Valuing Energy Options in a One Factor Model Fitted to Forward Prices

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1 Vag Enrgy Oons n a On acor Modl Clwlow and rcland Vag Enrgy Oons n a On acor Modl d o orward Prcs Ls Clwlow and Chrs rcland hs Vrson: 5 h Arl 999 chool of nanc and Economcs Unvrsy of chnology ydny Asrala h nancal Oons Rsarch Cnr Warwc Bsnss chool h Unvrsy of Warwc UK Cnr for nancal Mahmacs Asralan aonal Unvrsy Canbrra Asrala Inso d Esdos rors d Admnsracón Caracas Vnzla h ahors wold l o acnowldg h fnancal sor and hosaly of h chool of nanc and Economcs Unvrsy of chnology ydny. All commns wlcom. chrs_srcland@comsrv.com ls_clwlow@comsrv.com h ahors wold also l h acnowldg dscssons wh adma El-Hassan U and h rsarch asssanc of Chrsna oolos. All rrors rman or own. nrgy_sngl_facor

2 Vag Enrgy Oons n a On acor Modl Clwlow and rcland Vag Enrgy Oons n a On acor Modl d o orward Prcs Ls Clwlow and Chrs rcland Absrac In hs ar w dvlo a sngl-facor modg framwor whch s conssn wh mar obsrvabl forward rcs and volals. h modl s a scal cas of h ml-facor modl dvlod n Clwlow and rcland [999b] and lads o analycal rcng formla for sandard oons cas floors collars and swaons. W also show how Amrcan syl and oc nrgy drvavs can b rcd sng rnomal rs whch ar consrcd o b conssn wh h forward crv and volaly srcr. W dmonsra h alcaon of h rnomal r o h rcng of a Eroan and Amrcan Asan oon. h analyss n hs ar nds h rsls n chwarz [997] and Amn al. [995]. nrgy_sngl_facor

3 Vag Enrgy Oons n a On acor Modl Clwlow and rcland Vag Enrgy Oons n a On acor Modl d o orward Prcs Ls Clwlow and Chrs rcland Inrodcon In hs ar w dvlo a rcng framwor ha nabls h valaon of gnral nrgy conngn clams. hr ar crrnly wo srams o h rcng lrar. h frs sars from a sochasc rrsnaon of h nrgy so ass and ohr y varabls sch as h convnnc yld on h ass and nrs ras s for aml Gbson and chwarz [99] chwarz [997] and Hllard and Rs [998] and drvs h rcs of nrgy conngn clams conssn wh h so rocss. Howvr on of h roblms of mlmnng hs modls s ha ofn h sa varabls ar nobsrvabl - vn h so rc s hard o oban wh h roblms asrad f h convnnc yld has o b only smad. h scond sram of h lrar modls h volon of h forward or frs crv. orward or frs conracs ar wdly radd on many changs wh rcs asly obsrvd - ofn h nars mary frs rc s sd as a roy for h so rc wh longr dad conracs sd o mly h convnnc yld. h framwor of hs ar rsds n hs scond sram smlanosly modg h volon of h nr forward crv condonal on h nally obsrvd forward crv. As sch allows a nfd aroach o h rcng and rs managmn of a orfolo of nrgy drvav osons. Or Whn nrs ras ar drmnsc as w assm n hs ar frs rcs ar qal o forward rcs and so all or rsls for forward rcs also aly o frs rcs. h modl can b ndd o h cas of sochasc nrs ras sng h rsls of Amn and Jarrow [99]. nrgy_sngl_facor 3

4 Vag Enrgy Oons n a On acor Modl Clwlow and rcland framwor s hrfor closr o ha of Corazar and chwarz [994] and Amn al [995] alhogh as w show n hs ar h wo aroachs ar rlad. W nrodc or modl whch s a scal cas of h ml-facor modl n Clwlow and rcland [999b] n scon. h modl can b sn as an nson of h frs modl n chwarz [997] n h sam way ha h Hah Jarrow and Moron [99] framwor can b vwd as an nson of say h Vasc [977] modl. h volaly srcr of forward rcs s h sam and rflcs h man rvrng nar of nrgy rcs b h nal forward crv can b whavr h mar dcas nl h chwarz modl whr h crv s ndognosly drmnd. In scon 3 w drv analycal rcng formla for Eroan oons on h so ass oons on forward conracs cas floors collars and swaons. con 4 rsns or mhodology for bldng rcombnng rnomal rs for h so rc rocss conssn wh h forward crv. In scon 5 w show how Eroan and Amrcan syl ah dndn nrgy oons can b rcd sng h r wh Asan oons sd as an aml and wh mar daa for crd ol and gas. h analyss of hs ar sgnfcanly nds h analyss of boh h chwarz ar whch only loos a rcng frs conracs and h ar of Amn al. 995 whch brfly o how o rc Amrcan oons only whn h rm srcr of frs rcs has a fla volaly srcr. h Modl h sarng on for or analyss s h sochasc volon of h nrgy forward crv. In a rs-nral world nvsors rc all clams as h cd fr val dscond a h rslss ra. nc forward conracs do no rqr any nal nvsmn n a rs nral world h cd chang n h forward rc ms b zro. Also n ordr o oban a Marovan so rc rocss h volals of forward rcs ms hav a ngav onnal form. hs obsrvaons lad o h followng sochasc dffrnal qaon DE for h forward rc crv; Carvrhll [99] for h roof of hs n h con of h HJM framwor. nrgy_sngl_facor 4

5 Vag Enrgy Oons n a On acor Modl Clwlow and rcland d. hs s a mor gnral vrson of h facor vrson lood a by chwarz [997]. In ha ar h rooss a rocss for h so nrgy rc and drvs h forward rc crv and h volaly crv o hav arclar forms. h modl n qaon. has wo volaly aramrs; drmns h lvl of so and forward rc rrn volaly whls drmns h ra a whch h volaly of ncrasng mary forward rcs dc and s also h sd of man rvrson of h so rc. hs aramrs can b smad drcly from h rcs of oons on h so rc of nrgy or forward conracs sng h rsls n scon 3 of hs ar or alrnavly by bs fng o hsorcal volals of forward rcs an aroach w s n scon 5. Any scfcaon of h whol forward rc dynamcs mls a rocss for h so rc. or h scfcaon n qaon. h mld so rc rocss s shown n And A o b; d 4 d. h sngl facor modl for h so ass n chwarz [997] has h followng dfnng DE; d [ µ ] d.3 hrfor qaon. aans conssncy wh h nal forward crv by mang h long rm rs adsd drf µ h followng fncon of m; µ.4 4 W show n And B ha h forward crv a da s gvn by; nrgy_sngl_facor 5

6 Vag Enrgy Oons n a On acor Modl Clwlow and rcland [ ] 4.5 hs h forward crv a any fr m s smly a fncon of h so rc a ha m h nal forward crv and h volaly fncon aramrs. hs rsl s comaonally rmly sfl as mans ha whn rcng drvavs sng rs h ayoff of h drvavs can b valad analycally. I also allows s o oban an analycal formla for h rc of Eroan swaons n scon Prcng Eroan Oons In hs scon w dscss h rcng of Eroan oons on boh h so nrgy rc and on forward conracs. Rlad rsls for sandard Eroan oons hav rvosly aard n Amn and Jarrow [9999] and Amn al. [995]. 3. Oons on h o rom h sandard rs-nral rcng rsls Co and Ross [976] Harrson and Plsa [98] h rc of any conngn clam on h so rc C ; Θ s gvn by h caon of h dscond ayoff ndr h rs nral masr 3 [ P C ; ] C ; Θ E Θ 3. whr P r d and Θ s a vcor of consan aramrs. hrfor for a sandard Eroan call oon c ; K wh sr rc K and mary da on h ass. w hav [ P ma ] c ; K Ε K 3. 3 W ma h sandard assmons rgardng h flraon s for aml Amn and Jarrow [99]. nrgy_sngl_facor 6

7 Vag Enrgy Oons n a On acor Modl Clwlow and rcland nrgy_sngl_facor 7 Eqaon. can b ngrad o gv d 3.3 h rocss for h so can b oband by sng ; d 3.4 rom hs w can s ha h naral logarhm of h so rc s normally dsrbd; [ ] [ ] d d 4 ~ 3.5 nc nrs ras ar drmnsc and s normally dsrbd w can s h rsls of Blac and chols [973] o oban h followng analycal formla for a sandard Eroan call oon ] [ ; w h K h P K c 3.6 whr w w K h [ ] w A scal cas of qaon. s whr. hs s h rsrcon of Amn al. In hs cas w. h formla for sandard Eroan oons on h so can b asly oband by -call ary.

8 Vag Enrgy Oons n a On acor Modl Clwlow and rcland 3. Oons on orwards and rs Many oons n h nrgy mars ar on forward or frs conracs. In hs scon w drv h rc a m of a Eroan call oon wh sr rc K ha mars a m on a forward conrac ha mars a m s. Oons ar agan rcd sng h sandard mhods. A da h Eroan call has h rc [ P ma s ] c s; K s Ε K 3.7 Usng h mhodology of scon 3. s sraghforward o show ha h solon s c s; K s P [ s h K h w] 3.8 whr s K h w w w s now gvn by h ngral of h forward rc rrn varanc ovr h lf of h oon; w s s d s s 3.9 hs nds h rsls n chwarz [997] o rcng Eroan oons. o ha h rsls of scon 3. ar acally a scal cas of h rsls n hs scon wh s. 3.3 Cas loors and Collars Enrgy rc cas floors and collars ar olar nsrmns for nrgy rs managmn. An nrgy rc ca lms h floang rc of nrgy h holdr wll ay on a rdrmnd s of das ; o a fd ca lvl K. A ca s hrfor a orfolo of sandard Eroan call oons wh s rc gvn by nrgy_sngl_facor 8

9 Vag Enrgy Oons n a On acor Modl Clwlow and rcland nrgy_sngl_facor 9 ; ; K c K Ca 3. Convrsly an nrgy rc floor lms h mnmm rc h holdr wll ay and s hrfor a orfolo of sandard Eroan oons. A collar s smly a orfolo of a long oson n a ca and a shor oson n a floor. 3.4 Oons on was W dfn h m val of an nrgy swaon wh mary da o swa a srs of floang so rc aymns on das for a fd sr rc K o b Ε K P s K wn ma ; 3. W show n And C ha h val of h swaon dfnd n qaon 3. s gvn by ; ; K c s K wn 3. whr K... * and *s s h forward rc a m for mary s whn h so rc a m s * and s gvn by h solon o; K * Bldng rnomal rs for h o Procss In hs scon w roos a gnral robs and ffcn rocdr nvolvng h s of rnomal rs for modlg h so rocss. so ha s conssn wh nal mar daa. h rocdr s smlar o consrcng rnomal rs for h shor ra as od by Hll and Wh [994a 994b] and dscrbd n dal n Clwlow and rcland [998].

10 Vag Enrgy Oons n a On acor Modl Clwlow and rcland hs rs can hn b sd for rcng Amrcan syl and ah dndn oons. Amrcan oon valaon rqrs valaon of h followng rsson ~ θ C Ma E Ψ r d g θ 4. θ [ ] whr g θ s h ayoff of h oon whn s rcsd a da θ and Ψ[ ] s h class of all arly rcs srags song ms n []. h arly rcs sragy and hnc h oon rc can b asly drmnd from h r for h so nrgy rc. Amn al [995] show how o drv a bnomal r o b conssn wh h mld so rocss whn h volals of h forward rcs ar consan. hs scon nds hr analyss o h man rvrng modl of scon and o rnomal rs. 4. h r Bldng Procdr h so rc rocss. can b wrn n rms of s naral log afr an alcaon of Io s lmma as follows; d 4 d 4. whch w wr as d [ θ ] d 4.3 θ whr 4 h r bldng rocdr consss of wo sags. rs a rlmnary r s bl for assmng ha θ and h nal val of s zro. h rsg smlfd rocss for hs nw varabl s gvn by d d 4.4 nrgy_sngl_facor

11 Vag Enrgy Oons n a On acor Modl Clwlow and rcland h m vals rrsnd n h r ar qally sacd and hav h form whr s a non-ngav ngr and s h m s. h lvls of and consqnly ar qally sacd and hav h form whr s h sac s 4. Any nod n h r can hrfor b rfrncd by a ar of ngrs ha s h nod a h h m s and h lvl w rfr o as nod. h rnomal r chnq s bascally an lc fn dffrnc schm and from sably and convrgnc consdraons a rasonabl choc for h rlaonsh bwn h sac s and h m s s gvn by 5 : h rnomal branchng rocss and h assocad robabls ar chosn o b conssn wh h drf and volaly of h rocss 4.3. h hr nods whch can b rachd by h branchs manang from nod ar - and whr s chosn so ha h val of rachd by h mddl branch s as clos as ossbl o h cd val of a m. h cd val of s. L m and d dfn h robabls assocad wh h lowr mddl and r branchs manang from nod rscvly. W show n And C ha h robabls ar gvn by: m d d 4.6 h rocdr dscrbd so far als o h rocss wh θ and. 4 h mhodology gnralss n a sraghforward way o non-consan m and sac ss s Clwlow and rcland [998] Char 5. 5 Hll and Wh [993]. nrgy_sngl_facor

12 Vag Enrgy Oons n a On acor Modl Clwlow and rcland h scond sag n h r bldng rocdr consss of dslacng h nods n h smlfd r n ordr o add h ror drf and o b conssn wh h obsrvd forward rcs 6. W can nrodc h corrc m varyng drf by dslacng h nods a m by an amon a. h a s ar chosn o nsr ha h r corrcly rrns h obsrvd forward rc crv. h val of a nod n h nw r qals h val of a h corrsondng nod n h orgnal r ls a ; h robabls rman nchangd. h y o hs sag s h s of forward ndcon and sa rcs o nsr ha h r rrns h crrn mar forward rcs. Dfn h sa rc Q as h val a m of a scry ha ays n of cash f nod s rachd and zro ohrws. a rcs ar h bldng blocs of all scrs; n arclar h rc oday C of any Eroan clam wh ayoff fncon C a m s n h r s gvn by; C Q C 4.7 whr h smmaon as lac across all of h nods a m. h sa rcs ar oband by forward ndcon 7 : Q Q ' ' P 4.8 ' whr ' s h robably of movng from nod ' o nod and P dnos h rc a m of h r dscon bond marng a m. h smmaon as lac ovr all nods a m s whch branch o nod. In ordr o s h sa rcs o mach h forward crv w s h followng scal cas of qaon 4.7; 6 rom qaon 4.3 w hav an analycal solon for θ. Howvr w rfr no o s hs as s h connos m adsmn and wold fal o rrn h obsrvd forward rcs n h r acly d o dscrsaon nvolvd n h r consrcon. 7 Eqaon 4.8 s a dscr vrson of h Kolmogorov forward qaon. nrgy_sngl_facor

13 Vag Enrgy Oons n a On acor Modl Clwlow and rcland P Q 4.9 In And D w show ha h adsmn rm ndd o nsr ha h r corrcly rrns h obsrvd forward crv s gvn lcly as a P 4. Q 4. Eamls of rnomal rs d o Mar orward Crvs W hav fd h so ra r o a nmbr of dffrn mar forward crvs. gr 4. shows 3 mar crvs ha ar rrsnav of; a downward slong forward rc crv YMEX Lgh w Crd Ol rs Conracs Ocobr 997 an ward slong crv YMEX Lgh w Crd Ol rs Conracs 7 Dcmbr 997 and an aromaly fla forward crv whch hbs sasonaly YMEX Hnry Hb aral Gas rs Conracs 7 Dcmbr 997. wo yars worh of monhly mary conracs ar sd o consrc h crvs. gr 4. Mar orward Crvs.5 Ol //97 Ol 7//97 Gas 7// Crd Ol Prc $ Gas Prc $ Monhs o Mary.5 nrgy_sngl_facor 3

14 Vag Enrgy Oons n a On acor Modl Clwlow and rcland gr 4. shows h rsg rs wh m ss vry wo monhs. gr 4. o Prc rs d o Mar orward Crvs Downward slong Uward long and asonal 4 o Prc $ Mary Yars 4 o Prc $ Mary Yars 8 o Prc $ Ma ry Yars h volaly aramrs sd n h r consrcon wr chosn by bs fng n a las sqars sns h ngav onnal forward rc volaly fncon o saml sandard nrgy_sngl_facor 4

15 Vag Enrgy Oons n a On acor Modl Clwlow and rcland dvaons of on yars worh of hsorcal daly frs rrns. h rsg aramrs for h sd of man rvrson and so rc volaly wr.34 and.3 rscvly for crd ol and.4 and.69 for h gas daa. abl 4. shows h rsls of rcng a on yar a-h-mony forward oon on crd ol. h r was consrcd o f h downward slong forward crv of crd ol on h s Ocobr 997 from gr 4.. Prcs for Eroan and Amrcan rcs oons on boh h so and oons on a.5 yar forward conrac ar drmnd from h r for dffrn nmbrs of m ss. h volaly aramrs sd n h r consrcon wr chosn by a bs f o saml sandard dvaons for on yar of hsorcal daa ror o s Ocobr 997. Inrs ras ar assmd o b 6%. abl 4. Val of Eroan and Amrcan Oons Calclad rom h r Oons on o Oons on r s/ Ero Ero Amr Amr Ero Ero Amr Amr Yar Call P Call P Call P Call P Analycal W also comar h rcs of Eroan oons calclad from h r wh h analycal vals calclad va qaons 3.6 and 3.7. abl 4. llsras ha rcs calclad from h r convrg radly o h analycal rc. I can also b sn from abl 4. ha hr s an arly rcs rmm assocad wh boh oons on h so rc and on h forward rc d o h fac ha h downward slong forward crv mls a sgnfcan convnnc yld on h so ass. h nar of h consrcon of h r mls ha hdg aramrs can b qcly and asly calclad. If w calcla hdg aramrs wh rsc o som shf n h forward nrgy_sngl_facor 5

16 Vag Enrgy Oons n a On acor Modl Clwlow and rcland crv hn hs shf only affcs h dslacmn coffcns - dosn ffc h oson of h branchs rlav o h cnral branch or h robabls assocad wh h branchs. 5 Prcng Gnral Pah Dndn Oons n o Prc rs Havng consrcd rnomal rs for h so nrgy rocss w show n hs scon how o rc gnral ah dndn oons sng h chnqs dvlod n Hll and Wh [993] HW for a Blac and chols [973] world and ndd by Clwlow and rcland [999a] C o ml-facor nrs ra modls. 5. Prcng Gnral Pah Dndn Conngn Clams Assm w wsh o rc a gnral ah dndn oon whos ayoff dnds on som fncon G s; s of h ah of h forward rc crv. h rocdr dvlod n HW and C follows a nmbr of ss. rsly h sr drmns h rang.. h mnmm and mamm of h ossbl vals of G. whch can occr for vry nod n h r. hs s achvd by sng forward hrogh h r from h orgn o h mary da comng a ach nod h mnmm and mamm val of G. gvn h val a h nods a h rvos m s whch hav branchs o h crrn nod and h forward crv a h crrn nod. condly w choos an arora s of vals of G. bwn h mnmm and mamm ossbl for ach nod. In choosng hs s of vals w no ha h nods whch l on h r and lowr dgs of h r hav only on ah whch rachs hm and hrfor hr can b only on val of G.. h largs rang of vals wll ycally occr n h cnral scon of h r. h nmbr of vals w consdr shold n gnral ncras only arly wh h nmbr of m ss and also dcras arly from h cnral nods of h r down o on a h dgs of h r n ordr o conrol h comaonal rqrmns. L n b h nmbr of vals w sor a nod and G... n b h vals of G. whrg s h mnmm and G n s h mamm. Clwlow and rcland [998] sggs choosng n o b nrgy_sngl_facor 6

17 Vag Enrgy Oons n a On acor Modl Clwlow and rcland n abs β 5. so ha n wll always b on a h dgs of h r and β n h cnr of h r. In hs way w can ncras β o ncras h accracy of h aromaon by consdrng mor vals of G.. In choosng h acal s of n vals for ach nod w shold consdr h dsrbonal rors of h fncon G.. hs wll vary dndng on h nar of G. and hrfor ms b consdrd on a cas by cas bass. h hrd s n h rocdr s o s h val of h oon a mary a vry nod and for vry val of G. C C ; 5. nally w s bac hrogh h r comng dscond caons and alyng h arly rcs condon a vry nod and for vry val of G. C C C f C m m d d 5.3 whr f dnos h on rod forward ra from m s o m s and whr C C m C d ar h vals of h oon a m s gvn h crrn G for ward mddl and downward branchs of h so rc. hs ar oband by comng h val of G. gvn h crrn val afr ward mddl and downward branchs G G m G d. h vals G G m G d.and hrfor also h oon vals C C m C d wll no n gnral b sord a h ward mddl and downward nods and hrfor ms b oband by nrolaon. or aml sng ar nrolaon w hav C C G G 5.4 l C C l l G G l nrgy_sngl_facor 7

18 Vag Enrgy Oons n a On acor Modl Clwlow and rcland whr l and ar sch ha G G l G and. ha s h wo l vals of G. whch l closs o hr sd of G ar fond and a ar nrolaon bwn hs s don o oban an sma for C. h val of h ah dndn conngn clam s rad from h r as h val of C. 5. Prcng Asan Oons n a rnomal r As a scfc aml of h gnralsd mhodology od n scon 5. w rc Eroan and Amrcan vrsons of an avrag rc call oon whr h avrag s an ovr h so nrgy rc on h fng das l l L. L hr b a oal of m ss from h sar of h lf of h oon nl s mary. In ordr o fnd h rang of vals of h avrag a ach nod w s forward hrogh h r from o. If w hav fond h rang for all nods o m s - hn for any nod h mnmm avrag s drmnd by h mnmm avrag of h lows nod a m s - wh a branch o h crrn nod and h so rc a h crrn nod. h mnmm avrag s gvn by G G G l l m m f m ohrws.. a fng da 5.5 whr m s h nmbr of fng das whch hav occrrd o m s and nod s h lows nod wh a branch o nod. mlarly h mamm avrag s l drmnd by h mamm avrag of h hghs nod a m s - wh a branch o h crrn nod and h nrgy so rc a h crrn nod G n G G n n m m f m ohrws.. a fng da 5.6 nrgy_sngl_facor 8

19 Vag Enrgy Oons n a On acor Modl Clwlow and rcland whr nod s h hghs nod wh a branch o nod. ow snc h arhmc avrag of h so rc s ssnally a sm of lognormally dsrbd rcs wll also b aromaly lognormally dsrbd. W hrfor choos a log-ar s for h n vals of h avrag a ach nod whch gvs G h G 5.7 whr G n G h. n In ordr o drmn h oon vals of qaon 5.4 w frs com wha h avrag wold b gvn h crrn avrag afr ward mddl and downward branchs G G m G d G m f m.. a fng da G m G ohrws 5.8 G m f m.. a fng da G m m G ohrws 5.9 G m f m.. a fng da G d m 5. G ohrws 5.3 A mrcal Eaml In hs scon w rc Eroan and Amrcan vrsons of a fd sr avrag rc call oon on crd ol wh yar o mary and whr h rmnal ayoff s drmnd by h daly avrag of h crd ol rc drng h las monh of h lf of h oon. h valaon da s h s Ocobr 997 h r s consrcd o b conssn wh h nrgy_sngl_facor 9

20 Vag Enrgy Oons n a On acor Modl Clwlow and rcland downward forward crv n gr 4. sng h sam aramrs as sd for abl 4.. abl 5. conans h rsls. abl 5. Convrgnc of Eroan and Amrcan d r Avrag Ra Call Oons Eroan Ma. mbr of Vals for Avrag s/yar Amrcan abl 5. shows h convrgnc of boh h Eroan and Amrcan oon vals as w ncras boh h nmbrs of m ss r yar and also h mamm nmbr of avrags a ach nod s qaon 5.. A frhr ncras n hr of hs dmnsons dos no achv grar accracy of h oon val. 5 mmary and Conclsons In hs ar w hav dvlod a sngl-facor modg framwor whch s conssn wh mar obsrvabl forward rcs and volals. W drvd analycal formla for h forward rc crv a a fr da sandard Eroan oons on so and forward rcs cas floors collars and swaons. W hav also shown how Amrcan syl and oc nrgy drvavs can b rcd sng rnomal rs whch ar consrcd o b conssn wh h forward crv and volaly srcr. As an aml of h alcaon of h rnomal r chnq w dscrbd h rcng of Eroan and Amrcan Asan oons and gav an llsrav aml of h convrgnc rors of h rocdr. h analyss n hs ar nds h rsls n chwarz [997] and Amn al. [995]. nrgy_sngl_facor

21 Vag Enrgy Oons n a On acor Modl Clwlow and rcland nrgy_sngl_facor

22 Vag Enrgy Oons n a On acor Modl Clwlow and rcland nrgy_sngl_facor And A : Proof of h o Prc DE rom qaon. w hav ha forward rcs sasfy h followng DE; d A. hs lognormal scfcaon allows h followng solon for h forward rc; d A. h rocss for h so can b oband by sng ; d A.3 Dffrnang w oban; d d d A.4 or h scfc sngl facor modl of hs ar w hav; A.5 A.6 L d y d A.7 whr

23 Vag Enrgy Oons n a On acor Modl Clwlow and rcland nrgy_sngl_facor 3 d y A.8 hrfor w hav; d y A.9 rom A.3 w hav; d A. mlyng d A. hrfor d d y A. ow [ ] d and so afr rarrangng w oban; 4 d d A.3

24 d B. [ ] d B. rom qaon 3.4 w hav; d B.3 ow [ ] d whch mls ha [ ] 4 B.4 Also sbsng from qaon B.4 w oban; [ ] 4 B.5 bsng no qaon B. sng qaons B. and B.5 and smlfyng w oban; [ ] 4 B.6

25 Vag Enrgy Oons n a On acor Modl Clwlow and rcland nrgy_sngl_facor 5 And C : Proof of h Analycal ormla for a waon rom qaon 3. w hav; Ε K P s K wn ma ; C. L * b gvn by h solon o h followng; K * C. ow l K b gvn by; K... * C.3 nc h forward rc s s monooncally ncrasng n s qaon.5 hn w hav; ; ; K c s K wn C.4

26 Vag Enrgy Oons n a On acor Modl Clwlow and rcland nrgy_sngl_facor 6 And D : Proof of h ranson Probabls Undr h smlfd rocss for of scon 4. w hav E ] [ D. ] [ ] [ E E D. Rcall from scon 4. ha drmns h dsnaon lvl of of h mddl branch from nod hrfor qang h frs and scond momns of n h r wh h vals gvn by qaons D. and D. w oban; d m D.3 d m D.4 Also w rqr ha h sm of h robabls shold b qal o on; d m D.5 olvng h sysm of qaons D.3 D.4 and D.5 w oban; d D.6 d m

27 Vag Enrgy Oons n a On acor Modl Clwlow and rcland nrgy_sngl_facor 7 And E : Proof of h Adsmn rm for a[] rom qaon 4.9 w hav Q P E. Erssng h so rc n rms of w oban; a Q Q P E. a Q P E.3 Rarrangng qaon E.3 ylds; Q P a E.4

28 Vag Enrgy Oons n a On acor Modl Clwlow and rcland Rfrncs Amn K and R A Jarrow 99 Prcng orgn Crrncy Oons Undr ochasc Inrs Ras Jornal of Inrnaonal Mony and nanc Vol Amn K and R A Jarrow 99 "Prcng Oons on Rsy Asss n a ochasc Inrs Ra Economy" Mahmacal nanc Vol. o Amn K V g and C Prrong 995 Vag Enrgy Drvavs n Managng Enrgy Prc Rs Rs Pblcaons London. Blac and M chols 973 "h Prcng of Oons and Corora Labls" Jornal of Polcal Economy Vol Clwlow L and C rcland 998 Imlmnng Drvavs Modls John Wly and ons London. Clwlow L and C rcland 999a Prcng Inrs Ra Eocs n Ml-acor Marovan hor Ra rs Worng Par chool of nanc and Economcs Unvrsy of chnology ydny. Clwlow L and C rcland 999b A Ml-acor Modl for Enrgy Drvavs Rs Managmn" Worng Par chool of nanc and Economcs Unvrsy of chnology ydny. Corazar G and E chwarz 994 "h Valaon of Commody Conngn Clams" h Jornal of Drvavs Vol. o Co J C J E Ingrsoll and A Ross 98 h Rlaon bwn orward Prcs and rs Prcs Jornal of nancal Economcs Vol nrgy_sngl_facor 8

29 Vag Enrgy Oons n a On acor Modl Clwlow and rcland Co J C and A Ross 976 "h Valaon of Oons for Alrnav ochasc Procsss" Jornal of nancal Economcs Vol Gbson R and E chwarz 99 ochasc Convnnc Yld and h Prcng of Ol Conngn Clams Jornal of nanc Vol Hah D R Jarrow and A Moron 99 "Bond Prcng and h rm rcr of Inrs Ras: A w Mhodology for Conngn Clam Valaon" Economrca Vol. 6 o Harrson J M and Plsa 98 Marngals and ochasc Ingrals n h hory of Connos radng ochasc Procsss and hr Alcaons Vol Hllard J E and J Rs 998 Valaon of Commody rs and Oons ndr ochasc Convnnc Ylds Inrs Ras and Jm Dffsons n h o Jornal of nancal and Qanav Analyss Vol. 33 o Hll J and A Wh 993 Effcn Procdrs or Vag Eroan and Amrcan Pah- Dndn Oons h Jornal of Drvavs all. -3. Hll J and A Wh 994a mrcal Procdrs or Imlmnng rm rcr Modls II: wo-acor Modls h Jornal of Drvavs Wnr Hll J and A Wh 994b mrcal Procdrs or Imlmnng rm rcr Modls I: ngl-acor Modls h Jornal of Drvavs all chwarz E 997 h ochasc Bhavor of Commody Prcs: Imlcaons for Prcng and Hdgng h Jornal of nanc Vol. 5 o Vasc O 977 "An Eqlbrm Characrsaon of h rm rcr" Jornal of nancal Economcs Vol nrgy_sngl_facor 9