Pricing and Hedging of Long-term Futures and Forward Contracts by a Three-Factor Model

Transcription

1 CIRJE-F-68 Pricig ad Hdgig of Log-rm Fuurs ad Forward Coracs by a hr-facor Modl Kichiro Shiraya Mizuho-DL Fiacial chology Co. Ld. Akihiko akahashi Uivrsiy of okyo April 009 CIRJE Discussio Paprs ca b dowloadd wihou charg from: hp:// Discussio Paprs ar a sris of mauscrips i hir draf form. hy ar o idd for circulaio or disribuio cp as idicad by h auhor. For ha raso Discussio Paprs may o b rproducd or disribud wihou h wri cos of h auhor.

2 Pricig ad Hdgig of Log-rm Fuurs ad Forward Coracs by a hr-facor Modl Kichiro Shiraya + Akihiko akahashi * Absrac his papr shows pricig ad hdgig fficicy of a hr facor sochasic ma rvrsio Gaussia modl of commodiy prics usig oil ad coppr fuurs ad forward coracs. h modl is simad usig NYMEX WI ligh sw crud oil ad LME Coppr fuurs prics ad is show o fi h daa wll. Furhrmor i shows how o hdg basd o a hr-facor modl ad cofirms ha usig hr diffr fuurs coracs o hdg log-rm corac ouprforms h radiioal paralll hdg basd o a sigl fuurs posiio by im sris daa ad simulaio. I also fids ha h hr facor modl ouprforms is wo-facor vrsio i rplicaio of acual rm srucurs ad ha sochasic ma rvrsio modls ouprform cosa ma rvrsio modls i Ou of Sampl hdgs.. Iroducio h rm srucur of commodiis fuurs udrgos compl shap chags ad a umbr of diffr modls hav b proposd for is simaio. I his papr w propos a hr-facor modl o sima h rm srucur of commodiis fuurs ad h propos ad vrify ffciv hdgig chiqus for log-rm fuurs ad forwards simad wih h modl usig as hdgig isrums h shor ad mdium-rm fuurs ha ar radabl. Black 976 advocad h ida of radig commodiis as quiis wihou dividds ad mad us of gomric Browia moio. Howvr giv h compliy of h shaps associad wih h rm srucur of commodiis fuurs h simpl gomric Browia moio modl proposd by Black 976 is o a good fi. o rsolv his problm ma rvrsio has b iroducd. Ulik quiis wh commodiis prics ris hr is grally albi wih a im lag a icras i supply; covrsly wh prics dcli supply dcrass. h fac ha prics ar drmid by h supply ad dmad balac mas ha h supply sid adjuss supply volums which has h ffc of cosraiig h poial for commodiis prics o mov i a sigl dircio. ha is why i is grally cosidrd appropria o mploy ma rvrsio i commodiis pricig modls. Much mpirical rsarch has b do o his. For ampl i is vrifid i Bssmbidr al Mizuho-DL Fiacial chology Co. Ld. h viws prssd i his papr ar hos of h auhor ad do o cssarily rprs h viws of Mizuho-DL Fiacial chology Co. Ld. * Gradua School of Ecoomics h Uivrsiy of okyo - /5 -

3 Nohlss v if ma rvrsio is usd i a o-facor modl i is difficul o rprs h compl rm srucur of commodiis fuurs ladig Gibso - Schwarz 990 o propos a modl ha supplms h flucuaio of spo prics wih a covic yild sochasic procss ad Schwarz 997 o propos a modl ha plicily mploys covic yilds ad irs ras as h sochasic procss. O h ohr had diffr mhods hav b proposd ha do o amp o idividually modl commodiis spo prics covic yilds or irs ras bu isad amp dirc modlig usig sa variabls wih a ma rvrsio of spo prics. Eampls of dirc modlig of spo prics iclud Schwarz - Smih 000 s wo-facor ma rvrsio modl Casassus - Dufrs 005 s hr-facor ma rvrsio modl ad Corazar - Narajo 006 s N-facor ma rvrsio modl. W us hr-facor Gaussia modls wih cosa ma rvrsio or wihou cosa ma rvrsio. h modls paramrs ar simad usig a Kalma filr ad hav b cofirmd o rproduc acual fuurs prics o h NYMEX WI ligh sw crud oil ad LME Coppr marks. Whr our rsarch diffrs from prior rsarch is ha w sudy cass boh wih ad wihou a cosa ma rvrsio lvl i h commodiis pric modl ad provid a daild aalysis o oly of h modl s abiliy o rproduc fuurs prics bu also is uiliy i hdgig. Commodiis hdgig is a log-dbad opic. For ampl Culp - Millr 995 Mllo - Parsos 995 ad may ohr paprs hav discussd i i rms of h Mallgsllschaf cas. Culp - Millr 995 plais ha lik quiis c. h forward prics for commodiis ar drmid by h mchaism of cos of carry ad argus ha log-rm forward coracs ca b hdgd by holdig shor-rm fuurs ad rollig ovr h corac mohs. O h ohr had Mllo - Parsos 995 ackowldgs ha i is possibl o us shor-rm fuurs o hdg log-rm forward coracs bu criicizs h hdgig chiqu mployd by Mallgsllschaf which was o us h sam umbr of uis of shor-rm fuurs o hdg a ui of log-rm forward coracs. hy us h Gibso - Schwarz 990 modl o dmosra ha shor-rm prics ar mor ssiiv o spo pric chags ha log-rm prics ad ha h acual umbr of shor-rm fuurs rquird o hdg ui of log-rm forward coracs is approimaly 0.. Bcaus of his h radig of Mallgsllschaf whil havig hdgig lms is dmd o b primarily fuurs spculaio. Schwarz 997 also comms o his poi usig - facor modls o calcula hdg posiios ad plaiig ha wh o facor is usd h posiio is sigificaly lss ha approimaly ad v wih wo- ad hr-facors i is sill o a basis lss ha. Nubrgr 999 uss mulipl coracs o hdg log-rm posur ad shows h bfis of h simulaous us of diffr hdgig isrums. Eampls of rsarch aalyzig o oly hdg posiios bu also hdgig rrors iclud Bra - Crw 997 Kor 005 ad Buhlr - Kor - Schobl 004. Bra - Crw 997 amps o us a umbr of diffr pirig fuurs as hdgig isrums for hdgs udr a wo-facor modl bu all of h fuurs i uss as hdgs pir wihi 6 mohs ad h fuurs o b hdgd ar also rmly shor a o mor ha yars. Buhlr- Kor - Schobl 004 uss svral diffr modls o compar - /5 -

4 ad aalyz prformac wh hdgig 0-yar forward coracs. Howvr h fuurs usd as hdgig isrums ar rmly shor pirig i o mor ha mohs ad h daa also oly gos uil 996 so his aalysis dos o icorpora h rapid riss i commodiis prics s i rc yars. Kor 005 showd hdgig rror wih o ad wo-facor modls bu h did show i wih a hr facor modl. I his papr w compar hdgig rror prformd by Mallgsllschaf s paralll hdgig ad prformd by muli-facor modl basd hdgig. Mor spcifically w vrify h sabiliy of hdgs basd o wo- ad hr-facor modls ha do ad do o hav a cosa ma rvrsio lvl ad provid daild aalysis of h diffrcs i hdg ffcivss du o diffrcs i h way i which sa variabls ar calculad ad diffrcs i h rquird fuurs uis ad hdgig rror ra disribuio basd o is simulaios du o diffrcs i h corac mohs of h fuurs usd as hdgig isrums. W also us im sris daa o vrify hdgs for log-rm forward coracs for which irs ra facors hav b ak io accou. W fid ha h hr-facor modl wihou cosa ma rvrsio lvl is possibl o ffcivly hdg log-rm fuurs agais h compl chags i rm srucurs of rc yars. I scio w propos a hr-facor modl icludig a wo-facor modl as a spcial cas which dos o plicily icorpora irs ras or covic yilds ad us ha modl o driv a aalyic soluio for fuurs prics. scio maks us of Kalma filrs o sima h modl s paramrs. scio 4 gos o o mak us of shor ad mdium rm fuurs o cra a hdgig chiqu for log-rm fuurs ad o aalyz prformac wh his hdgig sragy is usd. scio 5 aks a mor pracical approach aalyzig hdgs o Ou of Sampl ad log-rm forward coracs. scio 6 uss a simulaio o aalyz how h form of disribuio chags for h hdg rror ra dpdig upo h slcio of fuurs corac mohs. I h appdi w provid h pcaio ad covariac of h modl prssd i fuurs prics ad os o h umbrs of uis of arr mauriy fuurs rquird o hdg log-rm fuurs.. Modl W firs dscrib a hr-facor Gaussia modl usd for pricig ad hdgig fuurs ad forward coracs. S rprss spo prics of commodiis a im. h logarihm of spo prics a his im is prssd by h followig quaio. log S. prsss a sa variabl corrspodig o h spo pric of h commodiy ad follows h sochasic diffrial quaio show blow. d d dw - /5 -

5 d d d dw d dw. prsss a sa variabl corrspodig o h diffrc bw mdium-rm ad log rm commodiy fuurs prics; is a sa variabl corrspodig o h log-rm porio of h rm srucur. W i muually hav h followig corrlaios i sadard Browig moio udr i quival Marigal masurs EMM. i j dw dw d i j. ij Paramr prsss s spd of rvrsio o ; prsss s spd of auaio. If 0 h is pulld back owards 0. prsss h spd wih which rvrs o spo pric: wh 0. hrfor iuiivly if 0 ovr h cours of im h spo pric is h rd prssd. mdium-rm pric log-rm pric. h sochasic diffrial quaios of idividual sa variabls ca b aalyically solvd ad prssd as follows: s dws 0 s s s s dws dws 0 s 0 dws 0 s 0 dws A his im h fuurs pric is prssd as show blow. horm Usig G o rprs h pric a im of a fuur wih piraio udr EMM:. - 4/5 -

6 5 G E S p. I his quaio s Appdi. E prsss h codiioal pcaio a im. For a discussio of ad Proof. S is a log-ormal disribuio ad h rsul ca hrfor b foud by calculaig h mom graig fucio of ormal disribuio. Q.E.D. N cosidr h mark pric of risk. is h mark pric of risk for sa variabls ad. A his im h followig rlaioship holds ru bw obsrvd masur P ad quival Marigal masur Q. Q P 6 W W u du. 0 hrfor udr masur P h sochasic diffrial quaios ha saisfy idividual sa variabls ar: d P 7 d d dw d d d dw P d P d d dw. I paricular rwriig wih h sa variabls ad a im fucio 8 a b : c d c 0. 0 h sochasic diffrial quaio dscribd abov ca hrfor b rwri as: d ˆ d dw 9 P d ˆ d dw P d P ˆ d dw ˆ whr - 5/5 -

7 Rmark.. ˆ a ˆ b ˆ c ˆ d. I h discussio abov wh 0 solvig for h limi will abl aalyic prssio. Also wh 0 dos hav a cosa ma rvrsio lvl ad h modl islf dos o hav a ulima ma rvrsio lvl. Blow his papr rfrs o cass i which 0 as h cosa ma rvrsio modl ad 0 as h sochasic ma rvrsio modl. Boh yps of modls ar ssially coaid by Corazar - Narajo 006 or Casassus - Dufrs 005. Rmark.. A wo-facor cosa ma rvrsio or wo-facor sochasic ma rvrsio modl ca b obaid by sig 0. hs modls ar ssially h sam as i Kor 005 ha usd h wo-facor modls for aalysis of hdgig. For a wo-facor modl i h subsqu aalysis w pu a rsricio 0 i our hr-facor modls.. Esimaio of paramrs his scio simas h paramrs i h modl. Usig v ad w as whi ois wih ma 0 ad variac h modl dscribd abov ca b prssd as h followig sysm modl ad obsrvaio modl. [Sysm modl] F C Q v F ˆ 0 0 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 0 ˆ ˆ 0 ˆ E 0 G CovQ Q ij C ˆ ˆ ˆ ˆ ˆ ˆ ˆ 0 ˆ ˆ ˆ ˆ. * prsss covariac. For spcific formulas s Appdi. ij - 6/5 -

8 - 7/5 - [Obsrvaio modl] y w R C H y 0 log 0 log m G G y m m m m m H 0 0 m h h R m m m m y C. R prsss obsrvaioal rrors whr i h m i do hos sadard dviaios. I ligh of h compuaioal burd h papr assums ha h obsrvaioal rror of fuurs a idividual mauriis is idpd. Paramrs ar simad usig h Kalma filr of his sa-spac rprsaio. Mor spcifically h followig prdicio ad filrig ar alraigly rpad ad a paramr s is obaid so as o maimiz h log-liklihood. [Prdicio] C F G F F V V. [Filrig] R H V H d d H V K y C H y K V H K I V. [Log-liklihood] N N u d u d mn l log d log y C H y u.

9 Ev if opimal valus ar o s for h iiial valus of ad V as calculaio procds usig h Kalma filr boh approach opimal valus. hrfor h iiial valu problm ca b avoidd by discardig svral sps of daa wh simaig paramrs wihou usig h liklihood calculaio. Esimaios of wo-facor modls ar obaid similarly.. Esimaio rsuls h cosa ma rvrsio modl ad sochasic ma rvrsio modl paramrs wr simad usig h procdur dscribd abov. h followig daa was usd for h simaios. NYMEX WI ligh sw crud oil Daa i 5-busiss day icrms was usd for h priods Jauary Ocobr 00 Jauary Ocobr 00 ad Jauary Novmbr 007; fuurs corac ar from h closs: Fro Moh s DEC d DEC rd DEC 4 h DEC 5 h DEC 6 h DEC ad 7 h DEC. Hr j-h DEC sads for h j-h corac pirig i Dcmbr. If Fro Moh = s DEC i was usd as fro moh. Daa uil 0 h DEC iss afr April 007. Howvr daa from 8 h DEC o 0 h DEC ar o usd i simaio du o lack of rliabiliy of h daa. LME Coppr Daa i 5-busiss day icrms was usd for h priods Spmbr 00 - Novmbr 004 ad Spmbr 00 - Dcmbr 007; fuurs corac ar from h closs: Fro Moh s DEC d DEC rd DEC 4 h DEC 5 h DEC ad 6 h DEC. If Fro Moh = s DEC i was usd as fro moh. hs ar liquid ad ypical asss of oil ad mal fuurs. h choic of im priod is h logs priod for which h daa has mid-rm 7 h DEC i WI 6 h DEC i Coppr fuurs. abls -4 show h paramrs ad obsrvaioal rrors R obaid usig h daa dscribd abov. abl : hr-facor modl WI Cosa ma rvrsio modl Sochasic ma rvrsio modl -Nov 07 Sd Err -Oc 0 Sd Err -Oc 0 Sd Err -Nov 07 Sd Err -Oc 0 Sd Err -Oc 0 Sd Err κ γ α β a b c d Sigma Sigma Sigma Rho Rho Rho Fro Moh sdc ddc rddc hDc hDc hDc hDc AIC /5 -

10 abl : hr-facor modl Coppr Cosa ma rvrsio modl Sochasic ma rvrsio modl -Dc 07 Sd Err -Nov04 Sd Err -Dc 07 Sd Err -Nov04 Sd Err κ γ α β a b c d Sigma Sigma Sigma Rho Rho Rho Fro Moh sdc ddc rddc hDc hDc hDc AIC abl : wo-facor modl WI Cosa ma rvrsio modl Sochasic ma rvrsio modl -Nov 07 Sd Err -Oc 0 Sd Err -Oc 0 Sd Err -Nov 07 Sd Err -Oc 0 Sd Err -Oc 0 Sd Err κ α β a c d Sigma Sigma Rho Fro Moh sdc ddc rddc hDc hDc hDc hDc AIC abl 4: wo-facor modl Coppr Cosa ma rvrsio modl Sochasic ma rvrsio modl -Dc 07 Sd Err -Nov04 Sd Err -Dc 07 Sd Err -Nov04 Sd Err κ α β a c d Sigma Sigma Rho Fro Moh sdc ddc rddc hDc hDc hDc AIC Hr w o ha obsrvaioal rrors i hr-facor modls ar vry small ad ha h modl rplicas h obsrvd fuurs prics vry wll. I h wo-facor cosa ma rvrsio modl simas of usig h WI daa up o 00 ad Coppr daa up o 007 wr 0. 4 ad hc was obsrvd o hav a cosa ma rvrsio lvl bu i all ohr priods boh WI ad Coppr had of virually 0. hrfor dos o flucua wih a cosa ma rvrsio lvl bu is rahr mor similar o a radom walk. h paramrs for Coppr ar sigificaly diffr bw h daa s - 9/5 -

11 up o 004 ad h daa s up o 007. Wh simaios ar mad usig h daa up o 004 hr is oly a lil mor ha yars daa usd ad prsumably h calculaio rsuls i biasd paramrs ha ar opimizd o hs yars. h mark pric of risk is prssd largly i paramr c ad d for ihr WI or Coppr. W also obsrv ha sadard rrors of Coppr s paramrs ar wors ha hos of WI s os i par du o shorag of daa usd i simaio. Fially hr-facor modls show br fiig rsuls ha wo-facor modls i rms of AIC Akaik s Iformaio s Cririo.. Compariso agais acual daa his scio vrifis h dgr of corrlaio bw h sa variabls calculad wih h Kalma filr usig daa hrough 007ad slm fuur prics for NYMEX WI ad LME Coppr. As plaid i scio h sa variabls corrspod o h rm srucur of fuurs. I his cas h sa variabls ar assumd o hav h corrspodcs od i abl 5 ad h aalysis sks o drmi h dgr of corrlaio bw hm. abl 5: Corrspodc of sa variabls facor facor WI Coppr X Fro Moh Fuur Pric Fro Moh Fuur Pric X rd DEC Fuur Pric - 6h DEC Fuur Pric* d DEC Fuur Pric - 5h DEC Fuur Pric* X 6h DEC Fuur Pric 5h DEC Fuur Pric X d DEC Fuur Pric Fro Moh Fuur Pric X 6h DEC Fuur Pric 5h DEC Fuur Pric * X is compard wih h sprad bw 6h Dc ad rd Dc for WI ad h sprad bw 5h Dc ad d Dc for Coppr. abl 6 coais corrlaios for sa variabls ad logarihmic prics calculad from hir 5-busiss day icrms. abl 6: Corrlaios facor facor Cosa ma rvrsio modl Sochasic ma rvrsio modl WI Coppr WI Coppr X X X X X Boh WI ad Coppr hav grally high corrlaios idicaig ha h movm of sa variabls roughly corrspods o acual daa. Also h hr-facor modls provid highr corrlaios ha h wo-facor modls. N w ami whhr modls ca rproduc h acual rm srucurs of fuurs prics. Figur shows h rm srucurs of wo-facor ad hr-facor modls agais mark prics of WI fuurs i Novmbr rd 00 Novmbr s 004 Novmbr s 005 Novmbr s 006 ad Novmbr s 007 rspcivly. Also Figur shows rsuls of Coppr i Dcmbr s 00 Dcmbr s 004 Dcmbr s 005 Dcmbr s 006 ad Dcmbr rd /5 -

12 Figur : WI Fuur rm srucur //007 // DEC07 DEC09 DEC DEC 56 DEC06 DEC08 DEC0 DEC Mark Pric Cosa ma rvrsio facor Sochasic ma rvrsio facor Cosa ma rvrsio facor Sochasic ma rvrsio facor Mark Pric Cosa ma rvrsio facor Sochasic ma rvrsio facor Cosa ma rvrsio facor Sochasic ma rvrsio facor //005 4 // DEC05 DEC07 DEC09 DEC 7 DEC04 DEC06 DEC08 DEC0 Mark Pric Cosa ma rvrsio facor Sochasic ma rvrsio facor Cosa ma rvrsio facor Sochasic ma rvrsio facor Mark Pric Cosa ma rvrsio facor Sochasic ma rvrsio facor Cosa ma rvrsio facor Sochasic ma rvrsio facor 5 // DEC0 DEC05 DEC07 DEC09 Mark Pric Cosa ma rvrsio facor Sochasic ma rvrsio facor Cosa ma rvrsio facor Sochasic ma rvrsio facor Figur : Coppr Fuur rm srucur //007 // DEC07 DEC09 DEC 4900 DEC06 DEC08 DEC0 Mark Pric Cosa ma rvrsio facor Sochasic ma rvrsio facor Cosa ma rvrsio facor Sochasic ma rvrsio facor Mark Pric Cosa ma rvrsio facor Sochasic ma rvrsio facor Cosa ma rvrsio facor Sochasic ma rvrsio facor - /5 -

13 //005 4 // DEC05 DEC07 DEC DEC04 DEC06 DEC08 Mark Pric Cosa ma rvrsio facor Sochasic ma rvrsio facor Cosa ma rvrsio facor Sochasic ma rvrsio facor Mark Pric Cosa ma rvrsio facor Sochasic ma rvrsio facor Cosa ma rvrsio facor Sochasic ma rvrsio facor 5 // DEC0 DEC05 DEC07 Mark Pric Cosa ma rvrsio facor Sochasic ma rvrsio facor Cosa ma rvrsio facor Sochasic ma rvrsio facor W ca obsrv i hos cass ha hr-facor modls ca rplica h acual rm srucurs wll whil wo-facor modls hav som difficuly i capurig h acual rm srucurs. I paricular h diffrc of fiig bw h wo-facor modl ad h hr-facor modl frquly occurs i 006 ad 007 o h obsrvd i h figurs ad of abl ad abl. 4. Fuurs hdgig chiqus his scio dscribs a mhod for buildig a hdgig sragy for ui of a log-rm fuurs corac ad obsrvs how h hr-facor modl dscribd i his papr ca b applid o his ask. h quaio prssig h fuurs pric uss sa variabls ad so ha h shap of h fuurs pric chags accordig o chags i hs sa variabls assumig o chag i h paramrs. hrfor i is possibl i hory o hdg agais log-rm fuurs pric flucuaios by calculaig h i h dlas of h sa variabls for h log-rm fuurs pric ad akig a posiio arr mauriy fuur ha cacls ou hos dlas. I a hr-facor modl hr ar facors o b hdgd ad hrfor fuurs wih diffr piraios will b rquird o build h hdg porfolio. G G ad prss arr G mauriy fuurs prics of diffr piraios ad G h log-rm fuurs pric o b hdgd. I his cas is h soluio o h followig simulaous quaio. whr A b 4 - /5 -

14 G G A G G G G G G G G 4 G 4 b. G 4 his papr rfrs o hdgig usig h hdgig porfolio as a dla hdg. For a wo-facor modl i is possibl o cosruc a dla hdg i h similar way by limiaig h scod facor i h corrspodig hr-facor modl. W vrify h dgr of hdgig rror agais his hdgig porfolio wh im sris daa is applid. For h purposs of his papr h hdgig rror ra is prssd as h fial cumulaiv hdgig rror dividd by h pric of h isrum o b hdgd a h im h hdg commcs. For compariso w calcula h hdgig rror raio for hdgs such as prformd by Mallgsllschaf i which a quival umbr of arr mauriy fuurs is hld agais h fuur o b hdgd. his papr rfrs o his hdgig mhod as h paralll hdg. Mallgsllshcaf hdgd is log-rm fuurs wih rmly shor-rm fuurs of - corac mohs. Howvr giv h icrasd liquidiy of curr commodiis fuurs marks io h mdium-rm rag w vrify h ffcivss of paralll hdgs usig fuurs of up o 6 yars for WI ad up o 5 yars for Coppr. Ulss spcifically sad o h corary h discussio blow rfrs o hdgs agais h 0 h DEC from h fro moh for h WI ad h 8 h DEC for Coppr of which prics ar simad by our modls. For h hdgig priod i is assumd ha h posiio will b closd wih a offsig rad of h 6 h DEC fuur for h WI. I ohr words a 4-yar hdg is rd io ha rducs h im o mauriy of h isrum o b hdgd from 0 yars o 6 yars. For Coppr i is assumd ha h posiio is closd wih a offsig rad of h 5 h DEC fuur rsulig i a -yar hdg ha rducs h im o mauriy of h isrum o b hdgd from 8 yars o 5 yars. For h paralll hdg fuurs for lisd DECs ar usd as hdg asss. For h dla hdgs of hr-facor modls h s - 4 h - 6 h DECs ad s - rd - 5 h DECs ar usd for WI ad Coppr rspcivly; For h dla hdgs of wo-facor modls 4 h - 6 h DECs ad rd - 5 h DECs ar usd for WI ad Coppr rspcivly. Posiios i ach fuurs corac mohs ar adjusd o h s busiss day of h moh afr rviwig hdgig raios ach moh. For boh h paralll hdg ad dla hdg upo h laps of -yar posiios ar rolld o h sam corac moh i h yar. For ampl if a DEC 6 posiio is usd o iiia a hdg o DEC afr h laps of -yar h DEC 6 posiio usd i h hdg will b rolld ovr o DEC 7. Liquidiy dclis h mor disa h fuur bu DEC fuurs hav comparaivly high liquidiy ad giv h ifrqucy wih which hdg raios ar chagd ad h small dgr of chag i h umbr of uis rquird for hdgig his is cosidrd a ralisic hdg. I slcig fuurs corac mohs his aalysis uss combiaios ha provid h rlaivly small hdgig rror ras obaid i scio 6. h similar procdur is ak i slcig fuurs corac mohs for wo-facor modls. - /5 -

15 4. Hdgig rror ra of h paralll hdg h papr firs vrifis h dgr of hdgig rror ra achivd usig h paralll hdg. h pric of h fuurs corac moh o b hdgd is calculad basd o h cosa ma rvrsio modl ad daa up o 007 usig paramrs ad sa variabls simad wih h Kalma filr. Figur shows cumulaiv hdgig rror ras h cumulaiv hdgig rror dividd by h pric of h isrum o b hdgd a h im h hdg commcs usig WI ad Coppr im sris daa. I his cas for h Fro Moh h cumulaiv hdgig rror ra is prssd for a paralll hdg rolld ovr o h -pirig corac moh ach moh; for ohrs h cumulaiv hdgig rror ra is prssd for a paralll hdg wih a o-yar roll usig DECs for ach yar. h fuurs o b hdgd ar h WI DEC ad h Coppr DEC ad h hdg rmias a h mos rcly availabl daa 007. h horizoal ais prsss h amou of im lapsd sic h commcm of h hdg; h vrical ais h cumulaiv hdgig rror ra. h sam oaio is usd for ohr graphs i his papr. Figur : Cumulaiv Hdgig rror ras of h paralll hdg WI DEC Coppr DEC 0% 00% 80% 60% 40% 0% 0% -0% Dc/0 Ju/04 Dc/04 Ju/05 Dc/05 Ju/06 Dc/06 Ju/07 FroMoh s DEC d DEC rd DEC 4h DEC 5h DEC 6h DEC 00% 60% 0% 80% 40% 0% Ja/05 Jul/05 Ja/06 Jul/06 Ja/07 Jul/07 FroMoh s DEC d DEC rd DEC 4h DEC 5h DEC As ca b obsrvd from Figur rror is lowr h mor disa h fuur usd o hdg. Coppr has a largr hdgig rror ra ha WI idicaig ha h compos i Coppr s rm srucur ha chag i paralll ar smallr ha WI s. Howvr v usig h mos disa fuur wih h smalls hdgig rror ra h hdgig rror ras wih a paralll hdg wr sill approimaly % for WI ad approimaly % for Coppr. 4. Hdgig rror ra of h dla hdg his scio obsrvs h hdgig rror ra for dla hdgs for boh h cosa ma rvrsio modl ad h sochasic ma rvrsio modl. For ach modl paramrs simad from daa up o 007 wr usd ad for vrificaio purposs mhods wr usd o sima sa variabls i ordr o sima log-rm fuurs prics. h firs is sa variabls wr simad usig h Kalma filr Kalma filr sa variabls hriafr; h scod simaio crad simulaous liar quaios for h sa variabls so ha h fuurs pric of h modl machs h fuurs pric of h fuurs corac moh o b hdgd allowig sa variabls o b calculad by solvig hs quaios simulaous quaio-basd sa variabls hriafr. o compar h rlaiv prcisio of hdgig usig h modl dscribd i his papr h vrificaios blow - 4/5 -

16 o h rsuls for mos disa fuur which was h mos prcis for h paralll hdg. Noaios follow h pracic usd for paralll hdgs. Vrificaios wr prformd wih diffr fuurs o b hdgd ad h obsrvd hdgig rror ras ar summarizd i abl 7. abl 7: Hdgig rror ras Cosa ma rvrsio modl Sochasic ma rvrsio modl Kalma Filr Equaios Basd Kalma Filr Equaios Basd facor facor facor facor Paralll facor facor facor facor Paralll DEC07.0%.% 0.7%.% 4.0% 0.7% 0.8% 0.4% 0.4%.8% DEC08.0% 0.6% 0.9% 0.6% 8.4% 0.7%.5% 0.5% 0.7% 8.% DEC09 0.8% 0.8%.0% 0.8%.4% 0.5% -0.8% 0.8% 0.6%.% WI DEC0 -.5% -.% -.5% -.5% -.5% -.7% -0.5% -.7% -.5% -4.0% DEC -.% -.8% -.% -.8%.7% -.% -.8% -.% -.6%.4% DEC -.% -4.% -.7% -4.% 8.0% -.0% 0.4% -.6% -0.9% 7.8% DEC -0.8% -.% 0.0% -.%.% -0.% 0.5% 0.5%.%.% DEC0-0.4% -.% -0.5% -.0%.4% -0.% 0.0% 0.0% -.8%.4% Coppr DEC 0.0% -5.% 0.% -4.8% 0.%.%.5%.4%.4% 0.% DEC -.% -0.8% -6.7% -0.8%.% 0.9% 4.8% -.%.7%.0% Figur 4: Cumulaiv hdgig rror ras of h dla hdg cosa ma rvrsio modl WI DEC 0% 8% 6% 4% % 0% -% Dc/0 Ju/04 Dc/04 Ju/05 Dc/05 Ju/06 Dc/06 Ju/07-4% -6% -8% -0% Kalma filr sa variabls facor Simulaous quaio-basd sa variabls facor Kalma filr sa variabls facor Simulaous quaio-basd sa variabls facor 0% 5% 0% 5% -0% -5% -0% Coppr DEC 0% -5% Ja/05 Jul/05 Ja/06 Jul/06 Ja/07 Jul/07 Kalma filr sa variabls facor Simulaous quaio-basd sa variabls facor Kalma filr sa variabls facor Simulaous quaio-basd sa variabls facor Figur 5: Cumulaiv hdgig rror ras of h dla hdg sochasic ma rvrsio modl WI DEC 0% 8% 6% 4% % 0% -% Dc/0 Ju/04 Dc/04 Ju/05 Dc/05 Ju/06 Dc/06 Ju/07-4% -6% -8% -0% Kalma filr sa variabls facor Simulaous quaio-basd sa variabls facor Kalma filr sa variabls facor Simulaous quaio-basd sa variabls facor 0% 5% 0% 5% -0% -5% -0% Coppr DEC 0% -5% Ja/05 Jul/05 Ja/06 Jul/06 Ja/07 Jul/07 Kalma filr sa variabls facor Simulaous quaio-basd sa variabls facor Kalma filr sa variabls facor Simulaous quaio-basd sa variabls facor h paralll hdg is abl o provid ffciv hdgig wh h ovrall rm srucur chags i paralll bu gras larg hdgig rror wh hr ar chags i h shap of h rm srucur. By coras h dla hdg works much br ha h paralll hdg s abl 7. wo- ad hr-facor modls provid rlaivly similar rsuls hough hr-facor modls works br for Coppr. Howvr wo-facor modls hav som difficuly i rplicaig acual rm srucurs as show i Figurs ad. - 5/5 -

17 Comparig Kalma filr sa variabls ad simulaous quaio-basd sa variabls wh prformig a dla hdg for Coppr simaio of Kalma filr sa variabls producs larg hdgig rror durig h rm of h hdg as ca b s i Figur 4 ad Figur 5. his is prsumably du o diffrcs i whhr h modl prics ar obaid i a mar cosis wih h ass prics usd i h hdg ad h pric of h asss o b hdgd. Wh usig simulaous quaio-basd sa variabls modl prics cludig rollovr imig mach h prics of h asss usd i h hdg ad of h asss o b hdgd. O h ohr had wh usig Kalma filr sa variabls h acual prics of h asss usd i h hdg diffrs from h modl prics rsulig i hdgig rror wh hdgig is prformd. For WI h obsrvaioal rror was small for h fuurs corac moh usd i h hdg ad virually quival o h simulaous quaio-basd sa variabls idicaig ha hr is lil diffrc du o h mhod by which sa variabls ar drmid. Bcaus of h rsul i h las paragraph h discussio blow uss oly sa variabls ha ar calculad by solvig simulaous quaios for boh WI ad coppr modls. 5. Sabiliy of h dla hdg h vrificaios so far hav simad paramrs basd o daa ha icludd h ir hdg priod. Howvr i acual pracic h paramr simaio priod ad h hdg priod diffr. Discussios so far hav also assumd ha h hdgs arg log-rm fuurs bu gral pracic is for log-rm coracs o b forwards rahr ha fuurs which rquirs ha irs-ra facors also b ak io accou. I his scio w cofirm h followig sigs so as o coduc vrificaios i a sa as clos as possibl o acual pracic.. No ovrlap bw h paramrs simaio priod ad h hdg priod. Hdgig agais forwards I his papr cass i which h ir hdg priod is icludd i h paramr simaio priod ar rfrrd o as I Sampl whil hdgs i which hr ar spara paramr simaio priods ar rfrrd o as Ou of Sampl. 5. Ou of Sampl hdgs o vrify h ffcivss of h Ou of Sampl hdg his scio uss h paramrs simad i scio wih h daa hrough 00 or 00 for WI ad hrough 004 for Coppr. h fuur o b hdgd is DEC hdg priod from 00 o 006 or DEC hdg priod from 00 o 007 for WI ad DEC hdg priod from 004 o 007 for Coppr. abl 8 summarizs rsuls of vrificaios usig im sris daa for hdgig rror ras wh hdgig log-rm fuurs prics as simad wih h modl usig hs paramrs. For purposs of compariso w hav also od h rsuls for h I Sampl simaios i h prvious scio. - 6/5 -

18 abl 8: Hdgig rror ras of h dla hdg Ou of Sampl WI Coppr Cosa ma rvrsio modl Sochasic ma rvrsio modl I sampl Ou of sampl I sampl Ou of sampl facor DEC -.7% -.% -.6% -.4% facor DEC -4.% -.5% -0.9%.5% facor DEC 0.0% -7.6% 0.5% 0.% facor DEC -.% -6.5%.% 5.% facor DEC -6.7% -6.9% -.% -.9% facor DEC -0.8% 0.0%.7%.0% For compariso bw h wo-facor modls ad h hr-facor modls i is obsrvd ha h modl producig h smallr absolu rror i I Sampl also cras h smallr absolu rror i Ou of Sampl for all cass of abl 8. For h hr-facor sochasic ma rvrsio modl w cofirmd ha h diffrcs i hdgig rror ras du o diffrcs i h paramr simaio priod wr o ha larg for WI. For Coppr hr wr diffrcs i I Sampl ad Ou of Sampl hdgig rror ras i par du o h diffrcs i h paramrs obaid for I Sampl ad Ou of Sampl. Howvr h rror is o rm ad v hough hr ar diffrcs i h paramrs obaid usig h maimum liklihood mhod hdgig udr h modl is cosidrd o b rlaivly sabl. O h ohr had for h wo-facor sochasic ma rvrsio modl hr ar mor diffrcs bw I sampl ad Ou of sampl hdgig rror ras: his dcy sms srogr for Coppr ha for WI. For h cosa ma rvrsio modls i h Ou of Sampl WI log-rm pric lvls chagd du o h sharp icrass i oil prics bgiig i 00 virually limiaig ma rvrsio. Nohlss h log-rm fuurs prics calculad by h modls rvr o h ma lvls obsrvd i h daa hrough 00 or 00 icrasig h hdgig rror ras. I ligh of his i is likly ha h cosa ma rvrsio modl is mor pro o hdgig rror wh hr ar chags i ma rvrsio lvls c. idicaig ha i is br o us h sochasic ma rvrsio modl wh prics ar basd o h hdg. Du o h rsuls abov h discussio i h subscio uss oly h hr-facor sochasic ma rvrsio modl. 5. Hdgig log-rm forward coracs h discussios o his poi hav assumd ha fuurs would b hdgd bu commo pracic is o rad forwards for h log-rm porio ha is o radd o chags. If irs ras ar drmiisic or mov idpdly from udrlyig asss prics ar h sam for fuurs ad forwards bu if irs ras ar o drmiisic hdgs mus ak accou of hir movms. For purposs of simpliciy his discussio assums ha irs ras ad udrlyig asss ar idpd dscribs hdgig chiqus wh h isrum o b hdgd is a forward ad h asss usd i h hdg ar fuurs. h uiliy of his hdgig chiqu is h vrifid usig im sris daa. W cosidr hdgig log-rm forwards wih shor-rm fuurs i h followig wo sps. Log-rm forwards ar hdgd usig log-rm fuurs wih h sam piraio. - 7/5 -

19 Log-rm fuurs ar hdgd usig h dla hdg wih h arr mauriy fuurs as dscribd i scio 4. Bcaus Sp is plaid i scio 4 w plai h hdgig chiqu of Sp. h oaios usd ar dfid as: F : Pric a poi i im of forward wih piraio. G : Pric a poi i im of fuur wih piraio. P : Pric a poi i im of zro-coupo bod wih piraio. I addiio 0. h amou of chag i h forward profi/loss a poi i 0 m im durig h priod from poi i im i hrough i is: PV a poi i im i is prssd as F F 0 P i i. hrfor h amou of chag i PV for h forward durig h priod from i hrough i is: 0 F F 0 P F F 0 P i i i i. I his cas 0 ca b rformd as follows: F F 0 P F F 0 P hus giv by: i i i i F F P F F 0 P P i i i i i. F F P P i i i i Cm ha dos h accumulaio valuad a im of h las rm o h righ sid is C m m j0 m ri i i j i F F P P j j j j whr h isaaous irs ra i ach priod i i is approimad as a cosa r i. Hrafr w igor Cm bcaus i prsss a gligibl amou corrspodig o h quadraic variaio. Assumig ha irs ra is idpd of udrlyig ass prics h forward pric ad fuurs pric ar quival F G ad quaio ca b prssd as show blow: F F P F F 0 P P i i i i i i G G P G G 0 P P i i i i i i. - 8/5 -

20 h firs rm o h righ sid prsss h chag i h fuur accordig o which a dla hdg is mad usig fuurs for h arr mauriy corac mohs udr h mhod dscribd i scio 4. As a rsul ui of forwards ca b giv a proima hdg usig a porfolio comprisig fuurs ad zro-coupo bods as show blow. A dla hdg usig fuurs for h arr mauriy corac mohs o hdg P uis of fuur G. Purchas of G G 0 uis of zro-coupo bod P. I is od ha his hdgig sragy is ssially h sam as Schwarz s hdgig sragy s Schwarz 997 p W aalyzd h hdgig rror ra wh a 0-yar WI forward corac is hdgd accordig o h mhod abov usig WI fuurs ad zro-coupo bods for 4 yars. Hr forward prics ar assumd o b qual o horical fuur prics calculad by our modl. No ha h fuds for purchas of zro-coupo bods ad cash flow grad by markig fuurs o mark ar ivsd/raisd i shor-rm irs ras. For purposs of h vrificaio w usd I Sampl paramrs ad o simplify h calculaios of h zro-coupo bods usd h 8-yar swap ra as spo yild; h calculaios of shor-rm irs ras for ivsms ad fudig usd h M LIBOR ad assum ha h irs ra is idpd of ass prics hus forward pric is qual o fuur pric. abl 9 coais hdgig rror ras du o diffrcs i h forwards o b hdgd. abl 9: Hdgig rror ras of h dla hdg DEC07 DEC08 DEC09 DEC0 DEC DEC DEC usig fuurs ad bods 0.% 0.% 0.4% -.% -.6% -.% -0.% usig fuurs 0.% -.% -.4% -9.% -5.8% -0.0% -5.6% Figur 6: Cumulaiv hdgig rror ras of h dla hdg WI DEC 0% 8% 6% 4% % 0% -% Dc/0 Ju/0 Dc/0 Ju/04 Dc/04 Ju/05 Dc/05 Ju/06-4% -6% -8% -0% usig fuur ad bods usig fuurs I will b od ha i his vrificaio which usd im sris daa for irs ras ad fuurs prics v assumig irs ras ad udrlyig asss o b idpd h us of zro-coupo bods ad fuurs o hdg forwards ad was abl o hdg virually all of h irs ra facors grad by h diffrc bw forwards ad fuurs. Howvr if irs ras ar o hdgd hr ar cass i which larg hdgig rrors ar grad durig h hdg priod as ca b s from Figur 6 so h - 9/5 -

21 ida ha hr dos o d o b a hdg o h irs-ra porio is o suppord. 6. Masurig h disribuio of hdgig rror ras h aalysis of hdgig rror ras basd o im sris daa ar limid o h o i a fw pahs. hus his scio provids a simulaio aalysis o masur h disribuio of hdgig rror ras rsulig from flucuaios i udrlyig asss. hr-facor sochasic ma rvrsio modls ar usd i simulaio whr I Sampl paramrs ar usd ad fuurs ar hdgd by fuurs wih shorr mauriis. Blow ar h spcific procdurs for h simulaio. Hisorical daily fuurs prics wr crad basd o paramrs ad sa variabls simad usig h Kalma filr. h rror ra bw h fuurs prics basd o h modl ad crad i Sp vs. acual fuurs prics quod o chags for WI h fro moh ad h s -6 h DEC; for Coppr h fro moh ad h s -5 h DEC was calculad Acual daa - Modl pric/modl pric ad h h ma ad covariac of h rror ra wr obaid. Hr i was assumd ha rror follows mulidimsioal ormal disribuio. hr-dimsioal ormal radom umbrs wr crad ad h sa variabls wr causd o flucua accordig o h modl so as o cra a rm srucur for fuurs. 4 Mulidimsioal ormal radom umbrs accordig o h disribuio dscribd i Sp wr crad for h fuurs rm srucur dvlopd i Sp muliplid as rror ad addd o h origial rm srucur. Sp modl prics + Sp modl prics * Radom umbrs followig h Sp rror disribuio. 5 Simulaous quaio-basd sa variabls wr calculad o h assumpio ha h rm srucur crad i Sp 4 was h rm srucur acually obsrvd i h mark. 6 h rm srucur crad i Sp 5 was usd o sima log-rm prics hdgs wr ak agais hos prics ad h fial hdgig rror ra masurd. 7 Sps -6 wr rpad for a cosa umbr of ims o fid h sampl ma ad sampl sadard dviaio of h hdgig rror ras obaid. 6. Disribuio of hdgig rror ras W prformd 5000 rials for ach combiaio of fuurs corac usd i h hdg accordig o h procdurs oulid abov ad calculad h avrag ad sadard dviaio of h hdgig rror ras. hr was lil diffrc i h hdgig rror ras du o diffrcs i iiial valus so as hdgd asss w usd DEC 7 for WI for a priod of 4 yars bgiig 007 ad DEC 5 for Coppr for a priod of yars bgiig 007. abl 0 coais h mas ad sadard dviaios for h obaid hdgig rror ras. h corac mohs colum i h abls rfrs o which DEC from h fro moh is usd for h hdg. For - 0/5 -

22 ampl -4-6 rfrs o h hdg usig h s 4 h ad 6 h DECs. Likwis 6Y Paralll 5Y Paralll prsss h hdgig rror ra wh a paralll hdg is rd io usig h 6 h 5 h DEC. h pric of h hdgd asss for h paralll hdg is i h pric foud usig -4-6 for WI ad --5 for Coppr. abl 0: Avrags ad sadard dviaios of hdgig rrors WI Corac Mohs Avrag Sadard Dviaio Corac Mohs Avrag Sadard Dviaio % 0.9% % 4.5% % 0.% %.% % 5.0% % 4.9% %.7% %.% % 8.0% %.9% %.9% % 6.0% %.0% %.5% % 4.% %.0% %.0% %.5% %.8% Paralll Hdg % 9.% 6Y Paralll -0.5% 6.0% Coppr Corac Mohs Avrag Sadard Dviaio Corac Mohs Avrag Sadard Dviaio %.5% % 8.5% % 5.9% % 4.% %.7% %.9% % 6.8% % 4.7% %.7% Paralll Hdg %.7% 5Y Paralll 0.4% 6.% S Appdi for mor o h umbr of fuur uis rquird du o diffrcs i h slcio of fuurs mohs a h im h hdg is commcd. Accordig o h rsuls from h simulaio appropria slcio of h fuurs corac mohs for h hdg porfolio wh rig io a dla hdg has h poial for a mor accura hdg ha h us of a paralll hdg. I paricular i rms of h rquird amous s Appdi ad also h rlaioship bw h mas ad sadard dviaios of hdgig rror ras hdgs for h WI ad Coppr hibid fficicy usig h s - 4 h - 6 h DECs; ad h s - rd - 5 h DECs rspcivly. As a mor gral rsul i was foud ha h fuurs usd o cra a hdg porfolio should o h possibl hav muually dispara corac mohs. his is bcaus wh h fuurs usd i h hdg ar clos o ach ohr h ad dla srucurs ar similar ad a grar umbr of fuurs uis is rquird o offs h dla of h isrum hdgd. h grar h umbr of fuurs uis usd i h hdg h largr h rror prssd as diffrcs i h prics of h modl ad acual fuurs. Covrsly wh h fuurs ar farhr away from ach ohr hy hav dispara dla srucurs makig i mor likly ha hdgig will o rquir as may uis. 7. Coclusio his papr dmosrad ha by hr-facor Gaussia modl wih appropria simaio of paramrs - /5 -

23 i was possibl o rproduc h rm srucurs of lisd commodiis fuurs NYMEX WI LME Coppr durig h im priod sudid ad ha log-rm fuurs prics could b obaid ha wr cosis wih liquid arr mauriy coracs. I was also foud ha wo-facor Gaussia modls hav som difficuly i capurig acual rm srucurs of fuurs. Furhrmor i w o o propos a hdgig chiqu for log-rm fuurs ad forwards coracs comparig h rsuls from his chiqu o h rsuls from h simpl shor-rm fuurs-basd hdgig sragy usd by Mallgsllschaf paralll hdg ad vrifid ha our proposd sragy was sabl i may diffr circumsacs backwardaio coago risig prics dcliig prics c.. I addiio i foud ha a sochasic ma rvrsio modl offrd mor sabl hdgig ha a modl wih a cosa ma rvrsio lvl. Also i obsrvd ha h modl producig h smallr absolu rror i I Sampl also cras h smallr absolu rror i Ou of Sampl. I h usd a simulaio o masur h hdgig rror ras obaid du o diffrcs i h corac mohs of h fuurs usd i h hdg. I was foud ha h fuurs usd o cra a hdg porfolio should o h possibl hav muually dispara corac mohs. I sum h hr-facor modl wih sochasic ma rvrsio sms usful i pracic for pricig log-rm fuurs/forward coracs ad for hdgig hm wih appropria slcd liquid isrums. Fuur issus iclud valuaio of opio valus usig h modl ad srucurig of rlva hdgig chiqus. Commodiis grally hav avrag-basd opios which maks calculaio compl. I would b usful o vrify hdgig chiqus ad hir fficicy. Ackowldgm W rally hak wo aoymous rfrs for hir ffors ad prcious comms. Rfrcs. Bssmbidr al.995. Ma Rvrsio i Equilibrium Ass Prics: Evidc from h fuur rm srucur Joural of Fiac Bjrksud P.99 Coig claims valuaio wh h covic yild is sochasic: aalyical rsuls Workig Papr Norwgia School of Ecoomics ad Busiss Admiisraio.. Black F.976 h pricig of commodiy coracs Joural of Fiacial Ecoomics Bra M.J. ad Schwarz E.S.985. Evaluaig aural rsourcs ivsms Joural of Busiss Bra M.J. ad Crw N.I.997 Hdgig Log-Mauriy Commodiy Commims wih Shor-Dad Fuurs Coracs i M. Dmpsr ad S. PliskaEds Mahmaics of - /5 -

24 drivaivs scuriis pp65-90 Cambridg Uivrsiy Prss 6. Bühlr W. Kor O. ad Schöbl R.004 Hdgig Log-rm Fowards Wih Shor-rm Fuurs: A wo-rgim Approach Rviw of Drivaivs Rsarch CasassusJ. ad Colli-DufrsP.005 Sochasic Covic Yild Implid from Commodiy Fuurs ad Irs Ras Joural of Fiac Corazar G. ad Narajo L.006 A N-Facor Gaussia Modl of Oil Fuurs Prics Joural of Fuurs Marks Culp C.L. ad Millr M.H.995 Mallgsllschaf ad h Ecoomics of Syhic Sorag Joural of Applid Corpora Fiac Gibso R. ad Schwarz E.S.990 Sochasic covic yild ad h pricig of oil coig claims Joural of Fiac Hah D. Jarrow R. ad Moro A.99 Bod pricig ad h rm srucur of irs ras: A w mhodology for coig claims valuaio Ecoomrica Hull J.C. ad A. Whi 990 Pricig irs-ra-drivaiv scuriis Rviw of Fiacial Sudis Jamshidia F. ad Fi M.990 Closd-form soluios for oil fuurs ad Europa opios i h Gibso-Schwarz modl Workig papr Mrrill Lych Capial Marks 4. Kor O.005 Drif Mars: A Aalysis of Commodiy Drivaivs Joural of Fuurs Marks Mllo A.S. ad Parsos J.E.995 Mauriy Srucur of a Hdg Mars: Lssos from h Mallgsllschaf Dbacl Joural of Applid Corpora Fiac Milrs K.R. ad Schwarz E.S.998 Pricig of Opios o Commodiy Fuurs wih Sochasic rm Srucurs of Covic Yilds ad Irs Ras Joural of Fiacial Ad Quaiaiv Aalysis Nubrgr A.999 Hdgig Log-rm Eposurs wih Mulipl Shor-rm Fuurs Coracs? Rviw of Fiacial Sudis Schwarz E.S.997 h Sochasic Bhavior of Commodiy Prics: Implicaios for Valuaio ad Hdgig Joural of Fiac Vol Schwarz E.S. ad Smih J.E.000 Shor-rm Variaios Log-rm Dyamics. i Commodiy Prics Maagm Scic /5 -

25 - 4/5 - Appdi Epcaio ad covariac mari of sa variabls E Cov.

26 Appdi h umbr of fuurs uis WI Coppr sdec ddec rddec 4hDEC 5hDEC 6hDEC sdec ddec rddec 4hDEC 5hDEC /5 -