongly mevaryng rend n he whole sample perod. We fnd ha fuures prces for emssons allowances show srongly me-varyng volaly, and hey have he hgher upward and downward ump. p r c e Fg.. EUA fuures prce for CO emssons allowances able Descrpve sascal evdence for EUA fuures prces fuures mean max mn Sd.dev sew ur F F F F F Seen from he above able, mean sascs shows mean values for EUA fuures prce gradually sep up wh me-o-maury ncrease. Emssons fuures prce wh long-erm delvery dae s hgher han he recen fuures conrac, herefore erm srucure of CO fuures prce s conango. o some exen, he volaly can measure changng speed n he mare prce. he ncreasng volaly n fuures prce for emssons allowances can be observed wh he rsng of me-omaury, and denoes mean-reverng speed n he prce of fuures conrac wh long-erm maury s faser han hose fuures conrac wh shor-erm maury. Wh he ncreasng of me-o-maury, hgher posve uross and sewness denoe he greaer devaon degree and long al draggng on he rgh of Gaussan dsrbuon. dae F F F F4 F5 Affne erm srucure model for CO fuures prces Fuures prces for CO emssons allowances are affeced by he observable and non-observable sae varables, for example emssons regulaon polcy, energy prces and energy effcency, low-carbon echnologes progress and applcaon, exreme clmae change and shor-me devaon n emssons allowances prce ec. Regulaon polcy, energy effcency, low-carbon echnologes progress and applcaon promoe long-erm equlbrum beween demand and supply n CO emssons allowances mares, hey drecly deermne longerm rend n emssons allowances prces. Exreme clmae change, neres rae flucuaon, energy prce and he oher facors nduce expeced changes beween demand and supply n CO emssons allowances mares, hey push shor-erm devaon n emssons allowances prces. herefore fuures prces are affeced by many sae varables. Manolu and ompads (00, Wang e al (00 propose ha he commodes spo prces are composed of general -facor sae varables ln S = x, and hey provde smple analycal valuaon formulas for fuures prces. Coarzar and arano (006 assume ha he commody spo prce can be descrbed as he arbrary number of sae varables and he long-erm consan growh rae, log S = h ' x + u.in our model, Le S denoe he spo prces for emssons allowances a me, and spo prces are assumed as non-observable sae varables. We consder ha he logarhmc process of spo prces for CO emssons allowances can be more precsely expressed as a sum of -facor sae varables[7][9]: ln S = x ( where he vecor of sae varables x follows meanreverng process wh he Ornsen- Uhlenbec ype. We assume ha a consan mare rs premum λ, he rs-adused process for he vecor of sae varables s equal o [8-9]: dx = ( Kx + λ + dz ( where mare rs premum λ= λ λ L s an n vecor of real [ ] λ n E-ISS: Issue, Volume 9, Aprl 0

4 WSEAS RASACIOS on BUSIESS and ECOOMICS Ka Chang, Su-Sheng Wang, Je-Mn Huang 0 L 0 consans, 0 L 0 K = and M M O M 0 0 L n σ 0 L 0 0 σ L 0 Σ = are n n dagonal marces M M O M 0 0 L σ n wh enres ha are posve consans and ermwse dfferen. dz s a n vecor of correlaed Brownan moon ncremens, such ha ( dz ( dz = Ωd,where he (, elemen of Ω s ρ [, ], he nsananeous correlaon beween sae varables and [8]. F(, denoes CO emssons allowances prce a me for fuures conrac wh he delvery dae. he fuures prce for emssons allowances F(, can be defned as he expeced value of spo prce a he delvery dae under he rs-neural measure [5]. F ( x,, = E ( S ( As shown n he Appendx A, he fuures prce for emssons allowances a me and maurng a n Equaon ( can be descrbed as: F ( x,, = exp( A( = = = = = ( e e ( ( e σ σ ρ ( + ( + x λ + + A( One mporan advanage of hs model s racable o oban smple analycal fuures prce formula for emssons allowances. he logarhm of he fuures prce s a lnear funcon of -facors sae varables, s useful when esmang he parameers of erm srucure model by usng he Kalman fler mehod. Because he sae varables have a mulvarae normal dsrbuon, any lnear combnaon of sae varables wll also dsrbue normal by allowng maxmum lelhood echnque [8]. 4 he Kalman fler As he above menon, he mos dffculy n he emprcal mplemenaon for EUA fuures prce s (4 he arbrary and non-observable sae varables n he affne model. he sae space form s he approprae procedure o deal wh suaons n whch he sae varables are no drecly observable. he Kalman fler s he mos approprae esmaon mehodology whch recursvely calculaes he model parameers and he me seres of unobservable sae varables. he form of sae varables s appled o a mulvarae me seres of sae varables. he measuremen equaon relaes a vecor of observable varables y wh a vecor of sae varables x. In our affne model, EUA fuures prces as npued observable varables are he me seres a several varyng maures. Measuremen equaon n he affne model s hen gven by equaon (4: y = H x + d + v v 0, R (5 = ln F(, m ( where y [ ] s a m vecor of EUA fuures conracs wh varyng maures, x = x x, L, x s a n vecor of sae [ ], L n τ τ nτ e e L e τ τ nτ varables, = e e L e H s a m n M M L M τ m τ m nτ m e e L e marx, d = [ A( ] s a m vecor, and v m s a n vecor of serally uncorrelaed Gaussan dsurbances wh E (v =0 and cov(v =R [8-9]. Based on equaon, he ranson equaon n he affne model s descrbed as he sochasc process followed by he sae varables: x = G x + c + w, w 0, (6 ( τ e 0 L 0 τ where = 0 e L 0 G s a n n vecor of M M M M τ n 0 0 L e dagonal marx, we assume τ =, c [ ] = 0,0, LL, 0 s a n vecor, and w s a n vecor of serally uncorrelaed Gaussan dsurbances wh E(w =0 and cov(w = [9]. 5 Parameers coeffcens esmaon he daa samples used n hs emprcal sudy are observable daly selemen prce for EUA fuures conracs wh he varyng maures gong from December 00 o December 04, and fve fuures conracs are used n he parameers esmaon. he emprcal perod of daa samples covers he hsorcal me seres gong from Aprl 8, 008 o E-ISS: Issue, Volume 9, Aprl 0

5 WSEAS RASACIOS on BUSIESS and ECOOMICS Ka Chang, Su-Sheng Wang, Je-Mn Huang December 0, 00. EUA fuures prces as observable varables are npued no he measuremen and ranson equaons, he esmaed parameers n he affne model are obaned by usng he Kalman fler and maxmum lelhood echnques. When he number of sae varables n s equal o one, wo and hree, he followng esmaed parameers coeffcens n he affne model are lsed n he followng able, and 4 by usng he same dae samples. able : Esmaed parameers coeffcens n onefacor affne model Model parame Coeffc en Sd. Error Z- Sas Probab ly ers value c λ ln σ Log le lhood able : Esmaed parameers coeffcens n wofacor affne model Model parame Coeffc en Sd. Error Z- Sas Probab ly ers value c λ ln σ λ lnσ cov(x, x Log le lhood As s shown n he above able, and 4, all mean-reverson speed parameers,,, are sgnfcanly unequal o zero a he sgnfcan level %, and her correspondng sandard devaons of he measure errors approxmaely go o zero when he number of unobservable sae varables are equal o one, wo and hree, hose sgns show ha he sae varables x, x, x follow mean-reverng process. he hgher value of mean-reverson speed parameers ndcaes he exsence of srong meanreverson process and he shorer me-varan of he correspondng sae varables for EUA fuures prce. he parameers of mare rs premum λ, λ, λ and volaly σ, σ. σ are almos sgnfcan a he sgnfcan level %, and her sandard devaons of he measure errors are lower. Covarance sascs descrbes ha he sae varables x and x have negave correlaon, and he sae varables x and x have posve correlaon, however he sae varables x and x have negave correlaon. able 4: Esmaed parameers coeffcens n hreefacor affne model Model parame Coeffc en Sd. Error Z- Sas Probab ly ers value c λ ln σ λ lnσ cov(x, x λ ln σ cov(x, x cov(x, x Log le lhood Parameers robusness he performance and esmaon procedure n he affne model are measured by esmang erm srucure of observable fuures prce and he emprcal volaly of erm srucure n fuures reurns [8]. hese daly predcon errors represen he dfference beween observable fuures prces and predcable fuures prces by usng he Kalman fler echnque. he predcon errors for EUA fuures conracs wh varyng delvery daes are shown n he followng Fgure,, 4. e,e,e.e 4.e 5 denoe he predcon errors for CO fuures conracs wh varyng maures, e s he predcon error for CO fuures conrac wh he closes o maury, e s he predcon error for CO fuures conrac wh he second closes o maury, and so on. E-ISS: Issue, Volume 9, Aprl 0

6 WSEAS RASACIOS on BUSIESS and ECOOMICS Ka Chang, Su-Sheng Wang, Je-Mn Huang p r e d c o n e r r o r dae Fg.. he predcon errors n EUA fuures prce (one-facor model, 00% p r e d c o n e r r o r dae Fg.. he predcon errors n EUA fuures prce (wo-facor model, 00% p r e d c o n e r r o r Fg. 4. he predcon errors n EUA fuures prce (hree-facor model, 00% Based on hsorcal me seres for EUA fuures conracs, we compare he capables of fng n he affne model. Seen from he above fgures,, 4, we can obvously fnd he predcon errors n CO fuures prce are he bgges n he one-facor dae e e e e4 e5 e e e e4 e5 e e e e4 e5 model, and he predcon errors n CO fuures prce are sgnfcanly lower n he wo-facor and hreefacor model. Mean absolue errors (MAE and roo mean square errors (RMSE n predcon errors for fve EUA fuures conracs are shown n he above able 5. he MAE and RMSE n predcon errors from he one-facor model are he larges, hey reflec ha he very large devaons from he onefacor model canno accuraely f observable fuures prce n me-varan dae samples. Compared wh one-facor model, MAE and RMSE n predcon errors from he wo-facor model s lower, and he lower devaons from he wo-facor model appear o sgnfcanly forecas for he observable fuures prce. Compared wh wo-facor model, MAE and RMSE n predcon errors from he hree-facor model s he lowes, and her lowes devaons ndcae ha her fng ably slghly ncreases. herefore MAE and RMSE for fve CO fuures conracs from he wo-facor and hree-facor model are less han %, hey ndcae ha wo-facor and hree-facor model can more accuraely descrbe he erm srucure of EUA fuures prces. able 5 Comparson resuls of he predcon errors 4 sascs ( 0 lnf One-facor MAE RMSE wo-facor MAE RMSE hree-facor MAE RMSE lnf lnf lnf lnf lnf Conclusons In recen years, fuures mares for emssons allowances have become lqud and poenal mares whn he EU emssons radng scheme (EU ES. Alhough he CO spo prce s a radable and observable asse n he EU ES, we assume ha spo prce s an unobservable sae varable, and our affne model provdes ha fuures prce can be expressed as mul-facor arbrary sae varables. Based on he affne model of fuures prce [8-9], we propose a new -facor affne erm srucure model of EUA fuures prce and he correspondng fuures valuaon. Based on he sae space formulaon of E-ISS: Issue, Volume 9, Aprl 0

7 WSEAS RASACIOS on BUSIESS and ECOOMICS Ka Chang, Su-Sheng Wang, Je-Mn Huang fuures prce, we can esmae parameers n he affne model by usng he Kalman fler and maxmum lelhood echnques. We fnd ha fuures prces for CO emsson allowances show srongly me-varyng moon rend. Our emprcal resuls show ha all unobservable sae varables follow sgnfcanly mean-reverson process, herefore fuures prces for CO emssons allowances also follow meanreverng process. he parameers coeffcens of mean-reverson speed, mare rs premum, volaly and correlaon are of almos sgnfcan level. Compared wh one-facor model, MAE and RMSE n predcon errors from wo-facor and hree-facor model are lower; accordngly wo-facor and hree-facor model can accuraely descrbe he erm srucure of EUA fuures prce. In general, our affne model can wor well n esmang he erm srucure of EUA fuures prces. he drecon of fuure wor s o sudy erm srucure of volaly and he mplcaons of affne model n fuure opons prcng for fuures conracs. Appendx A hs Appendx deduces Equaon (4 wh he use of Equaon (. Because he condonal normal dsrbuon of he spo prce S s he lognormal, follows ha E ( S = exp( E (ln S + VAR (ln S (A Where E (ln S = l E ( x VAR (ln S = l cov ( x l Where [ ] l =,, LL, s a n vecor, E x s he expeced value of he sae ( x = x, x, LL, xn, cov ( x s a covarance marx of he sae varable x From equaon (, he condonal momens of x are varable [ ] 0 K ( K E ( x = e x λ e τ dτ (A cov Kτ Kτ ( x = e ΣΣ ( e dτ (A 0 Where dz ( dz = Ωd E ( x = e (, hus λ, =,, LL n(a4 ( ( x ( e, ( + ( e cov ( x = ρ σσ +,, =,, LL, n(a5 he fuures prce for CO emssons allowances F(, can be defned as he expeced value of spo prce a he delvery dae under he rs-neural measure, F ( x,, = E ( S. he valuaon of formula (4 s obaned by nserng equaons (A, (A4 and (A5 no Equaon (4. F ( x,, = exp( A( = = = = = ( e e e σ σ ρ ( ( ( + ( + x λ + + A( (A6 References: [] Benz E., ruc S. CO emsson allowances radng n Europe: Specfyng a new class of asses, Problems and Perspecves n Managemen, Vol. 4, Issue, 006, pp [] Gbson R., Schwarz E.S. Sochasc convenence yeld and he prcng of ol conngen clams, Journal of Fnance, Vol. 45, o., 990, pp [] Schwarz E.S. he sochasc behavour of commody prces: mplcaons for valuaon and hedgng, Journal of Fnance, Vol. 5, o., 997, pp [4] Mlersen K.R., Schwarz E.S. Prcng of opons on commody fuures wh sochasc erm srucures of convenence yelds and neres raes, Journal of Fnancal and uanave Analyss, Vol., o., 998, pp [5] Lauer D. erm srucure models o f commody prces: a revew, Journal of Alernave Invesmens, Vol. 8, o., 005, pp [6] Schwarz E.S., Smh J.E. Shor-erm varaons and long-erm dynamcs n commody prces, Managemen Scence, Vol. 46, o.7, 000, pp [7] Manolu M., ompads S. Energy fuures prces: erm srucure models wh Kalman fler esmaon, Appled Mahemacal Fnance, Vol. 9, Issue, 00, pp. -4. [8] Corazar G., arano L. An n-facor Gaussan model of ol fuures prces, Journal of Fuures Mares, Vol. 6, o., 006, pp [9] Wang S.S, Wang L., L Z.C, Xang J. Affne erm srucure models of fuures prces based E-ISS: Issue, Volume 9, Aprl 0

8 WSEAS RASACIOS on BUSIESS and ECOOMICS Ka Chang, Su-Sheng Wang, Je-Mn Huang on Kalman fler, Journal of Sysems Engneerng, Vol. 5, o., 00, pp [0] Benz E., ruc S. Modelng he prce dynamcs of CO emsson allowances, Energy Economcs, Vol., 009, pp [] Wagner M.W., Homburg M.U. Fuures prce dynamcs of CO emsson cerfcaes An emprcal analyss, Journal of Dervaves, Vol.7, o., 009, pp [] Dasalas G., Psychoyos D., Marellos R.. Modelng CO emsson allowance prces and dervaves: Evdence from he European radng scheme, Journal of Banng & Fnance, Vol., 009, pp [] Chevaller J. Modelng rs prema n CO allowances spo and fuures prces, Economc Modellng, Vol. 7, 00, pp [4] Monagnol A., Vres F.P. Carbon radng hcness and mare effcency, Energy Economcs, Vol. 4, 00, pp. -6. [5] Cox J.C., Ingersoll J., Ross S. he relaon beween forward prces and fuures prces, Journal of Fnancal Economcs, Vol. 9, 98, pp. 46. E-ISS: Issue, Volume 9, Aprl 0