CONTINUOUS TIME KALMAN FILTER MODELS FOR THE VALUATION OF COMMODITY FUTURES AND OPTIONS

Transcription

1 CONTINUOUS TIME KALMAN FILTER MODELS FOR THE VALUATION OF COMMODITY FUTURES AND OPTIONS ANDRÉS GARCÍA MIRANTES DOCTORAL THESIS PhD IN QUANTITATIVE FINANCE AND BANKING UNIVERSIDAD DE CASTILLA-LA MANCHA DEPARTAMENTO DE ANÁLISIS ECONÓMICO Y FINANZAS ADVISORS: GREGORIO SERNA AND JAVIER POBLACIÓN SEPTEMBER

2 To all popl who ma h world h marvllous plac i is. L hm find h happinss hy giv and dsrv. A habi of basing convicions upon vidnc, and of giving o hm only ha dgr or crainy which h vidnc warrans, would, if i bcam gnral, cur mos of h ills from which h world suffrs Brrand Russll

3 ACKNOWLEDGES I is a somwha unforuna fac ha almos no on rads h acnowldgs scion bu h popl who bliv hy dsrv o b hand. As a rsul, wriing a lis of conribuors bcoms a rahr ricy businss. Thr is no spac o han vryon wih sufficin innsiy and usually popl appar firs according o h auhor s ida of how imporan hir hlp was, which of cours could b a bi unfair somims. Nvrhlss, s d bin nacidos sr agradcidos and h plac o rcogniz hlp is hr. I jus wand o poin ou my difficulis in haning vryon as much as hy dsrv and apologiz for any failur in doing so. This PhD hsis has an many yars, much mor ha i should. And hr is no on o blam for i apar from myslf. In such a long im span, many popl hav hlpd m, on way or anohr. L his b a small ribu for hir painc wih m. To my dircors, Grgorio Srna and Javir Población, for hir suppor in his long journy and spcially for bliving in his projc whn vn I did no. I is a commonplac o say ha his hsis would hav nvr bn raliy wihou hm, bu, bliv m, I do no hin his was vr rur han in my cas. To Crisina Suárz and Javir Suárz, for hir hlp in saring all his. To María Dolors, for hr suppor in im of crisis, vn if finally I oo a diffrn road o my PhD. And of cours o Mrcds Carmona, who gav m anohr chanc o sar ovr and, vry spcially, for hr uncondiional frindship xcding all our acadmic rlaionship. To my family, for hir undrsanding and for chring m up in h many criss I facd. And hir mri is doubl, bcaus hy wr as scpic on his projc as m. To ma pi ami chrè Vroniqu, for anding m in h mor ns momns I facd in his long journy. I could no xpc a br company in ha crisis. 3

4 To all my frinds (and pas girlfrinds) vrywhr, who hlpd m in solving a mar complly arcan o pracically all of hm (if no all) jus by polily lisning o unnding mahmaical nonsns and bing always supporiv. I would li o giv a spcial mnion o Carlos and Danil, as I am sur his projc would hav ndd yars bfor wr no bcaus of hm. Paradoxical as i may sm, hs dlays mad m h prson I am now and I fl rally graful. I would also li o includ among my frinds all my sudns who suffrd my unbarabl classs wih painc and all my collagus in Ovido Univrsiy, IES Juan dl Enzina and IES Vadinia. Thy gav m all h hlp in vry aspc I could nd. I would li o mnion spcially Visiación Rodriguz, for swapping urns vn whn vn I did no dsrv and Juanjo Monsinos for spaing abou childrn and PhD 4

5 INDEX ACKNOWLEDGES.. 3 INDEX 5 INTRODUCTION.. 7 A HISTORICAL BACKGROUND 7 GENERAL SETUP. 7 SUMMARY OF CHAPTER ONE SUMMARY OF CHAPTER TWO.. SUMMARY OF CHAPTER THREE. REFERENCES.. CHAPTER : ANALYTIC FORMULAE FOR COMMODITY CONTINGENT VALUATION. 4.. INTRODUCTION THEORETICAL MODEL.. 6 Conrac Valuaion 6 Volailiy of Fuur Rurns DISCRETIZATION AND ESTIMATION ISSUES PRECISE ESTIMATION OF THE SCHWARTZ (997) TWO- FACTOR MODEL...5. SIMPLIFIED DEDUCTION OF THE FUTURES PRICES IN THE TWO-FACTOR MODEL BY SCHWARTZ AND SMITH () CONCLUSIONS. 3 APPENDIX A: MATHEMATICAL REFERENCE RESULTS 3 APPENDIX B: FUTURES CONTRACT VALUATION. 36 APPENDIX C: VOLATILITY OF FUTURES RETURNS. 39 REFERENCES.. 4 TABLES AND FIGURES 43 CHAPTER : COMMODITY DERIVATIVES VALUATION UNDER A FACTOR MODEL WITH TIME-VARYING RISK PREMIA 48. INTRODUCTION 48. DATA. 5.3 PRELIMINARY FINDINGS.. 53 Mar prics of ris simaion using h maximum-lilihood mhod 53 Mar Prics of Ris Esimaion using h Kalman Filr Mhod A FACTOR MODEL WITH TIME-VARYING MARKET PRICES OF RISK DEPENDING ON THE BUSINESS CYCLE. 6.5 OPTION VALUATION WITH TIME-VARYING MARKET PRICES OF RISK DEPENDING ON THE BUSINESS CYCLE. 63 Opion Daa.. 63 Opion Valuaion Mhodology 64 Opion Valuaion Rsuls 65.6 CONCLUSIONS.. 67 APPENDIX. 7 REFERENCES.. 7 TABLES AND FIGURES 74 CHAPTER 3: THE STOCHASTIC SEASONAL BEHAVIOR OF ENERGY COMMODITY CONVENIENCE YIELS INTRODUCTION 9 3. DATA AND PRELIMINARY FINDINGS.. 93 Daa dscripion. 93 Prliminary Findings THE PRICE MODEL.. 98 Gnral Considraions.. 98 Thorical Modl.. 99 Esimaion Rsuls 3 5

6 3.4 THE CONVENIENCE YIELD MODEL 4 Thorical Modl 5 Esimaion Rsuls CONCLUSIONS 9 APPENDIX A. ESTIMATION METHODOLOGY. APPENDIX B. STOCHASTIC DIFERENTIAL EQUATIONS (SDE) INTEGRATION. 3 APPENDIX C. CANONICAL REPRESENTATION. 5 Inroducion 5 Gnral sup. 5 Invarian ransformaions 6 Rlaionship wih A (n). 6 Firs canonical form.. 7 Complx ignvalus. 8 Scond canonical form.. Maximaliy.. Ris prmia. 3 REFERENCES 4 TABLES AND FIGURES. 7 6

7 INTRODUCTION A HISTORICAL BACKGROUND Th hisory of Kalman filr is long and broad, and so is h liraur of is applicaions o h fild of Economics. I was firs drivd by Kalman in a clbrad aricl in 96, following a prvious and mor horical wor of Sraonovich (959). Is imporanc was rcognizd in h Enginring liraur from h vry sar. Economics laggd a fw yars in following his approach, as i was dominad by a mor aniqu ARIMA approach. Howvr, as arly as 989, Andrw J. Harvy, in his now classical boo Forcasing, Srucural Tim Sris and h Kalman Filr alrady xposs pracically all now mainsram chniqus in daling wih Kalman filr simaion. Coninuous-im Financ, bing a rahr mor rcn fild (w can no vn spa proprly of Coninuous-im Financ unil h svnis, wih h pionr wors of Blac and Schols) had o wai a bi mor. W can sablish h im whn his approach bcam dominan in h influnial wor of Schwarz (997). Howvr, sinc his da, h fild has rally bcom xubran. Kalman Filr dals rouinly, in h blacboards of acadmics and h worsaions of praciionrs wih housands of ral world financial sris and is implicaions sm o b far from xhausd. This hsis ris o b a conribuion, humbl as may b, o his rsarch. GENERAL SETUP Th framwor whr all hs hsis rsuls ar s is a coninuous-im sa spac sysm ha xhibis a dynamics givn by: dx S xp ( b+ AX ) ( cx ) d+ RdW (MR) 7

8 whr S is h spo pric of a givn financial ass commodiy, X is a vcor of n sas which ar usually no obsrvabl, W is uniary Brownian moion and b and A,R and C ar marics of appropria siz, ha in mos applicaions nd o b idnifid. Following Schwarz (997), in h spiri of h Blac-Schols ris nural valuaion, anohr ficiious dynamics is inroducd via a vcor calld ris prmium. W hus obaind a ris nural dynamics, which is usd o valu opions and fuurs conracs: dx S xp ( bλ+ AX ) ( cx ) d+ RdW (MN) I is worh rmaring why modls xhibi hiddn dynamics. In fac, classical coninuous-im financial modls ar dircly obsrvabl. In h blac-schols world, dynamics is jus givn by: dx µ S xp( X ) + X d+ dw And w jus hav o a logarihms o rcovr sa from spo pric. Howvr, as nod by Schwarz (997), his modl implis prfc corrlaion among diffrn fuurs, which is conrary o xising vidnc. As a rsul, h proposd a paricular vrsion of gnral modl (MR)-(MN) whr h spo pric was h sum of wo hiddn componns, on coninuous-im random wal (h classical modl for financial asss) and ransiory shor run componn. A numbr of gnralizaions following modl srucur (MN)-(MR) hav bn proposd sinc. As xampls, h radr can consul Corazar and Naranjo (3) or García, Población and Srna (). Going bac ino h quaions, w shall s ha hy can b solvd xplicily, giving a compl discr im modl o b idnifid dircly from obsrvabl daa. Alhough full dails will b givn in h hsis, l us brifly oulin how his is don. A dirc applicaion of h rsuls in Osndal (99) givs us h soluion of quaion 8

9 (MR) as X + As X + bds+ A As RdW + s which mans w can xacly compu sa dynamics. Dfining b D A As bds, A and A D A As RdW+ s η w hav a fully spcifid quaion X Α bd + AD X + η +. Howvr, w do no usually (and nvr in h modls considrd in his wor) obsrv spo prics bu insad hav daa on fuurs or opions. Rgarding fuurs, which is h daa w shall us o sima modls (opions ar an ino accoun lar for valuaion purposs), in h Blac Schols world hy ar simply h ris nural xpcaion of spo prics or, in symbols, F E [ S / I ] wih mauriy T (i.. wih dlivry im h informaion availabl a., T Q + T whr T F, is h fuur conracd a + T ), Q is h ris nural masur and I is Undr ris nural masur, w hav o us quaions (MN) and hrfor, condiional o, T AT F, is lognormal and As c X ( b ) T + λ ds is is logarihm s man whil c T A ' ' [ ] ds c ( Ts) ' A ( Ts) RR. Th boom lin is ha log F, T d( T) + c( T) X for nown marics d ( T) and c ( T) whras X has a nown discr dynamics so w arriv o a fully spcifid discr modl ha can b simad from ral daa via Kalman filr : X +Α bd log F, T d + A D X ( T) + c( T) Diffrn chaprs of his hsis dscrib diffrn aspcs of his modl, using i o sima paramrs and valu opions in diffrn commodiis. + η X + ε 9

10 SUMMARY OF CHAPTER ONE This chapr dals wih a mahmaically gnral vrsion of (MR) and (MN). As financial daa ar nvr obsrvd in coninuous im (vn ulra high frquncy daa is obsrvd a inrvals of ns of millisconds), in ordr o sima paramrs a discr im vrsion of h modl has o b achivd. In h liraur, h dominan approach was o dvlop discr im formula from ad hoc procdurs, involving limi sps and parial diffrnial quaions. W hav shown ha hs idas ar unncssary and hav dvlopd a gnral mhod o achiv discr im forms which is applicabl o all modls proposd in h liraur. Morovr, w hav also sablish a gnral, dircly programmabl, compur fficin mhod o obain his formula, which w hav conrasd agains horical alrnaivs, rducing compuaion im in an ordr of magniud. In his par, w hav also usd our formula o conras our approach wih Schwarz (997) formula using Ws Txas Inrmdia (WTI) fuurs daa. W show ha his mhod was an approximaion ha nds o (slighly) ovrsima h paramrs and incras rror. SUMMARY OF CHAPTER TWO This chapr ras a modificaion of modl (MN)-(MR) whr ris prmium is allowd o vary ovr im, ha is: dx S xp ( bλ + AX ) ( CX ) d+ RdW (MN ) This problm was vry appaling, as smd vry rasonabl o assum ha h sa of world conomy should hav a dirc implicaion in h prmium an invsor dmands o purchas a risy ass.

11 Esimaing his prmium via a Kolos and Ronn (8) algorihm and a moving window w obaind a im sris, which w compard wih svral conomic indicaors. Rsuls wr vry inrsing as w obsrvd, among ohr findings fully dscribd in h chapr, ha hr was a posiiv rlaion bwn h simad long-rm mar pric of ris and h avrag NAPM indx, h avrag S&P 5 indx and an indicaor of conomic xpansion. This rlaion was rvrsd whn w compard hs conomic indicaors wih shor rm ris prmium. In addiion, w proposd a modl wih im varying ris prmium, and showd how i could b simad via xacly h sam discr Kalman filr, by modifying h way discr im quaions wr obaind. This modl was simad (sparaly) wih ral WTI Oil, Haing Oil, Gasolin and Hnry Hub (HH) Naural Gas ouprforming consan ris prmium modls. Finally, w applid h nw modl was usd o valua a sampl of Amrican WTI opions, obaining br rsuls han mor sandard approachs. SUMMARY OF CHAPTER THREE This final chapr sudis convninc yild dynamics. Convninc yild can b dfind as h valu of owing a commodiy physically insad of having a financial ass ha guarans is possssion in a crain da. Mor formally, rmmbr ha in a Blac-Schols world, fuurs prics ar givn by ris * nural xpcaion of spo prics or F E [ S / I ]. Convninc yild (δ,t ) is h, T + T diffrnc, in coninuous im bwn his pric and h spo pric incrasd du o ral inrs ra, ha is F, T δ T r, T S r, T T. Wha w did in his par was o driv h disribuion of convninc yild from firs principls whn spo prics followd a sochasic sasonal modl. W showd ha his

12 implis, in convninc yild sris, a sasonal componn dircly rlad o h spo pric original. Morovr, his finding was confirmd whn simaing a modl for convninc yild dircly from ral world (WTI Oil, Haing Oil, Gasolin and HH Naural Gas) daa. In addiion, w also showd ha our sasonal modl was maximal in a sns rlad o Dai-Singlon () and gav a canonical, globally idnifiabl form for his modl, which can acually b applid o all consan volailiy modls in h liraur. REFERENCES Corazar, G. and Naranjo, L. (6), An N-Facor gaussian modl of oil fuurs prics, Th Journal of Fuurs Mars, 6, pp Corazar, G. and Schwarz, E.S. (3), Implmning a sochasic modl for oil fuurs prics, Enrgy Economics, 5, pp Dai, Q. and Singlon, K.J.(), Spcificaion analysis of affin rm srucur modls, Journal of Financ 55, pp García A., Población J. and Srna, G., (). Th sochasic sasonal bhavior of naural gas prics. Europan Financial Managmn 8, pp Harvy, A.C. (989), Forcasing Srucural Tim Sris Modls and h Kalman Filr Cambridg Univrsiy Prss, Cambridg, 989. Kalman, R.E. (96). A nw approach o linar filring and prdicion problms. Journal of Basic Enginring 8 () pp Kolos S.P and Ronn E.U. (8), Esimaing h commodiy mar pric of ris for nrgy prics. Enrgy Economics 3, Schwarz, E.S. (997), Th sochasic bhavior of commodiy prics: Implicaion for

13 valuaion and hdging, Th Journal of Financ, 5, pp Sraonovich, R.L. (959). Opimum nonlinar sysms which bring abou a sparaion of a signal wih consan paramrs from nois. Radiofizia, :6, pp

14 CHAPTER : ANALYTIC FORMULAE FOR COMMODITY CONTINGENT VALUATION.. INTRODUCTION Iô calculus has bcom h main approach in drivaivs valuaion hory sinc i was firs usd in Financ (Blac and Schols, 97). Th sam mhodology was firs usd in h valuaion of commodiy coningn claims (s for xampl Brnnan and Schwarz, 985, Paddoc al., 988, among ohrs), i.. by assuming ha ass prics follow a gomric Brownian moion, h classical Blac-Schols formula can b usd wih sligh modificaions (if any). Subsqunly svral auhors, such as Laughon and Jacobi (993) Ross (997) or Schwarz (997), hav considrd ha a man-rvring procss is mor appropria o modl h sochasic bhaviour of commodiy prics, poining ou ha h gomric Brownian moion hypohsis implis a consan ra of growh in h commodiy pric and a varianc of fuurs prics incrasing monoonically wih im, which ar no ralisic assumpions. Th ida bhind manrvring procsss is ha h supply of h commodiy, by incrasing or dcrasing, will forc is pric owards an quilibrium (or long-rm man) pric lvl. In spi of hir aracivnss, hs on-facor man-rvring modls ar no vry ralisic sinc hy gnra a consan volailiy rm srucur of fuurs rurns, insad of a dcrasing rm srucur, as obsrvd in pracic. Gibson and Schwarz (99) and Schwarz (997) propos a wo-facor modl, whr h scond facor is h convninc yild, which is also assumd o follow a man-rvring procss. Schwarz and Smih () propos a wo-facor modl allowing for man rvrsion in shor-rm prics and uncrainy in h quilibrium (long-rm) pric o which prics rvr, which S Schwarz (997) and Schwarz and Smih () for an xclln discussion of hs issus. 4

15 is quivaln o h Schwarz (997) on. Schwarz (997) also considrs a hr-facor modl, xnding h Gibson-Schwarz (99) modl o includ sochasic inrs ras. Corazar and Schwarz (3) propos a hr-facor modl, which is an xnsion of h Schwarz (997) wo-facor modl, whr all hr facors ar calibrad using only commodiy prics. Mor rcnly Corazar and Naranjo (6) xnd wo and hr facor modls o an arbirary numbr of facors (N-facor modl). Unforunaly, h applicaion of h sandard Blac-Schols valuaion framwor is no asy in h conx of commodiy coningn valuaion, givn h complx dynamics of commodiy prics. This is h rason why h sudis on commodiy coningn valuaion usually prsn vry complx ad-hoc soluions and somims includ approximaions or limi sps. In his aricl w show how o simplify formula and dducions, compuing h xplici, dircly implmnabl gnral formula, basd on wll nown rsuls in sochasic calculus. Spcifically, afr dscribing h gnral horical modl for commodiy coningn valuaion, w prsn wo spcific applicaions. Firsly, w show how his gnral framwor can b implmnd in h conx of h wo-facor modl by Schwarz (997), obaining simplr xprssions and mor prcis simas han h approximaions givn by h auhor. I is also shown ha h approximaions by Schwarz nd o ovrsima h paramrs, a fac ha, as w will s, bcoms imporan in h valuaion of commodiy coningn claims. Scondly, w shall show how o obain h xprssion for h fuurs pric and volailiy of fuurs rurns givn by Schwarz (997) and Schwarz and Smih () in a simplr way, avoiding unncssary parial diffrnial quaions or limi sps. This chapr is organizd as follows. Th gnral mhodology for commodiy coningn valuaion and volailiy simaion is prsnd in Scion. Scion 3 5

16 dscribs how hs formula can b usd in pracic and proposs a rady-oimplmn algorihm o sima any linar modl which is valuad in rms of compur im. Scion 4 shows how o obain mor prcis simaors of h paramrs in h wo-facor modl by Schwarz (997). Scion 5 shows how o simplify h dducion of h fuurs pric in h wo-facor modl by Schwarz and Smih (), avoiding unncssary limi sps. Finally, scion 6 concluds wih a summary and discussion... THEORETICAL MODEL Conrac Valuaion Mos of h modls proposd in h liraur for h sochasic bhaviour of commodiy prics can b summarizd by mans of h following sysm: dx Y ( b+ AX ) cx d+ RdW () whr Y is h commodiy pric (or is log), b, A, R and c ar drminisic marics n n x n n indpndn of ( b R, A, R R, c R ) and W is a n-dimnsional canonical Brownian moion (i.. all componns uncorrlad and is varianc qual o uniy). Usually, h simaion of hs marics can b simplifid, as hy can b assumd o dpnd in a prdfind way of som simabl valus, calld srucural paramrs or hyprparamrs (for xampl, if A is x, insad of compuing four valus on may assum, as in Schwarz, 997, ha A whr κ is h hyprparamr o b κ simad). R dos no hav o b compud, as all formula shall us RR. 6

17 As i shall b provn in appndix B h soluion of his problm is: X A A s A s X + + bds RdW s () W shall assum now ha A is diagonalizabl wih A PDP and D D diagonal 3. L us dfin h auxiliary quaniis: [ xp( D) I] ( ) I J P D P (3) ( ) xp( A) Pvc xp( Ds) xp( Ds) ds vc( P RR' P' ) ( P )'xp( A)' G (4) This ingral can b compud xplicily, bu dpnds on h ignvalus (s appndix A). Using () and h rsuls in Appndix A abou ingrals, i is vidn ha, givn X, is Gaussian, wih man and varianc: E A [ X ] X J( )b, [ X ] G( ) + X Var. (5) Which yilds ha Y is also Gaussian wih E[ Y ] ce[ X ], Var[ Y ] cvar[ X ] c ' Undr h ris-nural masur, h dynamics ar xacly h sam as in () bu changing b ino a diffrn sam) so, using his masur and condiional o X, * b which conains h ris prmia (all ohr marics say h X is Gaussian. To compu h ris-nural man and varianc of X and Y w mus subsiu b for * b in (5), hus providing a valuaion schm for all sors of commodiy coningn claims such as financial drivaivs on commodiy prics, ral opions, invsmn dcisions, c. 3 To h bs of h auhors nowldg all modls in h xising liraur fulfil his rsricion, mos of hm dircly by imposing A o b diagonal. Noabl xcpions whr A is no diagonal bu diagonalizabl ar h Schwarz (997) modl or h cycls in Harvy (99). 7

18 If Y is h log of h commodiy pric (S ), i is asy o prov (jus by h propris of h log-normal disribuion) ha h pric of a fuurs conrac radd a im wih mauriy a im +T is: F AT * (, T) xp c X + cj( T) b + cg( T ) c' (6) This mhodology is gnral, fasibl for all ind of problms, a las whn h paramrs in () ar indpndn of, and much simplr han h ad-hoc soluions prsnd in h liraur, ha can only b usd in h concr problm for which hy wr dvlopd and nd complx procdurs such as parial diffrnial quaions (Schwarz 997) or limi sps (Schwarz-Smih ). Evn mor, hs formula can b implmnd dircly in any mahmaical orind compur languag, such as Malab or C++ rgardlss on h siz of h marics or hir dpndnc of h hyprparamrs, using h marics dircly as inpus. So hr is no nd o compu xplici formula ach im w us a diffrn modl. I possibl o us h sam scrip (changing h way h marix dpnd on h hyprparamrs) for any modl. Volailiy of Fuur Rurns W can dfin h squard volailiy of a fuurs conrac radd a im wih mauriy a im +T as 4 : Var lim h [ log F log F ] + h, T h, T. In appndix C i is provd ha i is h xpcd valu of h squar of h cofficin of h Brownian moion ( ) in d log F h xpansion ( ) F, T µ ds+ dw, whr W F is a scalar canonical Brownian 4 Th sam rsuls would b obaind if h volailiy wr dfind as: Var lim h [ log F log F ] + h, Th h, T. 8

19 moion, as long as µ is man squard boundd in an inrval conaining (i dos no mar whhr i is a funcion of F, or no) and [ ] T E is coninuous in. In h gnral problm of his aricl hs condiions ar saisfid. Thrfor, afr aing logarihms and diffrnials on boh sids of Equaion (6), w can obain ha: AT AT AT ( log F, T) c dx c [ b+ AX ] d c R dw d + So, h squard volailiy is simply 5 : c AT RR' AT ' c'. (7).3. DISCRETIZATION AND ESTIMATION ISSUES This scion is dvod o provid a praciionr s guid o h us of h abov rsuls. Suppos ha w obsrv a forward curv ( T) F, of N fuurs prics and wish o sima a linar mulifacor modl as in (). Firs of all, w nd a discr vrsion of (). L b h inrval of discrizaion. A As sad abov E[ X ] X J( )b and [ X ] G( ) prov ha: + Var. Consqunly, i is asy o X y +Α b D + A d+ c d D X X + η + ε (8) whr [ log( F(, T )),,log( F( T ))] ' y is h log of h full forward curv, ( A ) A D, xp, J( )b b D N, Eη [ ], Var( ) G( ) η, * di cj( Ti) b + cg( Ti) c' i N and c D ( AT ) c xp c xp. ( AT ) N 5 No again ha R dos no nd o b compud as ' RR is h nois covarianc marix. 9

20 Of cours, h masurmn nois ( ε ) is usr-dfind. Th mos usual convnion, followd by Schwarz (997), Schwarz and Smih (), Corazar and Naranjo (6) among ohrs, is E[ ε ], Var[ ε ] N Th procss o sima a modl is as follows:. Givn a s of hyprparamrs φ, ma xplici h dpndnc of h coninuous im sysm marics ( φ) c( φ) A, and so on in (). Compu h discr-im sysm (8). This can b don using h formula (3) and (4) or dircly via h ingrals in appndix B. Th asis way is obviously o compu hm by hand and insr hm in h program. Howvr, h compur can do i, using h formula (3) and (4) ach iraion a a modra addiional compuaional cos (hus allowing h usr o wri a singl program for all modls, insad of changing i ach im). 3. Esima h paramrs in h modls by a log-lilihood algorihm. S Hamilon (994) for dails on simaing a sa-spac modl. From h auhors poin of viw, unlss h usr always dals wih h sam ind of modl, h incrasing complxiy of using formula (3) and (4) in ach iraion is a pric worh paying by having a singl gnral program. W would li o srss h imporanc of formula (3) and (4). Wihou hm, unlss h praciionr wris a spara scrip for ach modl, h would hav o compu (via a symbolic procssor such as Malab Symbolic Toolbox) an ingral in ach iraion. Th compuaional cos of ha is burdnsom, approximaly ims h on wih h formula, which is wo ordrs of magniud highr.

21 To proof his, w hav simad h Schwarz and Smih () and Corazar and Schwarz (3) modls wih diffrn daa ss, rprsnaiv of h ind of sris a praciionr is lily o wor wih. Hr, i suffics o say ha hy ar a wo facor (Schwarz and Smih, ) and a hr facor (Corazar and Schwarz, 3) modl wih 8 and 3 idnifiabl hyprparamrs rspcivly. Th daa s mployd consiss on wly obsrvaions of Hnry Hub naural gas, WTI crud oil fuurs prics (boh of hm radd a NYMEX) and Brn crud oil fuurs prics (radd a ICE). Th daa s for Hnry Hub naural gas is mad of conracs F, F5, F9, F3, F7, F, F5, F9, F33, F37, F4, F44 and F48 whr F is h conrac closs o mauriy, F is h scond conrac closs o mauriy and so on. This daa s conains 33 quoaions of ach conrac from /3/ o 3/4/8. Th daa s for WTI crud oil is mad of conracs F, F4, F7, F, F3, F6, F9, F, F5 and F8. This daa s conains 654 quoaions of ach conrac from 9/8/995 o 3/4/8. Th daa s for Brn crud oil is mad of conracs F, F4, F7, F, F, F6-8, F-4 and F3-36. This daa s conains 537 quoaions of ach conrac from /5/997 o 3/4/8. Ths daa ss hav bn chosn aing ino accoun ha fuurs conracs wih long-rm and shor-rm mauriis ar ncssary o sima proprly h paramrs of h long-rm and h shor-rm facors. In Tabl a brif summary of h im ndd for an valuaion of h log-lilihood funcion is givn, spcifying h daa and modl usd (wo facors mans Schwarz and Smih,, modl, hr facors mans Corazar and Schwarz, 3). No ha, as all quaniis ar givn in millisconds, a 3% lss for h formula (implmning ach cas sparaly) is no a big rward. All xprimns wr mad wih an x86 Inl Clron (Family 6 Modl 8 Spping 3, 6.66 Kb RAM).

22 In ordr o illusra his fac, w hav also includd anohr Tabl (numbr ) whr h simaion im is givn for h gnral cas and h simaion for ach cas sparaly (using h horical formula for ingrals would b oo slow). As h radr can s, h diffrnc is small nough and, from h auhors poin of viw, i is no worh h ffor o compu formula by hand cas by cas insad of using marix forms. No ha h diffrnc is simaing a modl in a minu and a minu and a half, vn wih a rahr old compur..4. PRECISE ESTIMATION OF THE SCHWARTZ (997) TWO- FACTOR MODEL L us considr h wo-facor modl in Schwarz (997). L S and δ b h spo pric of a commodiy and is insananous convninc yild a im. Th modl can b xprssd as: ds dδ ( µ δ ) Sd+ S κ( α δ ) d+ dz dz Th sandard Brownian moions, dz and dz, ar assumd o b corrlad, i.. dz dz ρd. Th paramr µ is h long-rm oal rurn on h commodiy, κ is h manrvring cofficin, α is h long-rm convninc yild, and finally and ar h volailiis of h spo pric and h convninc yild rspcivly. Dfining Y ln(s ) and applying Iô s Lmma, h modl, undr h ris-nural masur, can b xprssd as: dy dδ ( rδ / ) d+ dz * [ κ( α δ ) λ] d+ dz *

23 Whr * d z and * d z ar h Brownian moions undr h quivaln maringal masur, which ar assumd o b corrlad, i.. * * d z dz ρ d, λ is h mar pric of ris associad o h convninc yild and r is h ris-fr inrs ra. Y, If w dfin h sa vcor as ( ) ' i is asy o prov ha following xprssions 6,7 : X δ and afr applying h rsuls in scion, X is normally disribud wih a man and varianc givn by h E * ( r / α) + α( [ X ] + X α( ) ) λ / ( ) / * Var [ X ] + ρ( )/ (34 + )/ ρ( )/ + ( + )/ 3 ρ( )/ + ( ( )/ + )/ Thrfor, Y ln( S ) is also Gaussian, undr h ris-nural masur, wih man: Y δ ( α α * * ) / + ( r Y / ) + ( ) / * whr α α λ / κ, and varianc: κ ( ) ( ρ κ ρ κ. κ 3 + / / ) + ( ) / + ( / ) / Finally, givn ha h spo pric S is lognormal, h fuurs pric can b xprssd as: xp{ Y + ( F, T δ ( κ E ) * * * [ S ] xp E [ Y ] + Var [ Y ] T * ) / + ( rα + / 3 * / 4κ + ( α + ρ T / ) T ρ / ) T κ ( )/ κ } 6 E*[] and Var*[] ar h man and varianc undr h ris nural masur. 7 Hr, in his scion, w shall us h formulas in ingral form, wihou rsoring o (3) and (4). 3

24 4 This is h rsul alrady obaind in Schwarz (997), quaion, bu avoiding unncssary parial diffrnial quaions. Using h rsuls in scion, h squard volailiy of fuurs rurns can b xprssd as: ( ) ( ) ( ) / / T T T T κ κ κ κ κ ρ ρ κ ( ) ( ) κ ρ κ κ κ / / T T + which is h sam formula as in Schwarz (997), quaion 4. Now l us xprss h modl in is discr-im vrsion. Following Schwarz s noaion h modl can b xprssd as 8 : X M c X ψ + + whr: + ) ( / ) ( ) / ( c α λ α α µ, ( ) M κ κ κ / (9) and h rror rm vcor, dnod as ψ, is a n-vcor of srially uncorrlad Gaussian disurbancs wih zro man and varianc givn by h following xprssion: [ ] Var ) ( ) ( ) ( ) ( ) ( ) 4 (3 ) ( 3 ρ ρ ρ ψ () 8 No ha hs xprssions ar jus h discr-im counrpar of xprssions (8) wih D M A and c d in our noaion.

25 If w prform a Taylor xpansion whn nds o zro and drop all rms of ordr highr han on, w g xprssions 35 in Schwarz (997): c ( µ / ), M α κ ρ and Var[ ψ ] ρ Thrfor, w can conclud ha Schwarz (997) uss a discr-im vrsion of h modl which is an approximaion o h prcis on prsnd abov, which is givn by xprssions (9) and (). As w will s blow, hs divrgncs, spcially h mor accura simaor of h varianc of h rsidual, Varψ [ ], givn by xprssion (), ar imporan in h valuaion of commodiy coningn claims. Nx w ar going o compar h mpirical prformanc of boh simaion procdurs, i.. h prcis vrsion of h simas givn in his chapr and h approxima vrsion in Schwarz (997), using h sam daa s as in Schwarz (997). Spcifically, h daa s is composd of wly obsrvaions of NYMEX WTI crud oil fuurs conracs, wih mauriy, 3, 5, 7, and 9 monhs, from //985 o /3/995. W hav a oal of 59 obsrvaions 9. WTI fuurs prics wih on monh o mauriy ar dpicd in Figur. Th rsuls of h simaion of h wo facor modl by Schwarz obaind wih boh simaion procdurs ar conaind in Tabl 3. Th main diffrncs bwn h rsuls obaind wih boh procdurs ar found in h valus of κ (h man-rvring paramr), (h volailiy of h convninc yild) and λ (h mar pric of ris associad o h convninc yild). Spcifically, h valu of κ found wih h prcis vrsion,.5433, is considrabl lowr han h valu found wih h Schwarz approximaion, Morovr, h valu of λ found wih h prcis vrsion is also 9 This is on of h daa ss usd in Schwarz (997). Howvr in ha papr h daa s includs 5 obsrvaions, insad of 59. Tha is h rason why h rsuls prsnd hr for Schwarz approximaion ar no xacly h sam as h ons prsnd in Schwarz (997). 5

26 lowr han h valu found wih h Schwarz approximaion (.8 and.558 rspcivly). Finally, h valu of obaind wih h prcis and approxima vrsions is.3967 and.46 rspcivly. In gnral looing a h Tabl w can apprcia ha all h valus found wih h approxima vrsion usd by Schwarz (997) ar highr han h corrsponding valus found wih h prcis vrsion. Thrfor, w can conclud ha, a las wih his daa s, h approxima vrsion by Schwarz (997) nds o ovrsima h paramrs. Figurs and 3 prsn h diffrncs bwn on monh WTI fuurs prics and h spo pric calculad wih boh h prcis and h approximad simas. Spcifically, Figur compars h prdiciv abiliy of boh simas in rms of h man rror (ME), dfind as h avrag of h sris of on monh fuurs pric minus simad spo prics, whras in Figur 3 i is usd h roo man squard rror (RMSE). In h full sampl priod, , h prcis simas ouprform h approximaion by Schwarz (997), using h wo mrics. This is also h cas in all h annual priods considrd in h Figurs. Howvr, i is inrsing o no ha h bs prformanc of h prcis simas is found in 985 and 99, yars which ar characrizd by high volailiy, as can b apprciad in Figur. This fac is no surprising sinc, as poind ou abov, on of h main advanag of h prcis mhodology is ha i provids a mor accura simaion of h varianc of h rsidual, Varψ [ ], which is givn by xprssion (). Finally, i is worh noing ha h man rror is ngaiv in h whol sampl priod, implying ha boh simas nd o To h bs of our nowldg, hr is no rliabl indx which rflcs h WTI crud oil spo pric. Thrfor, h bs availabl approximaion for i, NYMEX WTI crud oil fuurs conracs wih on monh o mauriy, is usd. 6

27 ovrsima spo prics. I is also h cas in all h annual priods, xcp for 986, 993 and 994. Figurs 4 and 5 show h diffrncs bwn on monh WTI fuurs and spo prics calculad wih boh h prcis and h approximad simas, by monh. Th rsuls ar similar o hos obaind in Figurs and 3, i.. h prcis simas ouprform h approximaion by Schwarz (997), using h wo mrics (man rror and roo man squard rror), in all monhs, xcp for March wih h man rror masur. Finally, Tabl 4 compars h improvmn (xprssd in prcnag) in h RMSE and h sandard dviaion of on-monh fuurs pric, by monh. Inrsingly, h highs improvmn in h RMSE is obaind in Ocobr and Novmbr, which ar ha h monhs characrizd by h highs dgr of varianc. As poind ou abov, his rsul can b rlad wih h fac ha on of h main advanags of h prcis simaion procdur is ha i provids a mor accura simaion of h varianc of h rsidual, Varψ [ ], which is givn by xprssion (). I should b nod, howvr, ha hr ar also monhs wih no such high varianc showing a high improvmn in h RMSE (January and Dcmbr)..5. SIMPLIFIED DEDUCTION OF THE FUTURES PRICES IN THE TWO-FACTOR MODEL BY SCHWARTZ AND SMITH () L us considr h wo-facor modl in Schwarz and Smih (). Thy assum ha h spo log-pric of a commodiy a im, ln(s ), can b dcomposd as h sum of a shor-rm dviaion,, and h quilibrium pric lvl, ξ : ln( S ) + ξ. Dfind as h RMSE compud wih h Schwarz approximaion minus h RMSE compud wih h prcis vrsion of h simas. 7

28 Th shor-rm dviaion and h quilibrium lvl ar assumd o follow a manrvring procss (oward zro) and a sandard Brownian moion rspcivly, i..: d κd+ d z dξ µ ξ d+ ξ dzξ Whr dz and dzξ ar sandard Brownian moions wih corrlaion ρ, i.. d z dzξ ρ d, κ rprsns h ra a which h shor-rm dviaions rvr oward zro (h man-rvring cofficin), µ ξ is h quilibrium oal rurn and and ξ ar h volailiis of h shor-rm dviaion and h quilibrium lvl rspcivly. Th ris-nural vrsion of hir modl is givn by h following SDE: d ( κ λ ) d+ d z * * dξ µ ξ d+ ξ dzξ * Whr * dz and * dz ξ ar again sandard Brownian moions wih corrlaion ρ, i.. * d z dzξ ρ d, µ ξ µ ξ λξ, and λ and λ ξ ar h mar prics of ris associad o h shor-rm dviaion and h quilibrium lvl rspcivly., Dfining h sa vcor as ( ) ' X ξ, h modl can b xprssd as : λ κ dx * + X d+ RdW µ ξ whr R is h Cholsi dcomposiion of h nois covarianc marix 3 : ρ ξ ρ ξ ξ S Appndix B. 3 ' No again ha R dos no nd o b calculad as RR is h nois covarianc marix. 8

29 9 Now, w will us xprssions (3) and (4). No ha, as A is diagonal, I P so w can safly drop P and P from all xprssions. I is asy o s ha (no ha, in ordr o comply wih Schwarz and Smih s noaion, D D, h null par is in h boom of h marix): ( ) J κ κ ( ) xp A κ ( ) vc G κ ξ ξ ξ κ κ κ κ ρ ρ κ κ κ κ ξ ξ κ ξ κ κ κ ρ κ ρ κ κ ξ ξ κ ξ κ κ ρ κ ρ κ κ Now, h man and varianc of X ar: [ ] * * / ) ( X X E + κ ξ µ λ [ ] ( ) ( ) ( ) ( ) G X Var * / / / ξ ξ κ ξ κ κ κ ρ κ ρ κ

30 In his modl, h log of spo pric, Y ln(s ), is givn by + ξ. Thus, ln(s ) is a Gaussian variabl wih man: κ * + ξ + µ ( ) λ / ξ and varianc: κ κ ( ) / κ + ( ) ρ / κ + ξ ξ. Finally, h spo pric, S, is lognormal disribud, and, hrfor, h fuurs pric can b wrin as: F, T E * * * [ S ] xp E [ Y ] + Var [ Y ] T κ κ ( ) / κ + ( ) ρ / κ + κ * ξ xp + ξ + µ ξ ( ) λ / + T T ξ W hav obaind h sam rsul as in Schwarz and Smih (), Equaion 9, bu in a simplr way, avoiding unncssary limi sps..6. CONCLUSIONS Th sochasic bhaviour of commodiy prics has bn a common opic of rsarch during h las yars. Howvr, h applicaion of h sandard Blac-Schols analysis is no sraighforward, du o h complx dynamics of commodiy prics. This is h rason why mos of hs sudis prsn ad-hoc soluions, which ar vry complx and somims includ approximaions. This aricl shows how o simplify formula and dducions, and vn compu an xplici marix gnral formula, using wll nown chniqus and rsuls in sochasic 3

31 calculus. This formula has bn sd on ral daa and is a ral alrnaiv o programming ach modl sparaly. Concrly, w show how o obain mor prcis simaors of h paramrs in h Schwarz (997) wo-facor modl conx, han h approximaions givn by h auhor. I is found ha, in gnral, h approximaions by Schwarz nd o ovrsima h paramrs. Ths divrgncs ar imporan in h valuaion of commodiy coningn claims. Morovr, w hav shown how o obain h xprssion for h fuurs pric givn by Schwarz and Smih () in a simplr way, avoiding unncssary limi sps. 3

32 APPENDIX A: MATHEMATICAL REFERENCE RESULTS In ordr o undrsand h rsuls, i is ncssary o inroduc som mahmaical prliminaris. All h concps and formula hr shall b prsnd in an inuiiv way, srssing h pracical implmnaion. Firs of all, w rmind h radr som wll nown concps. For an xnsiv rviw of marix algbra and marix drivaivs, w rcommnd Magnus and Nudcr (999). Th drivaiv and ingral of a im-dpndn marix (which w shall dno A ( ) or A indisincly) ar givn lmn by lmn: d d ( ) A d d d d a a m d ( ) a n( ) d ( ) ( ) s, A( ) r d a d mn d r r s s a a m s ( ) d a ( ) d. s ( ) d a ( ) d r r n mn Indfini ingrals A d ar dfind in h sam way. Linar propris, such d d as ( BA) B A mnioning hm. d, ar asy o prov and shall b usd wihou xplicily d Th marix xponnial of a diagonalizabl marix A PDP wih D diagonal is: xp d d ( A) ( d ) xp P P xp( ) d n ( A) Axp( A) xp. I is no hard o s h qualiy Givn wo marics A pxq mxn R, B R hir Kroncr produc is a pm x qn marix dfind as: ab ab A B a pb a a a p B B B a q B aq B. a B pq 3

33 Th vc opraor is dfind as: a a a a q a p vc. a a p a pq a p Ingrals wih a singl produc: W shall calcula ( A) s r xp H d whr H is an arbirary consan marix. L PDP P P A wih D diagonal and D D non-singular. Th prvious ingral is hrfor asily compud xplicily as: s s ( A) Hd P xp( D) xp d P H P P H r P ( s r) I r ( xp( D s) ( D r) ) D xp P I H xp ( D ) ' Ingrals wih doubl produc: W shall calcula U xp ( A) H xp( A) s r s r V d, whr U, H, V ar arbirary consan marics. As bfor: A PDP P P D, r s U xp ' s ' ' ( A) H xp( A) V d UP xp( D) P H( P )'xp( D) d P V r so w shall focus on h middl par. Using h vc opraor: s ' s ( ) ( ) ( )( ( )) ( ) ' xp D H xp D d vc vc xp D P H P ' xp D d r vc vc r s ' ( xp( D) xp( D) ) vc P H( P ) r s xp r ( ) d ' vc( P H P ) ( D) xp( D) d ( ) Th only hing lf is o compu h cnral ingral. Howvr, if D is diagonal, l 33

34 D d d n. Thn xp( D) d. Th Kroncr dn produc is hus givn by: xp ( D) xp( D) ( d I+ D ). If no ( ) dn+ D ignvalu is xacly h opposi of anohr ignvalu h ingral is givn by s r xp ( D) xp( D) ( r s) I ( di+ D) ( d I + D ) ( ) ( + ) dn+ D d I D If wo ingnvalus ar on h opposi of h ohr, mars ar no much mor difficul. L µ D µ including all zro and nonzro ignvalus. If µ w jus l γ ij µ i + µ j and subsiu in h formula, w hav xp ( D) xp( D) γ xp γ γ γ and is ingral is: 34

35 r s xp ( D) xp( D) s r γ d r s γ d s r γ d s r. Whr γ d obviously r s γ ij d s r γ ijs γ ii γ ijr for for γ γ ij ij. ( ) s No ha h xprssion ( ) ( ) ' vc xp D xp D d vc P H( P ) r can b don in a diffrn way, using h Hadamard produc insad of h Kroncr on and hus avoiding h us of diagonal marics. To do so, rmmbr ha h Hadamard produc of A and B dnod A B is dfind ach lmn a a im: s ( A B) ij AijBij. If w jus dfin Z vc xp( D) xp( D) d r or quivalnly Z ij s r γ ij d, hn i is asy o noic, jus by subsiuion, ha vc r s xp ( ) ( ) ( ) ' D xp D d vc P H( P ) quals ZP H ( P ) '. Th radr should no, howvr, ha du o h fac ha our Kroncr produc is diagonal, i dos no hav o b sord in full, so an fficin implmnaion of h algorihm will us only h diagonal All opraions ar asily implmnd in any mahmaically adapd compur languag such as Malab. 35

36 APPENDIX B: FUTURES CONTRACT VALUATION Mos of h modls proposd in h liraur assum ha h ris-nural dynamics of a commodiy pric (or is log) is givn by a linar sochasic diffrnial sysm: dx Y ( b+ AX ) cx d+ RdW whr Y is h commodiy pric (or is log), b, A, R and c ar drminisic paramrs 4 n n x n n indpndn of ( b R, A, R R, c R ) and W is a n-dimnsional canonical Brownian moion (i.. all componns uncorrlad and is varianc qual o uniy) undr h ris-nural masur. L us s ha h soluion of ha problm is 5 : X A A s A s X + + bds RdW s (B) In ordr o proof i, w shall apply h gnral rul for h drivaion of h produc of sochasic componns (Osndal, 99): dx A ( d ) X As As A bds RdWs + d X + ( ) + + A As As d d X bds RdWs As bds + As RdW s + I is asy o show ha: As X bds+ d + As RdW s A bd+ A RdW Th firs diffrnial only has lmns of yp d, hnc h produc of h firs diffrnial ims h scond diffrnial is zro. Thus: 4 Again no ha R dos no nd o b compud. 5 Evn in h cas ha b, A and R wr funcion of, if A and A ds commu, h soluion of ha problm is (B). s 36

37 + Consqunly w obain xprssion (B): A A [ bd+ RdW] A X d+ bd RdW A As As A dx A d X + bds RdWs + + X A A s X + + bds A s RdW. s I is asy o prov ha h soluion is uniqu (Osndal, 99). An lmnary rul of h sochasic calculus sas ha if J s is a drminisic funcion, J s dws is normally disribud wih man zro and varianc: Var T J sdws J s J s ds (Iô s isomry). Accordingly, X is normally disribud wih man and varianc 6 : E * Var [ ] + A A s X X bds [ X ] * A A s ' A s ' A ' RR ds (B) (B3) Thrfor, Y, undr h ris-nural masur, is also Gaussian and i asily follows ha * * * * is man and varianc ar: E [ Y ] ce [ X ], Var [ Y ] cvar [ X ] c ', providing a valuaion schm for all sors of commodiy coningn claims as financial drivaivs on commodiy prics, ral opions, invsmn dcisions and ohr mor. If Y is h log of h commodiy pric (S ), h pric of a fuurs conrac radd a im wih mauriy a im +T, F,T, can b compud as: F, T E * * [ S I ] E [ Y I ] + Var [ Y I ] xp + T T (B4) * + T + whr I is h informaion availabl a im. 6 E*[] and Var*[] ar h man and varianc undr h ris nural masur. 37

38 This mhodology can b usd in all ind of problms (vn if b, A and R ar funcions of, alhough, in his cas h xplici formula for h ingrals, givn in appndix A, do no apply). Morovr, his mhodology is much simplr han h ad-hoc soluions prsnd in h liraur ha can only b usd in h concr problm for which hy wr dvlopd and nd complx procdurs li limi sps (Schwarz and Smih, ) or parial diffrnial quaions (Schwarz, 997). 38

39 APPENDIX C: VOLATILITY OF FUTURES RETURNS Th squard volailiy of a fuurs conrac radd a im wih mauriy a im +T is dfind as 7 : Var lim h [ log F log F ] + h, T h, T. W will prov ha i is h xpcd valu of h squar of h cofficin of h d log F Brownian moion ( ) in h xprssion ( ) F, T µ ds+ dw, whr W F is a scalar canonical Brownian moion, as long as µ is man squard boundd in an inrval conaining (i dos no mar whhr i is a funcion of F, T or no) and [ ] E is coninuous in 8. Exprssing d log F µ d+ dw in h quivaln ingral form:, T is xpcd valu is E [ µ ] Var + h + h + h, T log F, T µ sds+ log F dw, +h s ds. Thrfor, is varianc is givn by: + h + h [ log F ] [ ] + + h, T log F, T E s E µ s ds sdws Using sandard propris: h E µ + s E as µ is non-anicipaing. s µ. h + h + h [ µ ] [ ] + s ds sdws E µ s E µ s ds E sdws + + s By Iô s isomry: E [ ] ds + h + h sdws E s 7 Th sam rsuls ar going o b obaind if h volailiy is dfind as: Var[ log F+ h, Th log F, T] lim. h h 8 In h gnral problm of his aricl hs condiions ar saisfid. 39

40 Taing limis and using h man valu horm of h ingral calculus: lim h h h + E [ ] ds E[ ] s. For h ohr rm i can b sn ha: + h h E µ s E µ + h [ ] [ ] + µ s ds µ s E µ s ds µ s E[ s] ds As for som δ >, µ is man squard boundd in h inrval (-δ, +δ), whn h, { } his ingral is lss or qual han h µ E[ µ ] : s ( δ + δ) { µ E[ µ ] : s ( δ, + δ) } M s s sup sup for som M. Hnc, which convrgs o whn h. h E s E s h + µ µ s s, h [ ] ds h M, and Thrfor: Var lim h [ log F log F ] + h, T h, T E [ ]. Hnc, aing logarihms and diffrnials on boh sids of Equaion (B4), i follows ha: AT AT AT ( log F T) c dx c [ b+ AX ] d c R dw d +, Thrfor, h squard volailiy is 9 : c AT RR' AT ' c'. 9 Again no ha R nds no o b compud as ' RR is h nois covarianc marix. 4

41 REFERENCES Blac F, Schols M S. 97. Th valuaion of opion conracs and a s of mar fficincy. Th Journal of Financ 7 (); Brnnan, M.J. and E. Schwarz. Evaluaing naural rsourc invsmns Journal of Businss 58; Corazar G, Schwarz E S. 3. Implmning a sochasic modl for oil fuurs prics. Enrgy Economics 5; Corazar G, Naranjo L. 6. An N-Facor gaussian modl of oil fuurs prics. Journal of Fuurs Mars 6 (3); Gibson, R. and E. Schwarz. 99. Sochasic convninc yild and h pricing of oil coningn claims. Th Journal of Financ 45; Hamilon, J.D. (994) Tim Sris Analysis. Princon Univrsiy Prss. Harvy, A.C. (99). Forcasing, Srucural Tim sris modls and h Kalman Filr. Cambridg Univrsiy Prss. Laughon, D.G. and H.D. Jacoby Rvrsion, iming opions, and long-rm dcision maing. Financial Managmn 33; 5-4. Magnus, J.R. and Nudcr (999) Marix Diffrnial Calculus wih Applicaions in Saisics and Economrics. JohnWily and Sons Chichsr/Nw Yor Osndal B. 99. Sochasic Diffrnial Equaions. An Inroducion wih Applicaions, 3rd d. Springr-Vrlag: Brlin Hidlbrg. Paddoc, J.L, D.R. Sigl and J.L. Smih Opion valuaion of claims on ral asss: Th cas of offshor prolum lass. Quarrly Journal of Economics 3:

42 Ross, S Hdging long run commimns: Exrciss in incompl mar pricing. Banca Mon Economics Nos. 6; Schwarz E S Th sochasic bhaviour of commodiy prics: Implicaion for valuaion and hdging. Th Journal of Financ 5; Schwarz E S, Smih J E.. Shor-rm variaions and long-rm dynamics in commodiy prics. Managmn Scinc 46;

43 TABLES AND FIGURES TABLE TIME (MILISECONDS) NEEDED FOR AN EVALUATION OF THE LOG- LIKELIHOOD FUNCTION Ingral sands for using a symbolic procssor o compu h ingral ach sp. Gnral mans using h sam scrip (formula (3) and (4) in marix form) for all modls and Paricular mans wriing down h formula for ach cas. Daa Brn Haing oil WTI Facors Ingral Gnral Paricular TABLE TIME (SECONDS) FOR A FULL ESTIMATION OF A MODEL Gnral mans using h sam scrip (formula (3) and (4) in marix form) for all modls and Paricular mans wriing down h formula for ach cas. Ingraing symbolically ach sp would b compuaionally burdnsom. Daa Brn Haing oil WTI Facors Gnral Paricular

44 FIGURE WTI FUTURES PRICE WITH ONE MONTH TO MATURITY //85 /5/86 3//87 4/5/88 5//89 6/4/9 7/9/9 9//9 /6/93 //94 /5/95 TABLE 3 THE TWO-FACTOR MODEL BY SCHWARTZ (997). PRECISE AND APPROXIMATE ESTIMATES Th Tabl shows h paramr simas obaind wih h Schwarz (997) approximaion and wih h prcis mhod dscribd in his chapr. Sandard rrors in parnhsis. Paramr µ α Prcis Mhod.69 (.75).5433 (.38).458 (.558).378 (.73) ρ λ.3967 (.3).873 (.4).8 (.864) Schwarz Approximaion.678 (.73).8855 (.356).496 (.545).393 (.7).46 (.9).884 (.7).558 (.9) 44

45 FIGURE MEAN ERROR BY YEAR Th Figur shows h diffrncs (man rror) bwn h on monh fuurs pric and h spo pric calculad wih prcis and approximad simas, by yar ME Prcis ME Schwarz FIGURE 3 ROOT MEAN SQUARED ERROR BY YEAR Th Figur shows h diffrncs (roo man squard rror) bwn h on monh fuurs pric and h spo pric calculad wih prcis and approximad simas, by yar RMSE Prcis RMSE Schwarz 45

46 FIGURE 4 MEAN ERROR BY MONTH Th Figur shows h diffrncs (man rror) bwn h on monh fuurs pric and h spo pric calculad wih boh prcis and approximad simas, by monh All Monhs Jan Fb Mar Apr May Jun Jul Aug Sp Oc Nov Dc ME Prcis ME Schwarz FIGURE 5 ROOT MEAN SQUARED ERROR BY MONTH Th Figur shows h diffrncs (roo man squard rror) bwn h on monh fuurs pric and h spo pric calculad wih boh prcis and approximad simas, by monh All Monhs Jan Fb Mar Apr May Jun Jul Aug Sp Oc Nov Dc RMSE Prcis RMSE Schwarz 46

47 TABLE 4 COMPARISON OF THE IMPROVEMENT IN THE RMSE AND ONE-MONTH FUTURES PRICE STANDAR DEVIATION BY MONTH Th Tabl shows h improvmn (xprssd in prcnag) in h RMSE, dfind as h RMSE compud wih h Schwarz approximaion minus h RMSE compud wih h prcis vrsion of h simas, and on-monh fuurs pric sandard dviaion, by monh. Improvmn RMSE (%) Volailiy All Monhs January Fbruary March April May Jun July Augus Spmbr Ocobr Novmbr Dcmbr

48 CHAPTER : COMMODITY DERIVATIVES VALUATION UNDER A FACTOR MODEL WITH TIME-VARYING RISK PREMIA. INTRODUCTION In quiy mars, h mar pric of ris is h xcss rurn ovr h ris-fr ra pr uni sandard dviaion (( µ r) ) ha invsors wan as compnsaion for aing ris, which is also calld h Sharp raio. This raio plays an imporan rol in drivaivs valuaion. If h undrlying ass is a radd ass, i is possibl o build a ris-fr porfolio by buying h drivaiv and slling h undrlying ass or vic vrsa. Consqunly, h mar pric of ris dos no appar in h drivaivs valuaion modl. Howvr, if h undrlying ass is no a radd ass, hr is no way of building a rislss porfolio by buying h drivaiv and slling h undrlying ass or vic vrsa; hrfor, w mus now how much rurn is ndd o compnsa h unhdgabl ris. This is why h mar pric of ris mus b simad o obain a horical valu for h drivaiv ass. In commodiy mars, h mar pric of ris has a slighly diffrn dfiniion. As nod by Kolos and Ronn (8), quiis rquir a cosly invsmn and, consqunly, rurn h ris-fr ra undr h ris-nural masur. In h cas of commodiis, i should b nod ha somims hr is a sorag cos associad wih soring h commodiy and also a convninc yild associad wih holding h commodiy rahr han h drivaiv ass. Nvrhlss, fuurs conracs ar coslss o nr ino; hrfor, hir ris-nural drif is zro. Thus, h mar pric of ris in commodiy mars is dfind as h raio of h ass rurn o is sandard 48

49 dviaion( µ ). Addiionally, whras h mar pric of ris mus b posiiv in quiy mars, i can b ngaiv in commodiy mars. Thr hav bn svral paprs ha hav analyzd h propris of mar prics of ris in commodiy mars and hir rlaion wih ohr variabls. Fama and Frnch (987 and 988) no h imporanc of allowing for im-varying ris prmia as ngaiv corrlaions bwn spo prics and ris prmia can gnra man rvrsion in spo prics. Bssmbindr (99) shows ha mar prics of ris in financial and commodiy mars ar rlad o h covarianc of h mar porfolio and h fuurs rurns. Rouldg al. () and Bssmbindr and Lmmon () rla mar prics of ris o svral masurs of uncrainy, such as pric volailiy, spis and uncrainy in dmand. Moosa and Al-Loughani (994), Sardosy () and Jalali- Naini and Kazmi-Mansh (6) find vidnc of variabl ris prmia in oil mars using GARCH modls. Mor rcnly, Kolos and Ronnn (8) sima h mar prics of ris for nrgy commodiis, finding posiiv long-rm and ngaiv shor-rm mar prics of ris. Lucia and Torro (8) find ha ris prmia in h Nordic Powr Exchang (Nord Pool) vary sasonally ovr h yar and ar rlad o unxpcd low rsrvoir lvls. Thr hav also bn svral paprs ha hav analyzd h imporanc of allowing for im-varying ris prmia from h poin of viw of ass valuaion. Following h idas in Fama (984) and Fama and Bliss (987), Duff () and Dai and Singlon () propos inrs ra modls whr ris prmia ar linar funcions of h sa variabls. Casassus and Collin-Dufrsn (5) propos and sima a hr-facor modl for commodiy spo prics, convninc yilds and inrs ras whr convninc yilds dpnd on spo prics and inrs ras, and im-varying (sa dpnding) ris prmia using a maximum lilihood mhod. Thy also s h 49

50 imporanc of h dpndnc of convninc yilds on spo prics and of inrs ras on h valuaion of a s of horical commodiy Europan call opions. Howvr, hy do no s h imporanc of im-varying ris prmia on h valuaion of commodiy drivaivs. In his chapr, w xnd hs idas by proposing and simaing a commodiy drivaiv valuaion modl wih im-varying ris prmia. Tim sris of mar prics of ris for nrgy commodiis (crud oil, haing oil, gasolin and naural gas) ar simad undr h mos widly usd modl for commodiy drivaivs valuaion, which is h Schwarz and Smih () modl, using h Kalman filr mhod on a moving windows basis. Th rsuls show ha mar prics of ris vary hrough im accordingly wih svral macroconomic variabls rlad o h businss cycl, such as crud oil prics, NAPM (Naional Associaion of Purchasing Managrs) and S&P 5 indics. Ths rsuls consiu prliminary vidnc ha h ris compnsaion ha invsors wan in a commodiy drivaiv conrac varis as mar condiions chang. Basd on hs rsuls, a facor modl wih mar prics of ris dpnding on h businss cycl (proxid by h undrlying ass shor- and long-rm facors) using h Kalman filr mhod is proposd and simad. Th proposd modl wih imvarying ris prmia is also maximal, in accordanc wih Dai and Singlon (). Th valuaion rsuls obaind wih an xnsiv sampl of commodiy Amrican opions, radd on h NYMEX, show ha h proposd modl wih im-varying ris prmia ouprforms sandard modls wih consan ris prmia. Ths rsuls confirm h prvious findings shown in h liraur of non-consan mar prics of ris. Morovr, in h prsn chapr, i is found ha allowing for variabl mar prics of ris has an imporan ffc in commodiy drivaiv valuaion. To h bs of our Conrary o prvious paprs, such as Casassus and Collin-Dufrsn (5), who us a maximum lilihood mhod, in h prsn chapr, h simaion is carrid ou using h Kalman Filr mhod, which mploys all h informaion availabl in h forward curv of commodiy fuurs prics. 5

51 nowldg, his is h firs im ha a modl wih im-varying (sa dpnding) ris prmia is applid o h valuaion of xchang-radd commodiy drivaivs. Th rmaindr of his chapr is organizd as follows. Scion prsns h daa ss usd in h chapr. Som prliminary findings rgarding h mar prics of ris simaion using h maximum-lilihood mhod proposd by Kolos and Ronn (8) and h Kalman filr mhod, and hir rlaion o h businss cycl ar prsnd in Scion 3. Th facor modl wih im-varying businss cycl rlad mar prics of ris is proposd and simad in Scion 4. Scion 5 prsns h opion valuaion rsuls obaind wih h modls wih im-varying and consan mar prics of ris. Finally, Scion 6 concluds wih a summary and discussion.. DATA In his scion, w brifly dscrib h daa ha will b usd in his and h following scions. Th daa s usd in his chapr consiss of wly obsrvaions of WTI (ligh sw) crud oil, haing oil, unladd gasolin (RBOB) and naural gas (Hnry Hub) fuurs prics radd on h NYMEX, as wll as a s of xognous variabls rlad o h businss cycl. Currnly, hr ar fuurs bing radd on NYMEX for WTI crud oil wih mauriis of on monh o svn yars, for haing oil from on monh o ighn monhs, for gasolin from on monh o wlv monhs and for Hnry Hub naural gas from on monh o six yars. Howvr, hr is no nough liquidiy for h fuurs wih longr mauriis, spcially in h cas of gasolin. Thrfor, in h cass of WTI crud oil and haing oil, our daa s is comprisd of fuurs prics from on o ighn monhs (,338 wly obsrvaions) bwn //985 and 8/6/. In h cas of RBOB gasolin, h daa s is comprisd of fuurs prics from on o nin monhs (,338 5

52 wly obsrvaions) bwn //985 and 8/6/. Finally, in h cas of Hnry Hub naural gas, h daa s is comprisd of fuurs prics from on o ighn monhs (,64 wly obsrvaions) bwn 4//99 and 8/6/. Th main dscripiv saisics of hs variabls ar conaind in Tabl. To asss h robusnss of h rsuls, wo diffrn daa ss hav bn mployd for ach commodiy. Th firs s conains mor windows bu fwr fuurs conracs, whil h scond s conains fwr windows bu mor fuurs conracs. In h cas of WTI crud oil, h firs daa s is comprisd of conracs F, F3, F5, F7 and F9 from //985 o 8/6/, wih 8 windows, yilding a im sris of 8 mar prics of ris. F is h conrac for h monh closs o mauriy, F is h conrac for h scond-closs monh o mauriy, and so on. Th scond daa s for WTI crud oil is comprisd of conracs F, F4, F7, F, F5 and F8 from 9/9/996 o 8/6/, wih 8 windows, yilding a im sris of 8 mar prics of ris. In h cas of haing oil, h firs daa s is comprisd of conracs F, F3, F6, F8 and F from /4/985 o 8/6/, wih 77 windows, yilding a im sris of 77 mar prics of ris. Th scond daa s for haing oil is comprisd of conracs F, F4, F8, F, F5 and F8 from 9/9/996 o 8/6/, wih 8 windows, yilding a im sris of 8 mar prics of ris. In h cas of RBOB gasolin, h firs daa s is comprisd of conracs F, F3, F4, F5 and F7 from 4/9/985 o 8/6/, wih 8 windows, yilding a im sris of 8 mar prics of ris. Th scond daa s for haing oil is comprisd of conracs F, F3, F5, F7 and F9 from 7/7/995 o 8/6/, wih 9 windows, yilding a im sris of 9 mar prics of ris. Finally, in h cas of Hnry Hub naural gas, h firs daa s is comprisd of conracs F, F4, F6, F9 and F from 4/6/99 o 8/6/, wih 35 windows, yilding a 5

53 im sris of 35 mar prics of ris. Th scond daa s for Hnry Hub naural gas prics is comprisd of conracs F, F4, F8, F, F5, F8, F, F6, F9, F3 and F35 from 5/8/997 o 8/6/, wih 76 windows, yilding a im sris of 76 mar prics of ris. Th s of businss cycl-rlad variabls is composd of wly obsrvaions from //985 o 8/6/ of WTI on monh fuurs prics and S&P 5 indx prics, as wll as monhly obsrvaions of h NAPM (Naional Associaion of Purchasing Managrs) indx and h indicaor of h xpansion of h conomy, which as h valu () if h NAPM indx is abov (blow) 5..3 PRELIMINARY FINDINGS In his scion, w prsn som prliminary findings rgarding h im sris voluion of mar prics of ris for crud oil, haing oil, gasolin and naural gas, as wll as h mar prics of ris rlaionship wih h businss cycl, using h maximum lilihood mhod proposd by Kolos and Ronn (8) and h Kalman filr mhod. Mar prics of ris simaion using h maximum-lilihood mhod Kolos and Ronn (8) obain shor- and long-rm simas of h mar pric of ris for svral nrgy commodiis assuming h wo-facor modl by Schwarz and Smih (). In his modl, h log-spo pric (X ) is assumd o b h sum of wo sochasic facors, a shor-rm dviaion ( ) and a long-rm quilibrium pric lvl (ξ ). Thus, X ξ + () Th sochasic diffrnial quaions (SDEs) for hs facors ar as follows: dξ µ ξ d+ ξ dwξ d κd+ dw () 53

54 whr dw ξ and dw can b corrlad (dw ξ dw ρ ξ d) and wih ρ ξ rprsning h cofficin of corrlaion bwn long- and shor-rm facors. To valu drivaiv conracs, w mus rly on h ris-nural vrsion of h modl. Th SDEs for h facors undr h quivaln maringal masur can b xprssd as: dξ ( µ ξ λξ ) d+ ξ dwξ d ( κ λ ) d+ dw (3) whr λ ξ and λ ar h mar prics of ris for h long- and shor-rm facors, rspcivly, and maringal masur. W ξ and W ar h facor Brownian moions undr h quivaln Schwarz and Smih () and Kolos and Ronn (8) obain h SDE for forward conracs (undr h hisorical masur): df F κτ κτ ( λ + λξξ) d+ dw + ξ dwξ (4) Discrizing quaion (4) and applying Io s Lmma, i is possibl o obain h loglilihood funcion, which is (afr omiing unssnial consans): n i ln Lnln n i ln κτi + ( ) + ln κτi Fi λ + λξ( ξ ) κτi + ( ξ ) ξ κτi ( ( ) ) ξ (5) Maximum lilihood simas of shor- and long-rm mar prics of ris (λ and λ ξ, rspcivly), oghr wih h rs of h modl paramrs, can b obaind by maximizing his log-lilihood funcion. S Kolos and Ronn (8) for h dails. 54

55 In his chapr, h maximizaion of h log-lilihood funcion has bn prformd subsqunly ovr moving windows of 4 ws, using wly obsrvaions of on monh fuurs prics for WTI crud oil, haing oil, RBOB gasolin and Hnry Hub naural gas. In his way, w obain im sris of mar prics of ris for h four commodiy sris (8 obsrvaions in h cas of WTI crud oil, haing oil and RBOB gasolin, and 34 obsrvaions in h cas of Hnry Hub naural gas). In Figur, w plo h im sris voluion of h simad mar prics of ris in h cas of WTI crud oil. Th simad sris show high volailiy, which is consisn wih h rsuls found by Kolos and Ronn (8). Th rsuls rgarding h cofficins of corrlaion among h simad mar prics of ris and h businss cycl-rlad variabls dscribd abov ar shown in Tabl. Th corrlaions bwn shor- and long-rm ris prmia ar ngaiv in all cass, alhough significan only in h cas of Hnry Hub naural gas. Posiiv and significan corrlaions ar found among mar prics of ris and WTI on monh fuurs prics 3, xcp for h long-rm ris prmium for RBOB gasolin and long- and shor-rm ris prmia for Hnry Hub naural gas. Morovr, posiiv and significan corrlaions among mar prics of ris and S&P5 (and is on w lag) ar found, xcp for h long-rm on in h cas of RBOB gasolin and long- and shor-rm ons in h cas of Hnry Hub naural gas. In h cas of h NAPM indx (and is on monh lag), posiiv and significan corrlaions wih shor-rm mar prics of ris for all four commodiis and wih long-rm on in h cas of WTI ar also found, alhough h magniud of h corrlaion is lowr han h magniud in h S&P5 cas (xcp for h Hnry Hub For h sa of brviy, only h plo of mar prics of ris simad wih WTI crud oil ar prsnd hr. Th plos for h ohr hr commodiis show a vry similar parn. 3 WTI on monh fuurs prics ar calculad as h man of h fuurs pric during h window usd o sima h mar pric of ris. 55

56 naural gas). Finally, vidnc of corrlaion among mar prics of ris and h xpansion indicaor of h conomy has no bn found. In fac, as can b vidncd in Figur, mar prics of ris sm o show a nois parn ha is no clar and ha is no dircly associad wih mar condiions. In summary, h prliminary analysis prformd wih h Kolos and Ronn (8) maximum lilihood mhod shows vidnc of som linar rlaionship mosly among shor-rm mar ris prmia and businss cycl-rlad variabls, such as S&P 5 and NAPM indics. As will b discussd hrin, h maximum lilihood mhod usd by Kolos and Ronn (8) and Casassus and Collin-Dufrsn (5) prsns som disadvanags whn compard o h Kalman filr mhod usd in h nx scion. Mar Prics of Ris Esimaion using h Kalman Filr Mhod Th Kalman filr mhod is, horically, suprior o h maximum lilihood mhod for svral rasons. Firs, h Kalman filr mhod simas all of h dynamic of h undrlying ass, whras h maximum lilihood mhod only uss mar prics of fuurs conracs wihou aing ino accoun h dynamics of h common undrlying ass. Scond, wih h Kalman filr mhod, w ar abl o us mor fuurs conracs (mor mauriis), which will rsul in mor sabl simas of h paramrs han hos obaind wih h maximum lilihood mhod, such as in Kolos and Ronn (8) and Casassus and Collin-Dufrsn (5). As sad in Scion 3. and in h conx of h Schwarz and Smih () wo-facor modl, h log spo pric (X ) is assumd o b h sum of wo sochasic facors, a shorrm dviaion ( ) and a long-rm quilibrium pric lvl (ξ ). Morovr, in h cass of commodiis, such as naural gas, haing oil and gasolin, a drminisic sasonal 56

57 componn is addd, as suggsd by Sornsn () 4. Thrfor, h log spo pric for haing oil, gasolin and naural gas (X ) is assumd o b h sum of wo sochasic facors ( and ξ ) and a drminisic sasonal rigonomric componn (α ), X ξ + + α. Th SDEs for ξ and ar givn by xprssions () and: dα πϕα d * dα πϕα d * (6) whr α * is h ohr sasonal facor, which complmns α, and ϕ is h sasonal priod. Th SDEs for h long- and shor- rm facors undr h quivaln maringal masur ar givn by xprssions (3). As sad in prvious sudis, on of h main difficulis in simaing h paramrs of h wo-facor modl is ha h shor- and long-rm facors (or sa variabls) ar no dircly obsrvabl. Insad, hy mus b simad from spo and/or fuurs prics 5. Th formal mhod o sima h modl is o us h Kalman filr mhodology, which is brifly dscribd in h Appndix 6. Th Kalman filr mhod has bn subsqunly prformd ovr moving windows of 4 ws, using wly obsrvaions of fuurs prics for WTI crud oil, haing oil, RBOB gasolin and Hnry Hub naural gas 7. Two diffrn daa ss (dfind in Scion ) hav bn mployd for ach commodiy. Th firs s conains mor windows bu fwr fuurs conracs, whil h scond s conains fwr windows bu mor fuurs conracs. 4 Sornsn () suggss inroducing ino h modl a drminisic sasonal componn for agriculural commodiis. Hr, w us Sornsn s proposal for haing oil, gasolin and naural gas, which prsn a srong sasonal bhavior (s, for xampl, Garcia al., a). 5 Th xac xprssion for h fuurs pric undr h Schwarz and Smih () wo-facor modl wih sasonal facors can b found in Garcia al. (a). 6 Daild accouns for Kalman filring ar givn in Harvy (989) and Puhnpura al. (995). 7 Dails abou implmning h Kalman filr in Malab can b found in Da and Bang (9). 57

58 In Figur, w plo h im sris voluion of h simad mar prics of ris in h cas of WTI crud oil wih h firs daa s, oghr wih svral businss cyclrlad variabls 8. Looing a h im-voluion of h simad ris prmia, i is clar ha w obain mor sabl simas wih h Kalman filr mhod han hos obaind wih h maximum lilihood mhod. Th rsuls show a ngaiv rlaionship bwn long- and shor-rm mar ris prmia. Morovr, a posiiv (ngaiv) rlaionship bwn h long- (shor-) rm mar pric of ris and h avrag pric of on monh WTI fuurs is found, suggsing ha h long- (shor-) rm ris compnsaion ha invsors wan o nr in a commodiy drivaiv is posiivly (ngaivly) rlad o crud oil prics 9. This finding suggss ha whn crud oil prics ar high, h ris associad wih h long-rm facor (which is h facor ha dos no disappar wih im) nds o no b divrsifiabl. Morovr, h volailiy of on monh WTI fuurs prics is ngaivly (posiivly) rlad o h long- (shor-) rm mar pric of ris. Concrning h simad mar prics of ris p-valus, i is found ha ris prmia ar significan (and hrfor no divrsifiabl) during xpansion priods or whn crud oil prics ris, whras hy ar no significan in conracion priods or whn crud oil prics dcras, alhough h parn is somwha clarr in h cas of h long-rm mar ris prmium, which confirms ha h crud oil ris is no divrsifiabl whn crud oil pric is high nough. If w considr h rlaionship bwn h avrag longand shor-rm facors and h simad mar prics of ris, w find ha long-rm (shor-rm) mar prics of ris ar posiivly (ngaivly) rlad o boh long- and shor-rm facors. Morovr, h simad mar pric of ris sms o b posiivly rlad o is rspciv (long- or shor-rm) facor sandard dviaion. 8 As bfor, for h sa of brviy, only h plo of mar prics of ris simad wih WTI crud oil ar prsnd hr. Th plos for h ohr hr commodiis show a vry similar parn. 9 As in h prvious scion, h fuurs prics avrag is h man of h fuurs pric during h window usd o sima h mar pric of ris. 58

59 Finally, a posiiv (ngaiv) rlaionship is found bwn h simad long-rm (shor-rm) mar pric of ris and h avrag NAPM indx, h avrag S&P 5 indx and h indicaor of xpansion 3, suggsing ha h ris associad wih h longrm facor nds o no b divrsifiabl during xpansion priods. Th rsuls rgarding h cofficins of corrlaion among h simad mar prics of ris and h businss cycl rlad variabls dscribd abov ar shown in Tabls 3 for WTI crud oil, 4 for haing oil, 5 for RBOB gasolin and 6 for Hnry Hub naural gas. Th rsuls confirm h graphical analysis of Figur. Th rlaionship bwn h long- and shor-rm mar prics of ris is found o b ngaiv and significan in h cas of WTI crud oil (Tabl 3) and posiiv and significan in h cass of haing oil (Tabl 4), RBOB gasolin (Tabl 5) and Hnry Hub naural gas (Tabl 6). I is also inrsing o obsrv h posiiv and significan rlaionship found bwn h long-rm mar pric of ris and WTI fuurs prics for WTI crud oil, haing oil and RBOB gasolin (h rlaionship is lss clar in h cas of Hnry Hub naural gas). This rsul suggss, onc again, ha h long-rm compnsaion ha invsors rquir o nr ino a commodiy conrac riss as WTI fuurs prics ris 3. Rahr ambiguous rlaionships ar found among h mar prics of ris and h volailiy of on monh WTI fuurs pric, h modl volailiy and h maximum lilihood, and h NAPM and S&P5 (and hir lags) indics. Howvr, h mos obvious rlaionship is h on found among h simad mar prics of ris and h undrlying long- and shor-rm facors, alhough h rlaionship is lss clar in h cas of Hnry Hub naural gas prics. Lss clar is h rlaionship 3 Th indicaor of h xpansion of h conomy as h valu () if h NAPM indx is abov (blow) 5. 3 Corazar, Milla and Svrino (8) and García, Población and Srna (b) show ha crud oil and is main rfind producs (haing oil and gasolin) shar common long-rm dynamics. Thrfor, i is no surprising ha h long-rm compnsaions associad wih crud oil, haing oil and gasolin ar (posiivly) rlad o WTI fuurs prics. 59

60 among h mar prics of ris and h volailiy of h undrlying long- and shor-rm facors. Ths findings confirm our prvious assumpion ha h ris compnsaion ha invsors wan o nr ino a commodiy drivaiv conrac varis as mar condiions chang. Spcifically, i is qui inrsing o obsrv how h mar prics of ris vary according o h undrlying long- and shor-rm facors. Thrfor, i sms naural o propos a facor modl wih mar prics of ris dpnding on h businss cycl, proxid by h undrlying long- and shor-rm facors, along h lins suggsd by Casassus and Collin-Dufrsn (5), alhough hr w us h Kalman filr mhod insad of h maximum lilihood mhod..4 A FACTOR MODEL WITH TIME-VARYING MARKET PRICES OF RISK DEPENDING ON THE BUSINESS CYCLE Basd on h prvious rsuls, in his scion, a facor modl wih im-varying mar prics of ris dpnding on h businss cycl is proposd and simad. Th proxy for h businss cycl will b h Schwarz and Smih () long- and shor-rm facors, ξ and, rspcivly. Ths wo facors ar found o b h businss cycl rlad variabls wih highr cofficins of corrlaion wih h simad mar prics of ris. Th modl wih im-varying ris prmia will b an xnsion of h wo-facor modl dscribd in Scion 3, whr h log spo pric for haing oil, gasolin and naural gas (X ) is assumd o b h sum of wo sochasic facors ( and ξ ) and a drminisic sasonal rigonomric componn (α ), X ξ + + α ( X ξ + for crud oil), whr α is dfind in xprssions (6). Th SDEs for h long- and shor- rm facors undr h quivaln maringal masur, wih im-varying ris prmia, can b xprssd as: 6

61 dξ ( µ ξ λξ ) d+ ξ dwξ d ( κ λ ) d+ dw (7) whr, as bfor, W ξ and W ar h facor Brownian moions undr h quivaln maringal masur, and λ ξ and λ ar im-varying mar prics of ris for h long- and shor-rm facors, rspcivly. Following Duff (), Dai and Singlon () and Casassus and Collin-Dufrsn (5), h mar prics of ris ar xprssd as linar funcions of h undrlying long- and shor-rm facors: λ λ + λ ξ + λ λ ξ ξ λ ξ + λ ξ + λ ξ (8) Th paramrs of h modl ar simad, as in Scion 3., using h Kalman filr mhod rahr han h maximum lilihood procdur usd by Casassus and Collin- Dufrsn (5). Th rsuls of h simaion of his facor modl wih im-varying mar ris prmia, oghr wih h rsuls of h sandard wo-facor Schwarz and Smih () modl wih consan ris prmia for h four commodiy sris using boh h firs and h scond daa ss dscribd in Scion ar shown in Tabl 7 (WTI crud oil), Tabl 8 (haing oil), Tabl 9 (RBOB gasolin) and Tabl (Hnry Hub naural gas). Th rsuls in Tabls 7, 8, 9 and confirm h prsnc of h man rvrsion ffc, ypically obsrvd in commodiy mars (paramr κ is significan in all cass). Morovr, as xpcd, boh long- and shor-rm facors ar found o b sochasic (hir corrsponding sandard dviaions, ξ and, rspcivly, ar significan), alhough h shor-rm sandard dviaion is found o b highr han h corrsponding long-rm sandard dviaion, suggsing ha shor-rm ffcs hav a highr impac on 6

62 spo prics han long-rm ffcs 3. Howvr, as xplaind abov, i mus b p in mind ha shor-rm ffcs nd o disappar wih im (h shor-rm procss is saionary), whras long-rm ffcs do no disappar wih im (h long-rm procss is ingrad). Howvr, h mos imporan issu in Tabls 7, 8, 9 and from h poin of viw of h goal of his chapr, is ha h paramrs associad wih h mar prics of ris (λ ξ, λ ξ, λ ξ, λ, λ and λ ) ar significan in mos of h cass, confirming ha ris prmia vary hrough im dpnding on h conomic condiions (proxid in his chapr by h modl long- and shor-rm facors). If w dfin h Schwarz informaion cririon (SIC) as ln( L ML ) q ln( T), whr q is h numbr of simad paramrs, T is h numbr of obsrvaions and L ML is h valu of h lilihood funcion using h q simad paramrs, hn h fi is br whn h SIC is highr. Th sam conclusions ar obaind wih h Aai informaion cririon (AIC), which is dfind as ln( L ML ) q. I is worh noing ha in Tabls 7, 8, 9 and, h valus of h SIC and h AIC ar highr in h modl wih im-varying ris prmia. This finding confirms h rsuls obaind by Casassus and Collin-Dufrsn (5), in ha allowing for im-varying mar ris prmia improvs h simaion rsuls. Howvr, in his chapr, h simaion is carrid ou using h Kalman filr mhod, which is horically suprior o h maximum lilihood mhod usd by Casassus and Collin-Dufrsn (5). In h nx scion, w us hs rsuls for commodiy opion valuaion purposs. Spcifically, w show h imporanc of allowing for im-varying mar ris prmia in valuing a s of mar radd commodiy opions. I should b nod ha Casassus and Collin-Dufrsn (5) also propos a modl wih im-varying ris prmia, bu 3 This fac is also found in Schwarz and Smih () and Garcia al. (b), among ohrs. 6

63 hy do no s h imporanc of im-varying mar prics of ris on h valuaion of commodiy drivaivs..5 OPTION VALUATION WITH TIME-VARYING MARKET PRICES OF RISK DEPENDING ON THE BUSINESS CYCLE As sad abov, in his scion, w apply our modl wih im-varying ris prmia o h valuaion of an xnsiv s of commodiy mar radd opions. Opion Daa Th daa s usd in h simaion procdur consiss of daily obsrvaions of WTI, haing oil, RBOB gasolin and Hnry Hub naural gas Amrican call and pu opions quod a h NYMEX and corrsponding o h yars from 6 unil. Th numbr of sris is,93 call and,53 pu (3,7 and 8,36 obsrvaions, rspcivly) in h cas of WTI crud oil;,567 call and 3 pu (77,97 and 45,75 obsrvaions, rspcivly) in h cas of haing oil;,633 call and 938 pu (45,354 and 59,576 obsrvaions, rspcivly) in h cas of RBOB gasolin; and 68 call and 758 pu (79,957 and 99,88 obsrvaions, rspcivly) in h cas of Hnry Hub naural gas. In h NYMEX, WTI opion conracs maur ach monh for h currn yar and for h nx fiv yars. Addiionally, h Jun and Dcmbr monhs ar lisd byond h sixh yar. Sri prics ar h on a-h-mony sri pric, wny sri prics in incrmns of $.5 pr barrl abov and blow h a-h-mony sri pric, and h nx sri prics in incrmns of $.5 abov h highs and blow h lows xising sri prics for a oal of a las 6 sri prics. In h cas of haing oil and RBOB gasolin opions, hr ar lisd conracs for h nx 36 conscuiv monhs, and availabl sri prics ar h a-h-mony, wny 63

64 sri prics in $. pr gallon incrmns abov and blow h a-h-mony sri pric, and h nx sri prics in $.5 incrmns abov h highs and blow h lows xising sri prics for a oal of a las 6 sri prics. Finally, in h cas of Hnry Hub naural gas opions, hr ar lisd conracs for h conscuiv monhs for h balanc of h currn yar plus 5 addiional yars. Sri prics ar h on a-h-mony sri prics, wny sri prics in incrmns of $.5 pr mmbu abov and blow h a-h-mony sri pric in all monhs, plus an addiional sri prics in incrmns of $.5 pr mmbu abov h a-h-mony pric will b offrd in h firs hr narby monhs, and h nx sri prics in incrmns of $.5 pr mmbu abov h highs and blow h lows xising sri prics in all monhs, for a oal of a las 8 sri prics in h firs hr narby monhs and a oal of a las 6 sri prics for four monhs and byond 33. In all cass, h undrlying ass is h corrsponding WTI, haing oil, RBOB gasolin or Hnry Hub naural gas fuurs conrac. Opion Valuaion Mhodology Th compuaion of Amrican opion prics is a challnging problm which implis solving an opimal sopping problm. Th problm can b simplifid mploying Mon Carlo chniqus. Th saring poin of hs mhods is o rplac h im inrval of xrcis das by a fini subs. Th soluion of h corrsponding discr opimal sopping problm rducs o an ffciv implmnaion of h dynamic programming principl. Howvr, h condiional xpcaions involv in h iraions of h dynamic programming caus h main difficuly for h dvlopmn of h Mon Carlo chniqus. On way of raing his problm is h mhod prsnd in Longsaff and 33 Addiional dails abou h conracs can b found on h CME Group wb pag. 64

65 Swcharz (), which is on of h mos popular Amrican opion valuaion mhods and will b h mhod usd in his scion o valu commodiy Amrican opions. Spcifically, h mhod proposd by Longsaff and Schwarz () consiss of simaing h condiional xpcd pay-off o h holdr of h opion from coninuaion using las squars rgrssion chniqus. For h purpos of opion valuaion, w nd a full dscripion of h modl. In marix form, h sa dynamics can b dscribd as follows: ( + AZ ) d dw dz µ +. (9) To clarify, l us a U o b a uni of Brownian moion (i.., T du du I d ) and rwri (9) as: ( + AZ) d RdU dz µ +. () For paramr simaion purposs, w us Kalman filr quaions o sima Z E[ Z Z Z ], and as an inrmdia rsul, Z E[ Z Z Z ] /,, /,,. This procss (simaing using currn or vn fuur informaion) is rmd aliasing in h Kalman filr liraur. Th sris Z is usd as iniial sas for opion valuaion. Opion Valuaion Rsuls Tabl prsns svral mrics o analyz h prdiciv powr abiliy of h modls for h daa s of WTI, haing oil, RBOB gasolin and Hnry Hub naural gas Amrican opions. Th modls considrd ar h im-varying ris prmia and h sandard consan (wo-facor) ris prmia. Morovr, h rsuls shown in h abl ar basd on h simaion rsuls obaind from boh h firs and h scond daa ss dscribd in Scion. 65

66 Th saisics prsnd in Tabl ar h roo man squard rror (RMSE), h prcnag roo man squard rror (PRMSE) and h man absolu rror (MAE), which ar dfind as: RMSE PRMSE n n n i ( ) f i f, m i, n ( fi, m fi, ) i n n i f i, m MAE n n i f i f, m i, whr f i,m and f i, ar h mar and h horical prics, rspcivly, of opion i. Th valus shown in h abl ar h mdian of h diffrn mans for ach opion sris. I can b obsrvd ha w achiv br rsuls wih h im-varying ris prmia modl for all commodiis undr sudy wih all hr saisics (xcp in h cas of haing oil using h RMSE wih h scond daa s). I is also worh noing ha, in gnral, w achiv br rsuls using h firs daa s (a las in h cas of WTI crud oil, haing oil and RBOB gasolin). Furhrmor, i can b apprciad ha h bs rsuls of h im-varying modl ar achivd wih RBOB gasolin, followd by haing oil. Ths rsuls confirm ha h consan ris prmia assumpion in sandard opion valuaion modls has an imporan ffc in rms of valuaion rrors. Thrfor, h fac ha mar prics of ris vary ovr im according o h businss cycl mus b an ino accoun in opion valuaion modls. Spcifically, w hav sn ha h ris ha invsors fac whn hy nr in a drivaiv conrac canno somims b divrsifid, dpnding on h mar condiions, which has imporan implicaions in rms of drivaiv valuaion. In paricular, i is found ha h ris associad wih h long-rm 66

67 facor nds o no b divrsifiabl during xpansion priods. Thrfor, i sms naural ha h ris associad wih h modl facors is somims no divrsifiabl, dpnding on h mar condiions, and can somwha affc opion valus. In his chapr, w hav sn ha, in fac, by allowing for im-varying (sa-dpnding) mar prics of ris opion valuaion, rrors can b rducd compard o hos obaind wih sandard (consan mar prics of ris) modls. Finally, i should b nod ha hr hav bn svral paprs proposing facor modls wih im-varying (sa dpnding) ris prmia, such as Casassus and Collin-Dufrsn (5). Howvr, hs paprs do no s h imporanc of im-varying ris prmia on h valuaion of commodiy drivaivs. To h bs of our nowldg, his is h firs im ha a modl wih im-varying (sa dpnding) ris prmia is applid o h valuaion of xchang-radd commodiy drivaivs..6 CONCLUSIONS In his chapr, w no h imporanc of allowing for im-varying mar prics of ris in a commodiy drivaiv modl. Spcifically, w show ha h compnsaion ha invsors wan in a commodiy drivaiv conrac varis hrough im according o svral businss rlad variabls. Mor imporanly, his businss cycl dpndnc of mar prics of ris has an imporan ffc in rms of opion valuaion rrors. Th chapr bgins by simaing im sris of mar prics of ris for crud oil, haing oil, gasolin and naural gas undr h wo-facor modl proposd by Schwarz and Smih () and using h Kalman filr mhod. Th rsuls show ha h ris compnsaion ha invsors wan in a commodiy drivaiv conrac varis as mar condiions chang. Spcifically, clos rlaionships among mar prics of ris and svral variabls rlad o h businss cycl, such as NAPM and S&P 5 indics, 67

68 crud oil prics, crud oil pric volailiy and long- and shor-rm pric facors, among ohrs, ar found. Basd on hs rsuls, a facor modl wih mar prics of ris dpnding on h businss cycl and proxid by long- and shor-rm pric facors is proposd and simad. Th valuaion rsuls obaind wih a sampl of fuurs conracs on crud oil, haing oil, gasolin and naural gas show ha h proposd modl wih imvarying ris prmia dpnding on h businss cycl ouprforms h sandard wofacor modl wih consan ris prmia. This finding confirms h rsuls obaind by Casassus and Collin-Dufrsn (5) in ha allowing for im-varying mar ris prmia improvs h simaion rsuls. Nonhlss, in his chapr, h simaion is carrid ou using h Kalman filr mhod, which is horically suprior o h maximum lilihood mhod usd by Casassus and Collin-Dufrsn (5). Howvr, h mos imporan conribuion of his chapr is h applicaion of h modl wih im-varying ris prmia o h valuaion of an xnsiv sampl of xchangradd commodiy drivaivs. Spcifically, h daa bas is comprisd of Amrican opions on WTI, haing oil, RBOB gasolin and Hnry Hub naural gas fuurs conracs, radd a NYMEX and yilding br rsuls han hos obaind wih sandard (consan mar prics of ris) modls. Spcifically, w hav sn ha h ris ha invsors fac whn hy nr in a drivaiv conrac canno always b divrsifid, dpnding on h mar condiions. In paricular, i is found ha h ris associad wih h long-rm facor nds o no b divrsifiabl in xpansion priods. Consqunly, i is imporan o a ino accoun h dpndnc of ris prmia on h conomic condiions in valuing drivaiv conracs. 68

69 To h bs of our nowldg, his is h firs im ha a modl wih im-varying (sa dpnding) ris prmia is applid o h valuaion of xchang-radd commodiy drivaivs. 69

70 APPENDIX Th Kalman filr chniqu is a rcursiv mhodology ha simas h unobsrvabl im sris and h sa variabls or facors (Z ) basd on an obsrvabl im sris (Y ), which dpnds on hs sa variabls. Th masurmn quaion accouns for h rlaionship bwn h obsrvabl im sris and h sa variabls such ha: Y d + M Z + η,, N (A) whr Y d R M R, Z R n n x h h,,, h is h numbr of sa variabls, or facors, in n h modl, and η R is a vcor of srially uncorrlad Gaussian disurbancs wih zro man and covarianc marix H. To avoid daling wih a larg numbr of paramrs, w assum ha H is diagonal wih main diagonal nris qual o η. Th ransiion quaion accouns for h voluion of h sa variabls: Z c + T Z + ψ,, N (A) whr c R T R and ψ R h h x h h, ar vcors of srially uncorrlad Gaussian disurbancs wih zro man and covarianc marix Q. L Y b h condiional xpcaion of Y and l Ξ b h covarianc marix of Y condiional on all informaion availabl a im. Thn, afr omiing unssnial consans, h log-lilihood funcion can b xprssd as: Ξ ( Y Y )' Ξ ( Y Y l ln ) (A3) REFERENCES Bssmbindr, H. (99). Sysmaic ris, hdging prssur, and ris prmiums in fuurs mars. Rviw of Financial Suds, 5(4),

71 Bssmbindr, H., & Lmmon, M.L. (). Equilibrium pricing and opimal hdging in lcriciy forward mars. Th Journal of Financ, 57, Casassus, J., & Collin-Dufrsn, P. (5). Sochasic Convninc Yild Implid from Commodiy Fuurs and Inrss Ras. Th Journal of Financ, 6(5) Corazar, G., C. Milla, F., & Svrino. A (8). Mulicommodiy Modl for Fuurs Prics: Using Fuurs Prics of On Commodiy o Esima h Sochasic Procss of Anohr. Journal of Fuurs Mars, 8 (6), Corazar G., & Naranjo L. (6). An N-Facor gaussian modl of oil fuurs prics. Journal of Fuurs Mars, 6(3), Corazar, G., & E.S. Schwarz. (3). Implmning a sochasic modl for oil fuurs prics. Enrgy Economics, 5, 5-8. Dai, Q., & Singlon, K.J. (). Spcificaion analysis of affin rm srucur modls. Th Journal of Financ, 55, Dai, Q., & Singlon, K.J. (). Expcaions Puzzls Tim-Varying Ris Prmia, and Affin Modls of h Trm Srucur. Journal of Financial Economics, 63, Da, P., & Wang, C. (9). Linar Gaussian affin rm srucur modls wih unobsrvabl facors: Calibraion and yild forcasing. Europan Journal of Opraional Rsarch, 95, Duff, G.R. (). Trm Prmia and Inrs Ra Forcass in Affin Modls. Journal of Financ, 57, Fama, E.F. (984). Trm Prmiums in Bond Rurns. Journal of Financial Economics, 3,

72 Fama, E.F., & Bliss, R.R. (987). Th Informaion in Long-Mauriy Forward Ras. Amrican Economic Rviw, 77(4), Fama, E.F., & Frnch, E.R. (987). Commodiy fuurs prics: som vidnc on forcas powr, prmiums, and h hory of sorag. Journal of Businss, 6(), Fama, E.F., & Frnch, E.R. (988). Prmann and mporary componns of soc prics. Journal of Poliical Economy, 96, García A., Población J., & Srna, G. (a). Th sochasic sasonal bhavior of naural gas prics. Europan Financial Managmn, forhcoming. García, A. Población, J., & Srna, G. (b). Analyzing h dynamics of h rfining margin: Implicaions for valuaion and hdging. Woring Papr. Harvy, A.C. (989). Forcasing Srucural Tim Sris Modls and h Kalman Filr, Cambridg (U.K.): Cambridg Univrsiy Prss. Jalali-Naini, A., & Kazmi-Mansh, M. (6). Pric Volailiy, Hding and Variabl Ris Prmium in h Crud Oil Mar. OPEC Rviw, 3(), Kolos, S.P., & Ronn, E.I. (8). Esimaing h commodiy mar pric of ris for nrgy prics. Enrgy Economics, 3, Longsaff F., & Schwarz E. (). Valuing Amrican opions by simulaions a simpl las squars approach. Th Rviw of Financial Sudis, 4(), Lucia, J., & Torro, H. (). Shor-Trm Elcriciy Fuurs Prics: Evidnc on h Tim-Varying Ris Prmium. Inrnaional Rviw of Economics and Financ, (4), Moosa, I.A., & Al-Loughani, N.E. (994). Unbiasdnss and Tim Varying Ris Prmia in h Crud Oil Fuurs Mar. Enrgy Economics, 6(),

73 Puhnpura, S., Sinha, L. Fang, S.-C., & Saigal, R. (995). Solving sochasic programming problms via Kalman filr and affin scaling. Europan Journal of Opraional Rsarch, 83(3), Rouldg, B.R., Sppi, D., & Spa, C.W. (). Th spar sprad: crosscommodiy quilibrium rsricions and lcriciy. Woring Papr. Sardosy, P. (). Tim-varying ris prmiums in prolum fuurs prics. Enrgy Economics, 4, Schwarz, E.S. (997). Th sochasic bhavior of commodiy prics: Implicaion for valuaion and hdging. Th Journal of Financ, 5, Schwarz, E.S., & Smih, J.E. (). Shor-rm variaions and long-rm dynamics in commodiy prics. Managmn Scinc, 46(7), Sornsn, C. (). Modling sasonaliy in agriculural commodiy fuurs. Th Journal of Fuurs Mars,,

74 TABLES AND FIGURES TABLE DESCRIPTIVE STATISTICS Th abl shows h man and sandard dviaion (S.D.) of h four commodiy sris prics. F is h fuurs conrac closs o mauriy, F is h conrac scond-closs o mauriy and so on. In h cass of WTI crud oil and haing oil h daa s is comprisd of fuurs prics from on o ighn monhs (338 wly obsrvaions) from //985 o 8/6/. In h cas of RBOB gasolin, h daa s is comprisd of fuurs prics from on o nin monhs (338 wly obsrvaions) from //985 o 8/6/. In h cas of Hnry Hub naural gas, h daa s is comprisd of fuurs prics from on o ighn monhs (64 wly obsrvaions) from 4//99 o 8/6/. WTI Crud Oil Haing Oil RBOB Gasolin Hnry Hub Naural Gas Man S. D. Man S. D. Man S. D. Man S. D. F F F F F F F F F F F F F F F F F F

75 TABLE CORRELATION AMONG MAXIMUM LIKELIHOOD ESTIMATES OF MARKET PRICES OF RISK AND BUSINESS CYCLE RELATED VARIABLES Th abl shows h cofficins or corrlaion among h simad mar prics of ris wih h Kolos and Ronn (8) maximum lilihood mhod and svral businss cycl variabls. Cofficins of corrlaion ar rpord wih hir sandard rrors in parnhsis. Th simad valus ar rpord wih * dnoing significanc a %, ** dnoing significanc a 5%, and *** dnoing significanc a %. λ WTI CRUDE OIL λ λ ξ WTI F S&P 5 S&P 5 (-5) NAPM -. (.749) λ ξ λ.3957 *** (.6883).365 *** (.78).596 *** (.64).34 *** (.7) λ λ ξ WTI F S&P (.749) λ ξ λ.365 *** (.699).794 *** (.7).467 *** (.665).347 *** (.76) λ λ ξ WTI F S&P (.76) λ ξ λ.595 *** (.645).936 *** (.76) HEATING OIL S&P 5 (-5).4773 *** (.66).399 *** (.7) RBOB GASOLINE S&P 5 (-5).736 ** (.738).98 ** * (.736) NAPM.65 * (.746) -.65 (.75) NAPM.39 *** (.69).4746 *** (.66).4666 *** (.665).86 ** (.738) (.75) (.749) (.75) (.747) HENRY HUB NATURAL GAS λ λ ξ WTI F S&P *** (.749) λ ξ -.63 (.859) (.859).3 ** (.84) (.85) S&P 5 (-5).83 ** (.846) -.34 (.855) NAPM.5 ** (.84).36 (.86) NAPM (-).894 ** (.735).4 *** (.73) NAPM (-).49 ** (.743).45 (.75) NAPM (-).949 *** (.737).87 (.747) NAPM (-).59 ** (.84).4 (.86) Expans. Indicaor.85 (.749).436 (.749) Expans. Indicaor.58 (.75).4 (.75) Expans. Indicaor.9 (.747) -.93 (.75) Expans. Indicaor.98 (.855).98 (.857) 75

76 TABLE 3 CORRELATION AMONG WTI KALMAN FILER ESTIMATES OF MARKET PRICES OF RISK AND BUSINESS CYCLE RELATED VARIABLES Th abl shows h cofficins or corrlaion among h simad WTI crud oil mar prics of ris wih h Kalman filr mhod and svral businss cycl rlad variabls. Cofficins of corrlaion ar rpord wih hir sandard rrors in parnhsis. Th simad valus ar rpord wih * dnoing significanc a %, ** dnoing significanc a 5%, and *** dnoing significanc a %. WTI FRIST DATA SET WTI SECOND DATA SET λ ξ λ λ ξ λ *** *** λ ξ (.6) (.53) *** *** λ (.6) (.53) WTI F.549 *** (.66) *** (.394).389 *** (.33) *** (.553) VOLAT. F.8 ** (.744) -.34 (.749) *** (.538).3 (.) MODEL VOLAT..666 (.748).58 (.748) *** (.693).3445 *** (.5) LIKELIHOOD -.58 *** (.75).4568 *** (.667) -.89 *** (.58).76 ** (.76) ξ.6574 *** (.565) *** (.44).464 *** (.99) *** (.698).98 *** (.735) *** (.638).4989 *** (.969) *** (.754) ξ.834 *** (.45) *** (.566).864 *** (.56) *** (.984).37 (.449) -.46 (.75) *** (.85).336 *** (.53) NAPM.83 ** (.737) -.38 (.749).8479 *** (.593) -.58 ** (.8) NAPM(-).896 ** (.736) -.68 (.748).848 *** (.65) *** (.6) S&P *** (.539) *** (.544) ** (.98) *** (.7) S&P 5 (-5).6897 *** (.543) *** (.565).57 (.) *** (.93) 76

77 TABLE 4 CORRELATION AMONG HEATING OIL KALMAN FILER ESTIMATES OF MARKET PRICES OF RISK AND BUSINESS CYCLE RELATED VARIABLES Th abl shows h cofficins or corrlaion among h simad haing oil mar prics of ris wih h Kalman filr mhod and svral businss cycl rlad variabls. Cofficins of corrlaion ar rpord wih hir sandard rrors in parnhsis. Th simad valus ar rpord wih * dnoing significanc a %, ** dnoing significanc a 5%, and *** dnoing significanc a %. HEATING OIL FRIST DATA SET HEATING OIL SECOND DATA SET λ ξ λ λ ξ λ λ ξ.7896 ***.57 *** (.464) (.955) λ.7896 ***.57 *** (.464) (.955) WTI F.37 *** *** -.9 * (.7) (.756) (.) (.97) VOLAT. F ***.84 (.753) (.755) (.39) (.4) MODEL VOLAT..56 **.383 *.59 **.5875 *** (.747) (.749) (.8) (.95) LIKELIHOOD ** ***.8 (.756) (.743) (.5) (.4) ξ.66 ***.4365 ***.6884 ***.467 *** (.599) (.68) (.8) (.) -.64 *** *** *** *** (.6) (.584) (.97) (.497) ξ.9 *** *** -.69 (.76) (.755) (.) (.8) *** (.756) (.75) (.37) (.5) NAPM ***.748 (.755) (.75) (.53) (.5) NAPM(-) ***.496 (.755) (.75) (.5) (.7) S&P ** S&P 5 (-5) (.745).5 (.75) (.755) (.753) (.7).5 (.) (.98) -.97 * (.93) 77

78 TABLE 5 CORRELATION AMONG RBOB GASOLINE KALMAN FILER ESTIMATES OF MARKET PRICES OF RISK AND BUSINESS CYCLE RELATED VARIABLES Th abl shows h cofficins or corrlaion among h simad RBOB gasolin mar prics of ris wih h Kalman filr mhod and svral businss cycl rlad variabls. Cofficins of corrlaion ar rpord wih hir sandard rrors in parnhsis. Th simad valus ar rpord wih * dnoing significanc a %, ** dnoing significanc a 5%, and *** dnoing significanc a %. RBOB GASOLINE FRIST DATA SET RBOB GASOLINE SECOND DATA SET λ ξ λ λ ξ λ.868 ***.85 *** λ ξ (.379) (.55).868 ***.85 *** λ (.379) (.55) WTI F.99 *** (.73).34 (.747).644 *** (.7).68 (.5) VOLAT. F.94 *** (.733).67 (.747).3 (.46) -. (.54) MODEL VOLAT (.747) ** (.74) -.63 (.4) (.4) LIKELIHOOD -.9 (.74) (.745) -.5 (.54) -.35 (.46) ξ.64 *** (.596).484 *** (.655).645 *** (.89).554 *** (.878) *** (.545) *** (.539) -.75 *** (.73) *** (.6) ξ.66 (.746) (.745).66 ** (.9) (.5) -.63 ** (.738) -.9 (.743) -.99 (.5).44 (.54) NAPM -.49 (.747) -.67 (.746). * (.33).89 (.54) NAPM(-) (.747) -.65 (.746).63 ** (.7).365 (.53) S&P ** (.739) (.747).6 (.54) -.4 (.53) S&P 5 (-5).357 * (.74) -.45 (.747).4 (.53). (.54) 78

79 TABLE 6 CORRELATION AMONG HENRY HUB NATURAL GAS KALMAN FILER ESTIMATES OF MARKET PRICES OF RISK AND BUSINESS CYCLE RELATED VARIABLES Th abl shows h cofficins or corrlaion among h simad Hnry Hub naural gas mar prics of ris wih h Kalman filr mhod and svral businss cycl rlad variabls. Cofficins of corrlaion ar rpord wih hir sandard rrors in parnhsis. Th simad valus ar rpord wih * dnoing significanc a %, ** dnoing significanc a 5%, and *** dnoing significanc a %. HENRY HUB NATURAL GAS FRIST DATA SET HENRY HUB NATURAL GAS SECOND DATA SET λ ξ λ λ ξ λ λ ξ.34 ***.33 ** (.87) (.3) λ.34 ***.33 ** (.87) (.3) WTI F.456 *** -.34 *** *** (.774) (.774) (.6) (.895) VOLAT. F.379 *** ***.466 *** (.8) (.8) (.94) (.8) MODEL VOLAT..35 ***.3538 ***.385 ***.369 *** (.844) (.844) (.6) (.99) LIKELIHOOD *** (.867) (.867) (.55) (.898) ξ.676 *** (.643) (.643) (.6) (.6) *** *** *** (.64) (.64) (.6) (.685) ξ.437 *** **.77 ***.68 (.78) (.78) (.89) (.47) ***.383 *** (.867) (.867) (.73) (.98) NAPM *** -.43 ** (.867) (.867) (.895) (.8) NAPM(-) *** -.7 ** (.867) (.867) (.95) (.9) S&P *** -.83 ** *** -.39 ** S&P 5 (-5) (.756).49 *** (.755) (.756) ** (.755) (.36) -.65 ** (.) (.3) *** (.85) 79

80 TABLE 7 ESTIMATION RESULTS OF THE FACTOR MODELS WITH TIME-VARYING BUSINESS CYCLE RELATED AND CONSTANT MARKET PRICES OF RISK WTI CRUDE OIL Th abl shows h simaion rsuls of h modl wih im-varying mar prics of ris (MPR), dpnding on h modl long- and shor-rm facors, oghr wih hos obaind wih h sandard Schwarz and Smih () wo-facor modl wih consan ris prmia. Th abl shows h rsuls obaind wih boh h firs and h scond daa ss dscribd in scion 3. Sandard rrors ar in parnhss. Th simad valus ar rpord wih * dnoing significanc a %, ** dnoing significanc a 5%, and *** dnoing significanc a %. WTI Consan MPR Firs Daa S WTI Variabl MPR Scond Daa S WTI Consan MPR WTI Variabl MPR µ ξ.45 * (.7).44 (.68).8 *** (.334).868 ** (.4) κ.9748 ***.859 ***.54 ***.357 *** (.34) (.38) (.3) (.478) ξ.936 *** (.3).99 *** (.7).76 *** (.37).6 *** (.83).467 *** (.43).799 *** (.66).763 *** (.65).393 *** (.384) λ ξ.97 *** (.7).38 * (.99).669 *** (.334).4945 *** (.55) λ ξ *** - (.569) (.387) λ ξ -.83 *.9 * - (.484) (.6864) λ.453 (.346).38 *** (.3736) (.54).367 ** (.758) λ *** -. ** - (.84) (.473) λ -.8 *** (.647) (.7) ρ ξ.494 *** (.39).5775 *** (.949 ).445 (.3).7349 *** (.79) η.79 *** (.).78 *** (.).93 *** (.).93 *** (.) Log-L AIC SIC

81 TABLE 8 ESTIMATION RESULTS FOR HEATING OIL Th abl shows h simaion rsuls of h modl wih im-varying mar prics of ris (MPR), dpnding on h modl long- and shor-rm facors, oghr wih hos obaind wih h sandard Schwarz and Smih () wo-facor modl wih consan ris prmia. Th abl shows h rsuls obaind wih boh h firs and h scond daa ss dscribd in scion 3. Sandard rrors ar in parnhss. Th simad valus ar rpord wih * dnoing significanc a %, ** dnoing significanc a 5%, and *** dnoing significanc a %. Haing Oil Consan MPR Firs Daa S Haing Oil Variabl MPR Scond Daa S Haing Oil Cosnan MPR Haing Oil Variabl MPR µ ξ.659 *** (.3).3994 *** (.37).48 *** (.37).635 (.83) κ.85 *** (.9).3 *** (.5).78 *** (.58).46 *** (.) ξ.77 *** (.35).369 *** (.3).696 *** (.37).95 *** (.).3343 *** (.58).65 *** (.6).44 *** (.7).3758 *** (.) φ.9976 *** (.).9974 *** (.).997 *** (.).9974 *** (.) λ ξ.39 *** (.37).364 (.4).3 *** (.39).883 (.) λ ξ ***.6 *** - (.) (.) λ ξ *** *** - (.5) (.98) λ.6999 *** (.79).573 ** (.67).3546 *** (.496) *** (.) λ ***.46 *** - (.98) (.) λ ***.536 *** - (.378) (.) ρ ξ -.9 *** (.43) *** (.34).4488 *** (.37) *** (.) η.9 *** (.3).43 *** (.).9 *** (.).87 *** (.) Log-L AIC SIC

82 TABLE 9 ESTIMATION RESULTS FOR RBOB GASOLINE Th abl shows h simaion rsuls of h modl wih im-varying mar prics of ris (MPR), dpnding on h modl long- and shor-rm facors, oghr wih hos obaind wih h sandard Schwarz and Smih () wo-facor modl wih consan ris prmia. Th abl shows h rsuls obaind wih boh h firs and h scond daa ss dscribd in scion 3. Sandard rrors ar in parnhss. Th simad valus ar rpord wih * dnoing significanc a %, ** dnoing significanc a 5%, and *** dnoing significanc a %. RBOB Consan MPR Firs Daa S RBOB Variabl MPR Scond Daa S RBOB Cosnan MPR RBOB Variabl MPR µ ξ -.4 *** ().855 *** (.73).6 *** (.35).99 (.) κ 3.44 *** (.96).4 *** (.6).5 *** (.558).85 *** (.59) ξ.93 *** (.34).33 *** (.).877 *** (.45).458 *** (.).377 *** (.88).3 *** (.).384 *** (.84).567 *** (.) φ.9947 *** (.).994 *** (.).8 *** (.4). *** (.3) λ ξ *** (.4).6893 *** (.).3439 *** (.33).98 (.3) λ ξ *** (.4) (.396) λ ξ *** *** - (.) (.) λ *** (.88).64 * (.).379 *** (.548).9855 *** (.98) λ *** -.64 * - (.) (.9) λ -.39 *** (.48) (.778) ρ ξ.764 ** (.3) -.5 *** (.).7 *** (.44) *** (.) η.6 *** (.).5 *** (.).6 *** (.).59 *** (.) Log-L AIC SIC

83 TABLE ESTIMATION RESULTS FOR HENRY HUB NATURAL GAS Th abl shows h simaion rsuls of h modl wih im-varying mar prics of ris (MPR), dpnding on h modl long- and shor-rm facors, oghr wih hos obaind wih h sandard Schwarz and Smih () wo-facor modl wih consan ris prmia. Th abl shows h rsuls obaind wih boh h firs and h scond daa ss dscribd in scion 3. Sandard rrors ar in parnhss. Th simad valus ar rpord wih * dnoing significanc a %, ** dnoing significanc a 5%, and *** dnoing significanc a %. Hnry Hub Consan MPR Firs Daa S Hnry Hub Variabl MPR Scond Daa S Hnry Hub Cosnan MPR Hnry Hub Variabl MPR µ ξ *** (.37) *** (.6).79 *** (.5).84 *** (.) κ.858 *** (.).846 *** (.67).63 *** (.38).458 *** (.) ξ.55 *** (.).334 *** (.95).97 *** (.4).335 *** (.).5547 *** (.87).5475 *** (.85).4779 *** (.55).74 *** (.) φ.9957 *** (.).9997 *** (.).9999 *** (.).999 *** (.) λ ξ *** (.488).489 *** (.84).36 *** (.53) *** (.) λ ξ *** (.539) (.56) λ ξ ***.999 *** - (.768) (.) λ.8 (.994).466 * (.9) -.77 ** (.98) -.5 *** (.) λ (.466) (.) λ ***.45 *** - (.) (.) ρ ξ *** (.) -.75 *** (.677) -. (.47).966 *** (.) η.96 *** (.6).94 *** (.6).399 *** (.).383 *** (.) Log-L AIC SIC

84 TABLE AMERICAN OPTION VALUATION RESULTS ERROR DESCRIPTIVE STATISTICS Th abl prsns svral mrics, roo man squard rror (RMSE), prcnag roo man squard rror (PRMSE) and man absolu rror (MAE), o analyz h prdiciv powr abiliy of h modls undr sudy: h im-varying ris prmia modl and h sandard (wo-facor) modl wih consan ris prmia. Th daa s is comprisd of daily obsrvaions of WTI Amrican call and pu opions quod a NYMEX during h yars 6 o. For ach sris, w hav calculad h corrsponding saisic. Ths rsuls corrspond o h mdian valu of hs mulipl mans. Th oal numbr of obsrvaions is 34588, 365, 493 and for WTI crud oil, haing oil, RBOB gasolin and Hnry Hub naural gas rspcivly. PANEL A: WTI AMERICAN OPTIONS CONSTANT RISK PREMIA TIME-VARYING RISK PREMIA RMSE PRMSE MAE RMSE PRMSE MAE FIRST DATA SET SECOND DATA SET PANEL B: HEATING OIL AMERICAN OPTIONS CONSTANT RISK PREMIA TIME-VARYING RISK PREMIA RMSE PRMSE MAE RMSE PRMSE MAE FIRST DATA SET SECOND DATA SET

85 TABLE (CONT.) AMERICAN OPTION VALUATION RESULTS ERROR DESCRIPTIVE STATISTICS Th abl prsns svral mrics, roo man squard rror (RMSE), prcnag roo man squard rror (PRMSE) and man absolu rror (MAE), o analyz h prdiciv powr abiliy of h modls undr sudy: h im-varying ris prmia modl and h sandard (wo-facor) modl wih consan ris prmia. Th daa s is comprisd of daily obsrvaions of WTI Amrican call and pu opions quod a NYMEX during h yars 6 o. For ach sris, w hav calculad h corrsponding saisic. Ths rsuls corrspond o h mdian valu of hs mulipl mans. Th oal numbr of obsrvaions is 34588, 365, 493 and for WTI crud oil, haing oil, RBOB gasolin and Hnry Hub naural gas rspcivly. PANEL C: RBOB GASOLINE AMERICAN OPTIONS CONSTANT RISK PREMIA TIME-VARYING RISK PREMIA RMSE PRMSE MAE RMSE PRMSE MAE FIRST DATA SET SECOND DATA SET PANEL D: HENRY HUB NATURAL GAS AMERICAN OPTIONS CONSTANT RISK PREMIA TIME-VARYING RISK PREMIA RMSE PRMSE MAE RMSE PRMSE MAE FIRST DATA SET SECOND DATA SET

86 FIGURE TIME-SERIES EVOLUTION OF MAXIMUM LIKELIHOOD MARKET PRICES OF RISK FOR WTI CRUDE OIL WTI SHORT-TERM LAMBDA,5,5,5 -,5 Dc-88 Sp-9 Jun-94 Mar-97 Dc-99 Sp- May-5 Fb-8 Nov- - -,5 WTI LONG-TERM LAMBDA,5,5,5 -,5 Mar-86 Dc-88 Sp-9 Jun-94 Mar-97 Dc-99 Sp- May-5 Fb-8 Nov- - -,5 86

87 FIGURE TIME-SERIES EVOLUTION OF KALMAN FILTER MARKET PRICES OF RISK FOR WTI CRUDE OIL AND BUSINESS CYCLE RELATED VARIABLES WTI LAMBDAS AND AVERAGE WTI F,6 85,5,4 Long-Trm Lambda Shor-Trm Lambda 75,3 Avrag WTI F 65, 55, /3/997 4/7/998 6//999 9/4/ /9/ 4//4 8/5/5 //6 //8 6/7/9 45 -, 35 -, -,3 5 -,4 5 WTI LAMBDAS AND WTI F VOLATILITY,6 38%,5 36%,4 34%,3, 3% 3%, 8% /3/997 4/7/998 6//999 9/4/ /9/ 4//4 8/5/5 //6 //8 6/7/9 6% -, -, -,3 -,4 Long-Trm Lambda Shor-Trm Lambda WTI F Volailiy 4% % % LONG-TERM LAMBDA, AVERAGE WTI F AND LONG-TERM LAMBDA P-VALUE,6 85,5,4 75,3 65, 55, /3/997 4/7/998 6//999 9/4/ /9/ 4//4 8/5/5 //6 //8 6/7/9 45 -, 35 -, -,3 Long-Trm Lambda Long-Trm Lambda p-valu Avrag WTI F 5 -,4 5 87

88 FIGURE TIME-SERIES EVOLUTION OF KALMAN FILTER MARKET PRICES OF RISK FOR WTI CRUDE OIL AND BUSINESS CYCLE RELATED VARIABLES (CONT.) SHORT-TERM LAMBDA, AVERAGE WTI F AND SHORT-TERM LAMBDA P-VALUE,5 85,4 75,3 65,, /3/997 4/7/998 6//999 9/4/ /9/ 4//4 8/5/5 //6 //8 6/7/9 -, 35 -, -,3 -,4 Shor-Trm Lambda Shor-Trm Lambda p-valu Avrag WTI F 5 5 WTI LAMBDAS AND AVERAGE LONG- AND SHORT-TERM FACTORS,6 4,5,5 4,4 3,5,3 3,,5, /3/997 4/7/998 6//999 9/4/ /9/ 4//4 8/5/5 //6 //8 6/7/9,5 -, -,,5 -,3 Long-Trm Lambda Shor-Trm Lambda -,4 Avrag Shor-Trm Facor Avrag Long-Trm Facor -,5 WTI LAMBDAS AND LONG- AND SHORT-TERM FACRORS STANDARD DEVIATION,6,5,5,45,4,4,3,35,,3,,5 /3/997 4/7/998 6//999 9/4/ /9/ 4//4 8/5/5 //6 //8 6/7/9, -,,5 -,, -,3 -,4 Long-Trm Lambda Shor-Trm Lambda Shor-Trm Facor Sand. Dv. Long-Trm Facor Sand. Dv.,5 88

89 FIGURE TIME-SERIES EVOLUTION OF KALMAN FILTER MARKET PRICES OF RISK FOR WTI CRUDE OIL AND BUSINESS CYCLE RELATED VARIABLES (CONT.) WTI LAMBDAS AND AVERAGE NAPM,6 56,5 54,4 5,3, 5, /3/997 4/7/998 6//999 9/4/ /9/ 4//4 8/5/5 //6 //8 6/7/9 -, 44 -, -,3 -,4 Long-Trm Lambda Shor-Trm Lambda Avrag NAPM 4 4 WTI LAMBDAS AND EXPANSION INDICATOR,6,,5,,4,3,8,,6, /3/997 4/7/998 6//999 9/4/ /9/ 4//4 8/5/5 //6 //8 6/7/9,4 -,, -, -,3 -,4 Long-Trm Lambda Shor-Trm Lambda Expansion Indicaor - -, WTI LAMBDAS AND AVERAGE S&P 5,6.35,,5.3,,4.5,,3,,.,.5,., /3/997 4/7/998 6//999 9/4/ /9/ 4//4 8/5/5 //6 //8 6/7/9.5, -, -, -,3 -,4 Long-Trm Lambda Shor-Trm Lambda Avrag S&P 5., 95, 9, 89

90 CHAPTER 3: THE STOCHASTIC SEASONAL BEHAVIOR OF ENERGY COMMODITY CONVENIENCE YIELS 3. INTRODUCTION In consumpion commodiis (commodiis ha ar consumpion asss rahr han invsmn asss) h bnfi from holding h physical ass n of sorag cos is somims rfrrd o as h convninc yild providd by h commodiy (s for xampl Hull, 3). In ohr words, if w dno by F and S h fuurs and spo prics rspcivly, in h cas of consumpion commodiis w do no ncssarily hav qualiy in F S ( r+ u) T (whr r and u rprsn h ris fr ra and sorag coss rspcivly and T is h im o mauriy), bcaus usrs of a consumpion commodiy may fl ha ownrship of h physical commodiy provids bnfis ha ar no obaind by holdrs of fuurs conracs. For xampl, an oil rfinr is unlily o rgard a fuurs conrac on crud oil as quivaln o crud oil hld in invnory. Th crud oil in invnory can b an inpu o h rfining procss whras a fuurs conrac canno b usd for his purpos. In gnral, ownrship of h physical ass nabls a manufacurr o p a producion procss running and prhaps profi from mporary local shorags. A fuurs conrac dos no do h sam (s for xampl Brnnan and Schwarz, 985). Thrfor h convninc yild n of sorag coss, dnod by δ, is dfind so ha: F δ T S r T. Prvious sudis hav considrd h convninc yild as a drminisic funcion of im, such as Brnnan and Schwarz (985), or as a sochasic procss, such as Gibson and Schwarz (99) and Schwarz (987). Spcifically, Gibson and Schwarz (99) allow for sochasic convninc yild of crud oil in ordr o dvlop a wo-facor oil coningn claims pric modl. Morovr, Gibson and Schwarz (99) show ha 9

91 convninc yilds xhibi man rvrsion, which is consisn wih h hory of sorag (s, for xampl, Brnnan, 985) in which i is sablishd an invrs rlaionship bwn h n convninc yild and h lvl of invnoris. Schwarz (997) prsns and mpirically compars svral facor modls in which h convninc yild is assumd o b a sochasic facor. Hilliard and Ris (998) and Milrsn and Schwarz (998) us modls wih sochasic convninc yild o valu commodiy drivaivs (fuurs and opions). Mor rcnly, Casassus and Collin- Dufrsn (5) characriz a hr-facor modl, maximal in a sns of Dai and Singlon (), of commodiy spo prics, convninc yilds and inrs ras, which nss many xising spcificaions. Wi and Zhu (6) invsiga h mpirical propris of convninc yilds in h US naural gas mar, finding ha convninc yilds ar highly variabl and conomically significan, wih hir variabiliy dpnding on spo pric lvl, spo pric variabiliy and h variabiliy of laggd convninc yilds. In spi of hr hav bn many paprs analyzing h sasonal bhavior of som commodiy prics (Lucia and Schwarz,, Sornsn,, Manoliu and Tompaidis,, Garcia al.,, among ohrs), considrably lss anion has bn paid o h sasonal bhavior of convninc yilds. Basd on h finding of sasonaliy in h convninc yild mad by Fama and Frnch (987), Amin al. (994) propos a onfacor modl for h spo pric wih drminisic sasonal convninc yild. Mor rcnly, Borovova and Gman (6) prsn a wo-facor modl in which h firs facor is h avrag forward pric, insad of h commodiy spo pric, and h scond facor is h sochasic convninc yild. Ths auhors allow for a drminisic sasonal prmium wihin h convninc yild. 9

92 In his chapr, w go furhr by prsning a facor modl in which h (sochasic) convninc yild xhibis sochasic sasonaliy. Spcifically, w show ha h fourfacor modl prsnd by Garcia al. (), wih wo long- and shor-rm facors and wo addiional rigonomric sasonal facors, can gnra sochasic sasonal convninc yilds. An xprssion for h insananous convninc yild wihin his modl is obaind, showing ha h insananous convninc yild xhibis man rvrsion and sochasic sasonaliy. Morovr, i is found a π/ lag in h convninc yild sasonaliy wih rspc o spo pric sasonaliy. Basd on his vidnc, h nx sp is o prsn a horical modl o characriz h commodiy convninc yild dynamics which is cohrn wih h prvious findings. Spcifically, h modl as ino accoun man rvrsion and sochasic sasonal ffcs in h convninc yild. Th modl is simad using daa from a variy of nrgy commodiy fuurs prics: crud oil, haing oil, gasolin and naural gas. W also show ha commodiy pric sasonaliy can b br simad hrough convninc yilds rahr han hrough fuurs prics. Th rason is ha fuurs prics ar drivn for many hings, such as supply, dmand, poliical aspcs, spculaion, wahr condiions, c. Thrfor, somims i may b difficul o xrac h sasonal componn from fuurs prics. Howvr, as w will show in Scion, h convninc yild is simad hough a raio of wo fuurs prics, so many of hs non-sasonal facors nd o disappar, faciliaing h simaion of h sasonal componn. Th rmaindr of his chapr is organizd as follows. Scion prsns h daa and som prliminary findings rgarding sasonaliy in convninc yilds. W show ha convninc yilds show man rvrsion and sochasic sasonaliy, using daa from haing oil, gasolin and naural gas fuurs mars. In scion 3 w prsn h fourfacor modl accouning for sochasic sasonaliy in commodiis and h xprssion 9

93 for h insananous convninc yild drivd from his four-facor modl. In scion 3 w also discuss h propris of h modl simad convninc yilds for h four commodiis undr sudy, showing ha in fac hy xhibi man rvrsion, sochasic sasonaliy and a π/ lag wih rspc o spo pric sasonaliy. Basd on his mpirical vidnc, in scion 4 i is proposd and simad a facor modl characrizing h commodiy convninc yild dynamics, aing ino accoun man rvrsion and sochasic sasonal ffcs in h convninc yild. Finally, Scion 5 concluds wih a summary and discussion. 3. DATA AND PRELIMINARY FINDINGS In his scion, w prsn a daa dscripion of h fuurs prics for h four commodiis usd in h chapr, i.. WTI crud oil, haing oil, RBOB gasolin and Hnry Hub naural gas. Morovr, i is dscribd h procdur prsnd by Gibson and Schwarz (99) in ordr o obain h convninc yild daa. Th scion concluds analyzing h main mpirically obsrvd characrisics of h convninc yild daa. Daa dscripion Fuurs Prics Th daa s usd in his chapr consiss of wly obsrvaions of WTI (ligh sw) crud oil, haing oil, unladd gasolin (RBOB) and naural gas fuurs prics radd a NYMEX, during h priod 9/7/999 o 7/4/ (65 wly obsrvaions). Acually, hr ar fuurs bing radd on NYMEX wih mauriis from on monh up o svn yars for WTI crud oil, from on o ighn monhs for haing oil, from on o wlv monhs for RBOB gasolin and from on monh o six yars in h cas of 93

94 Hnry Hub naural gas. Howvr, liquidiy is scarc for h fuurs wih longr mauriis, mosly in h cas of gasolin. In h simaion of h modls prsnd blow a rprsnaiv s of mauriis has bn usd for ach commodiy. Thus, in h cas of WTI crud oil, h daa s is comprisd of conracs F, F4, F7, F, F4, F7, F, F4 and F7, whr F is h conrac for h monh closs o mauriy, F is h conrac for h scond-closs monh o mauriy, and so on. In h cas of haing oil, h daa s is comprisd of conracs F, F3, F5, F7, F, F, F4, F6 and F8. In h cas of RBOB gasolin, h daa s conains conracs F, F3, F5, F7, F9 and F. Finally, in h cas of Hnry Hub naural gas, h daa s conains conracs F, F5, F9, F4, F8, F, F7, F3 and F35. Th main dscripiv saisics of hs variabls ar conaind in Tabl. Convninc Yild Th simaion of h convninc yild sris is carrid ou using h procdur dfind in Gibson and Schwarz (99). Basd on h convninc yild dfiniion, F δ T S r T, w hav: F( S, x _ monhs) S xp{( r _ δ _ x monhs x monhs ) ( x /)} whr r x_monhs is h inrs ra of a zro coupon bond wih x monhs o mauriy and δ x_monhs is h convninc yild in x monhs for his commodiy. Analogously: F( S, x _ monh) S xp{( r ) (( + ) /)} + x+ monh δ x+ monh x whr r x+_monh is h inrs ra of a zro coupon bond wih x+ monhs o mauriy and δ x+_monh is h convninc yild in x+ monhs for his commodiy. From hs xprssions w hav: F( S, x+ _ monhs) xp F( S, x _ monh) {[ ( x+ ) r x r ) ( ( x+ ) δ x δ )] (/)} x _ monhs x _ monh x+ _ monhs x _ monh + () 94

95 On h ohr hand, by dfiniion: xp { ( + ) r x r ) (/)} xp{ r (/)} x x+ _ monhs x _ monh implici _ x _ o _ x+ _ monhs whr r implici_x_o_x+_monhs is h implici inrs ra from x monhs o x+ monhs, and xp { ( + ) δ x δ ) (/)} xp{ δ (/)} x x+ _ monhs x _ monh implici _ x _ o _ x+ _ monhs whr δ implici_x_o_x+_monhs is h implici convninc yild from x monhs o x+ monhs. Taing ino accoun hs dfiniions, xprssion () can b wrin as: F( S, x+ _ monhs) xp{( F( S, x _ monh) or quivalnly: r implici _ x _ o _ x+ _ monhs δ implici _ x _ o _ x+ _ monhs ) (/)} δ implici _ x _ o _ x+ _ monhs r implici _ x _ o _ x+ _ monhs F( S, x+ _ monhs) ln F( S, x _ monh) δ implici_x_o_x+_monhs can b usd as a proxy for h insananous convninc yild δ. Following his procdur w hav simad h convninc yild sris for h four commodiy fuurs prics sris dscribd abov. Th main dscripiv saisics of hs convninc yild sris ar summarizd in Tabl. In Figur w plo h im sris voluion of som of h simad convninc yilds for h four commodiis undr sudy. I can b apprciad in h figurs h man-rvring and sasonaliy ffcs, alhough h parn is lss clar in h cas of WTI crud oil. Ths issus ar furhr discussd blow. Prliminary Findings Prvious sudis found vidnc of man rvrsion in h convninc yild dynamics. From convninc yild daa obaind as in h prvious sub-scion, Gibson and Schwarz (99) show a srong man rvring ndncy in h convninc yild, which is consisn wih h hory of sorag (s, for xampl, Brnnan,985) in 95

96 which i is sablishd an invrs rlaionship bwn h lvl of invnoris and h rlaiv n convninc yild. Fama and Frnch (987) poind ou ha sasonals in producion or dmand can gnra sasonals in invnoris. Undr h hory of sorag, invnory sasonals gnra sasonals in h marginal convninc yild. Following his rasoning, Borovova and Gman (6) prsn a modl allowing for a drminisic sasonal prmium wihin h convninc yild. Hr, using h simad convninc yild sris from h prvious sub-scion for h four commodiis undr sudy, w will invsiga h xisnc of man rvring and sasonal ffcs in h convninc yild. Tabl 3 prsns h rsuls of h uni roo ss for WTI, haing oil, gasolin and Hnry Hub naural gas convninc yild sris. Th mpirical vidnc from prvious sudis of man rvrsion is confirmd in h prsn wor using h sandard Augmnd Dicy-Fullr s. Spcifically, w ar abl o rjc h null hypohsis of a uni roo in all h cass, wih h only xcpion of WTI crud oil (mosly as w go furhr in im). Ths rsuls ar cohrn wih h im voluion of h sris shown in Figur. Th prsnc of sasonaliy in h simad convninc yild sris is assssd hrough h Kursal-Wallis s. To prform h s w hav compud monhly avrags from h wly simad convninc yild sris. Th null hypohsis of h s is ha hr ar no monhly sasonal ffcs. Th rsuls of h s ar shown in Tabl 4. Th rsuls indica h rjcion of h null hypohsis of no sasonal ffcs in all cass, xcp for WTI crud oil. Th sasonal ffcs ar vn clarr in h cass of RBOB gasolin and Hnry Hub naural gas convninc yild sris. Ths sasonal ffcs ar vidn in Figur. 96

97 As xplaind abov, Borovova and Gman (6) allow for a drminisic sasonal prmium wihin h convninc yild. Howvr, i may b possibl ha sasonal ffcs in h convninc yild ar sochasic rahr han drminisic. Garcia al. () prsn a modl for h sochasic bhavior of commodiy prics allowing for sochasic sasonaliy in commodiy prics. Following his ida, w will chc for h xisnc of sochasic sasonal ffcs in h convninc yild sris. Th RBOB gasolin 34 convninc yild spcrum and is firs diffrncs ar dpicd in Figur, assuming ha h sris follows an AR() procss wih yarly sasonaliy, following h procdur dscribd in Garcia al. (). As xplaind by Garcia al. (), sharp spis in h spcrum ar lily o indica a drminisic cyclical componn, whil broad pas ofn indica a nondrminisic sasonal componn. Th asrics (*) shown in h Figur dno harmonic poins, calculad as π/ (pas) and π(-)/ (roughs), whr,, 3, 4, 5 and 6. Looing a Figur, i sms ha, mor or lss, h spcrum xhibis broad pas and houghs, suggsing ha sasonaliy in convninc yilds is sochasic rahr han drminisic. Howvr, hs rsuls mus b an wih car, as aliasing ffcs and simaion rrors can confus drminisic and sochasic parns. In Figur 3 w plo h forward curvs for h simad convninc yild sris on a rprsnaiv da (July 4, ) in h cas of Hnry Hub naural gas prics 35. Looing a h figur i can b apprciad ha boh fuurs and convninc yild sris prsn an vidn sasonal parn. Morovr, i is inrsing o obsrv how h sasonal pics in h convninc yild sris ar dlayd hr monhs compard o hos obsrvd in h fuurs sris. 34 Th par for h rs of commodiis is vry similar. 35 For shor only h figur for Hnry Hub naural gas is prsnd. Th parn is similar in h rs of h cass. 97

98 3.3 THE PRICE MODEL In his scion, w show ha a four-facor modl for h sochasic bhavior of commodiy prics, wih wo long- and shor-rm facors and wo addiional sasonal facors, can accommoda som of h mos imporan mpirically obsrvd characrisics of commodiy convninc yilds dscribd in Scion, such as man rvrsion, sochasic sasonaliy and a hr monhs dlay in h convninc yild sasonaliy wih rspc o h spo pric sasonaliy. Gnral Considraions Basd on h convninc yild dfiniion, F δ T S r T,aing ino accoun ha h spo pric (S ) and h convninc yild (δ ) ar sochasic if T >, h prvious quaion can b xprssd as an SDE in h following way: ds * ( rδ ) d+ dw S () which is h classical dfiniion of h convninc yild undr h Q-masur (s, for xampl, Schwarz, 997, or Casassus and Collin-Dufrsn, 5). Undr h P- masur h SDE can b xprssd in h following way: ds ( µ δ ) d+ dw S (3) To characriz h convninc yild dynamics, l X log( ) b h log of h spo S pric. If w assum a linar modl, li in h sudis lisd abov, is gnral dynamics is givn by: dx S xpφ ( m+ AX ) d ( + CX ) + RdW (4) As i shall b provn in appndix B, h modl abov has an xplici (uniqu) soluion (no ha i is nough o solv for X ): 98

99 X A As X + + mds RdW As. s No ha S ( + ) CX xpφ and w would li o sablish a sochasic diffrnial quaion for S. Taing diffrnials and using Io s lmma: ds xp ' ' ( φ + CX ) CdX + xp( φ + CX ) C( dx )( dx ) C S CdX + C( dx )( dx ) ' C Using h fac ha ddw dd and ( dw )( dw ) Id ' w obain: ds S C d ( m+ AX ) d+ CRdW + CRR' C' and finally: ds S C m+ RR' C' + AX d+ CRdW (5) µ If m is dfind as m, which is ncssary o h modl b maximal (or globally M idnifiabl), w g ha Cm µ and from (4): δ C RR' C' + AX (6) Thrfor, wih (6) w can obain h convninc yild dynamics from h modl facors dynamics. Thorical Modl Hr w ar going o prsn a modl o characriz h commodiy prics dynamics which as ino accoun h sasonal ffcs and which is cohrn wih h prvious findings. 99

100 In h four-facor modl in Garcia al. (), h log spo pric (X ) is h sum of hr sochasic facors, a long-rm componn (ξ ), a shor-rm componn ( ) and a sasonal componn (α ). X ξ + + α (7) Th fourh sochasic facor is h ohr sasonal facor (α * ), which complmns α. Th SDEs of hs facors ar: dξ µ ξ d+ ξ (8) dwξ d κ d+ dw (9) dα πϕα d+ dw () * α α * d α πϕαd+ α dw * α () Equaions (8) and (9) ar idnical o quaions () and (), rspcivly, in Schwarz and Smih (). This modl is maximal in a sns of Dai and Singlon (). Evn mor his modl is Dai-Singlon A (4) as can b sn in Appndix C. To s his, no ha in h canonical form givn by xprssions (4): a A α πϕ πϕ and h modl is globally idnifiabl. Th García al. () modl imposs h rsricion a and α >. And, as a rsricion of a globally idnifiabl modl imposing concr valus and inrvals o h paramrs, i is also globally idnifiabl and maximal.

101 As sad abov, in Garcia al. () modl w hav 36 : πϕ πϕ A, ( ) C, µ m And: * * ' α α α α α ξ α α α α α ξ ξ ξ ξ ρ ρ ρ ρ ρ RR Undr his modl, using xprssion (6), h convninc yild can b wrin in h following way: * * * ) ( πϕα ρ ρ ρ ρ ρ δ α α α α α α ξ α α ξ ξ ξ α ξ () As can b apprciad in h prvious xprssion, δ dos no dpnd on h longrm facor, ξ, nihr h sasonal facor, α. Howvr, i dpnds on h sum of facor variancs, h shor-rm facor,, (ims h spd of man rvrsion) and h sasonal facor ha complmns h on dfind in h spo pric, α *, (ims h sasonal frquncy). In ohr words, h convninc yild is h sum of a consan rm plus a shor-rm facor plus a sasonal facor. Th fac ha δ is saionary (dos no dpnd on h long-rm facor and dpnds on h shor-rm on) in h prvious xprssion is cohrn wih h fac ha h wo facor modl dfind in Schwarz-Smih () is quivaln o h on dfind in Schwarz (997) in which δ follows an Ornsin-Uhlnbc procss, which is a manrvring on. I is clar, hrfor, ha δ should dpnds on insad of ξ. I is also clar ha h dpndncy should b modulad by bcaus h highr h man- 36 As can b sn in García al. (), ρ αα* and α α*.

102 rvring spd, h highr h bnfi of holding h physical ass. Thin, for xampl, in a shorag, if h pric com bac o is quilibrium lvl in a shor-rm priod (high man-rvring spd) h ownr of h physical ass can sll h commodiy and buy i again in a shor-priod (consqunly wih a low cos) ging h bnfi. In h ohr hand, if h pric dlay in coming bac o hir quilibrium lvl (low man-rvring spd), h ownr of h physical ass has o buy h commodiy again a a highr pric or h is no going o b abl o p h producion procss running. Taing ino accoun xprssion (), and ging around h sochasic par of i, i is clar ha: ds S d δ. As dα * πϕα, i is no supprssiv ha * δ dpnds on α d insad of α, ha implis a π/ lag in h convninc yild sasonaliy wih rspc spo pric sasonaliy. As in h prvious cas, h dpndncy should b modulad by ϕ bcaus h highr h sasonal frquncy, h highr h bnfi of holding h physical ass. Th sam can b said abou h sum of facor variancs, h highr h varianc h highr is h convninc yild (in absolu valu) bcaus h bnfi of holding h physical ass is highr. I is inrsing o no ha h convninc yild dpnds on h sum of h facor variancs insad of h spo pric varianc, ha is, dpnds on h whol sysm varianc and no only h varianc of h facors which compos h spo pric. Finally, i is worh noing ha xprssion () for h convninc yild is cohrn wih h mpirical facs obsrvd for h convninc yild in Scion.: man rvrsion, (sochasic) sasonaliy and a hr monhs (π/) lag in h convninc yild sasonaliy wih rspc o h spo pric on.

103 Esimaion Rsuls Hr, w prsn h rsuls of h simaion of h four-facor modl for h four commodiis prsnd abov. Th modls prsnd in Scion 3. wr simad using h Kalman filr mhodology, which is brifly dscribd in Appndix A. Th rsuls ar shown in Tabl 5. I is found ha in all cass h sasonal facor volailiy ( α ) is significanly diffrn from zro and h sasonal priod (ϕ) is mor or lss on yar, implying ha sasonaliy in all four commodiy prics is sochasic wih a priod of on yar, which is consisn wih h rsuls obaind by Garcia al. (). Morovr, h spd of adjusmn () is highly significan, implying, man rvrsion in commodiy prics, which is cohrn wih h rsuls obaind by Schwarz (997). I is also found ha h long-rm rnd (µ ξ ) is posiiv and significanly diffrn from zro in all cass, implying long-rm growh in commodiy prics, spcially in h cass of RBOB gasolin, haing oil and WTI crud oil. I is also inrsing o no ha shor-rm volailiy ( ) is highr han long-rm volailiy ( ξ ) in all cass, which is cohrn wih h rsuls found by Schwarz (997) and Garcia al. (). Concrning h mar prics if ris, i is found ha h ris prmium associa wih h long-rm facor (λ ξ ) is significanly diffrn from zro in all cass, whras h ris prmium associad wih h shor-rm on (λ ) is no, suggsing ha h ris associad wih h long-rm facor is mor difficul o divrsify han h ris associad wih h shor-rm on. Morovr, h mar prics of ris associad wih h ral and complx pars of h sasonal componn (λ α and λ α* rspcivly) ar no significanly diffrn from zro in mos of h cass, suggsing ha h ris associad o h sasonal componn can b divrsifid in mos of h cass. 3

104 Howvr, from h poin of viw of h goal of his chapr i is inrsing o analyz h influnc of h simad paramrs for ach commodiy on is convninc yild. As sad abov, h spd of adjusmn () is rlaivly high and significanly diffrn from zro in all cass, implying high convninc yild, spcially in h cas of RBOB gasolin, followd by Hnry Hub naural gas. I is also found ha h highs valu of h sasonal priod (ϕ) is found in h cas of Hnry Hub naural gas, followd by RBOB gasolin, haing oil and WTI crud oil, implying highr convninc yild for Hnry Hub naural gas and lowr for WTI crud oil (in absolu valu). Finally, from h simad vals shown in Tabl 5 i is asy o compu h rm in parnhsis in xprssion (), involving h sandard dviaions and h corrlaions among h modl facors. I is found ha h highs valu for his rm, and hrfor h highs absolu valu for h convninc yild, corrsponds o Hnry Hub naural gas (wih a valu of.336), followd by RBOB gasolin (.96), WTI (.8) and haing oil (.993). Thrfor, w can conclud ha h highs simad valus of h convninc yild ar found in h cass of Hnry Hub naural gas and RBOB gasolin. Finally, Figur 4 shows h im sris voluion of h simad sasonal componns and h simad convninc yild, boh obaind wih h four-facor modl. I can b apprciad h hr monhs dlay of convninc yilds (grn lin) sasonaliy wih rspc o h commodiy pric sasonaliy (blu lin), alhough h parn is lss clar in h cas of WTI crud oil. Th sasonal parn is lss clar in h cas of WTI, which is cohrn wih h rsuls found in Scion. 3.4 THE CONVENIENCE YIELD MODEL Hr w prsn a modl for h sochasic bhavior of convninc yilds. This modl will accoun for sochasic sasonaliy. Morovr, i could b h cas ha in crain 4

105 commodiis li crud oil, in which hr wr no obsrv sasonaliy, i is possibl ha hr is a wa sasonal componn, which is hiddn by ohr facors, and his sasonal componn can b simad hrough h convninc yild. Spcifically, h proposd modl for h convninc yild is h hr-facor modl by Garcia al. (). This modl will allow us o sima crud oil sasonaliy hrough is convninc yild and o compar spo pric and convninc yild sasonaliy. Thorical Modl Hr w prsn a modl o characriz h commodiy convninc yild dynamics which as ino accoun h sasonal ffcs and which is cohrn wih h prvious findings. Th proposd modl for h sochasic bhavior of convninc yilds is h hr-facor modl in Garcia al. () 37. In his hr-facor modl h spo convninc yild (X ) is h sum of a drminisic long-rm facor (ξ ) and wo sochasic facors 38, a shor-rm componn ( ) and a sasonal componn (α ): X ξ + + α (3) Th hird sochasic facor is h ohr sasonal facor (α * ), which complmns α. Th SDEs of hs facors ar: dξ µ ξ d (4) d κ d+ dw (5) dα πϕα d+ dw (6) * α α 37 A four facor modl li h on prsnd in scion 3 has bn simad for h convninc yild, howvr h sochasic paramrs rlad wih h long-rm facor wr no significan, which confirms prvious vidnc rgarding h srong man-rvring bhavior of convninc yild sris. 38 I should b nod ha in h original hr-facor modl by Garcia al. () h log-spo pric is h sum of hr sochasic facors. Howvr, hr w modl dircly h convninc yild pric insad of is log, givn ha h convninc yild can a ngaiv valus. 5

106 * d α πϕαd+ α dw * α (7) As shown in h cas of h four-facor modl, his modl is maximal in h sns of Dai and Singlon (). Evn mor his modl is Dai-Singlon A (3), as can b sn in Appndix C. Esimaion Rsuls Th hr facor modl prsnd abov has bn simad hough h Kalman filr mhodology, using h convninc yild daa simad in Scion. Th rsuls of h modl simaion ar shown in Tabl 6. Th rsuls indica a high dgr of man rvrsion (high valu of κ), mosly in h cas of Hnry Hub naural gas, which is cohrn wih h prliminary rsuls obaind in Scion. Howvr, h mos imporan issu in Tabl 6, from h poin of viw of his chapr goal, is h fac ha h sandard dviaion of h sasonal facor ( α ) is significanly diffrn from zro for all four commodiis. This rsul is suggsing ha convninc yilds no only show sasonaliy, bu his sasonaliy is sochasic rahr han drminisic. Morovr, h valus of h sandard dviaion of h sasonal facor obaind in Tabl 6 for h convninc yild sris ar considrabl highr han hos obaind in Tabl 5 for h commodiy pric sris. This rsul is suggsing ha sasonaliy is vn clarr in h convninc yild sris han in h commodiy pric ons. I is inrsing o obsrv h high valus of α obaind in h cass of RBOB gasolin and Hnry Hub naural gas convninc yild sris, which is cohrn wih rsuls shown in Figur. I is also vry inrsing o obsrv ha h WTI convninc yild sris (and h WTI fuurs prics sris in Tabl 5) also shows 6

107 vidnc of sochasic sasonaliy, alhough h ss in Scion did no dcd vidnc of sasonaliy in h cas of WTI crud oil convninc yild sris. Looing a xprssion () i is clar ha h shor-rm componn in h convninc yild is qual o h shor-rm componn in h spo pric muliplid by h spd of adjusmn in h four-facor modl (κ). Givn ha h simad valus of κ in h fourfacor modl (Tabl 5) ar no vry far from on, h sandard dviaions of h shorrm componns in h convninc yild and h spo pric sris should b similar. This is h rsul found in h cass of RBOB gasolin and haing oil. Th valus of h sandard dviaions of h shor-rm componn in h WTI and Hnry Hub naural gas convninc yild sris (Tabl 6) ar highr han h corrsponding valus in h spo pric sris (Tabl 5) du o h high variabiliy found in hs convninc yild sris, as can b apprciad in Figur. Morovr, from xprssion () w can conclud ha h sasonal componn in h convninc yild is qual (in absolu valu) o h complmnary sasonal componn in h spo pric muliplid by πϕ. Givn ha h simad valus of h sasonal priod (ϕ) in Tabl 5 ar vry clos o on, h sandard dviaion of h spo pric complmnary facor 39 should b similar o h sandard dviaion of h convninc yild dividd by π. In h cas of WTI crud oil h sandard dviaion of h complmnary sasonal facor in h spo pric modl is.6, whras h sandard dviaion of h sasonal facor in h convninc yild modl (dividd by π) is.844. Th figurs in h cas of haing oil ar.8 and.5 rspcivly. In h cas of RBOB gasolin hs figurs ar.45 and.76 rspcivly. Finally, h figurs in h cas of Hnry Hub naural gas ar.385 and.6 rspcivly. 39 Rmmbr ha in h four-facor modl α α*. 7

108 This rsul can b corroborad looing a Figur 4. In his Figur h simad convninc yild (grn lin) shows a vry similar par o h complmnary sasonal facor (α * ) in h four-facor modl (rd lin), alhough as bfor h parn is lss clar in h cas of WTI crud oil. Tabl 7 prsns a summary of h influnc of h sasonal componns on h commodiy pric (four-facor modl for commodiy spo prics) and on h convninc yild (hr-facor modl for convninc yilds). Spcifically h abl shows h avrag wighs of h sasonal facors (α and α * ) in h log-pric of h commodiy (Panl A) and in h convninc yild (Panl B) 4. I is qui sriing o obsrv how h wighs of h sasonal componns ar considrabl highr in h modl for h convninc yild (Panl B). In boh panls h highs wighs ar achivd in h cass of RBOB, haing oil and Hnry Hub naural gas. Finally, i is also inrsing o obsrv h rlaiv high wigh of h sasonal parn on h convninc yild in h cas of WTI crud oil, suggsing ha in commodiis li crud oil, in which hr wr no obsrv sasonaliy, ha hr is a wa sasonal componn and his sasonal componn can b simad hrough h convninc yild. In summary, w can conclud ha h simad convninc yild sris show vidnc of sochasic sasonaliy and ha his sasonaliy is vn clarr han in h cas of commodiy spo prics sris. This rsul is suggsing ha commodiy pric sasonaliy can b br simad hrough convninc yilds rahr han hrough fuurs prics. Th rason is ha fuurs prics ar drivn for many hings, such as supply, dmand, poliical aspcs, spculaion, wahr condiions, c. Thrfor, somims i may b difficul o xrac h sasonal componn from fuurs prics. 4 Th wigh of h sum of h wo sasonal facors (α and α * ) ovr h convninc yild pric in Panl B of Tabl 7 is grar han %. This is du o h fac ha in h hr-facor modl h convninc yild is h sum of a long-rm (ξ, drminisic) componn, a shor-rm (, sochasic) componn and a sasonal (α, sochasic) componn. Th ohr sasonal componn, α *, dos no influnc h convninc yild pric. 8

109 Howvr, as shown in Scion, h convninc yild is simad hough a raio of wo fuurs prics, so many of hs non-sasonal facors nd o disappar, faciliaing h simaion of h sasonal componn. 3.5 CONCLUSIONS This chapr focuss on commodiy convninc yilds. Convninc yilds for four nrgy commodiis (WTI crud oil, haing oil, RBOB gasolin and Hnry Hub naural gas) ar simad using h procdur dfind in Gibson and Schwarz (99), finding, as in prvious sudis, ha convninc yilds xhibi sasonaliy and man rvrsion. Basd on his mpirical vidnc, w prsn a facor modl in which h convninc yild xhibis man rvrsion and sochasic sasonaliy. Spcifically, w show ha h four-facor modl prsnd by Garcia al. (), wih wo long- and shor-rm facors and wo addiional rigonomric sasonal facors, can gnra sochasic sasonal man-rvring convninc yilds. Morovr, i is found a π/ lag in h convninc yild sasonaliy wih rspc o spo pric sasonaliy. Basd on his vidnc, h nx sp is o prsn a horical modl o characriz h commodiy convninc yild dynamics which is cohrn wih h prvious findings. Spcifically, h modl as ino accoun man rvrsion and sochasic sasonal ffcs in h convninc yild. W also show ha commodiy pric sasonaliy can b br simad hrough convninc yilds rahr han hrough fuurs prics. Th rason is ha fuurs prics ar drivn for many hings, such as supply, dmand, poliical aspcs, spculaion, wahr condiions, c. Thrfor, somims i may b difficul o xrac h sasonal componn from fuurs prics. Howvr, h convninc yild is simad hough a raio of wo fuurs prics, so many of hs 9

110 non-sasonal facors nd o disappar, faciliaing h simaion of h sasonal componn.

111 APPENDIX A. ESTIMATION METHODOLOGY Th Kalman filr chniqu is a rcursiv mhodology ha simas h unobsrvabl im sris, h sa variabls or h facors (Z ) basd on an obsrvabl im sris (Y ) ha dpnds on hs sa variabls. Th masurmn quaion accouns for h rlaionship bwn h obsrvabl im sris and h sa variabls: Z M d Y η + +,, N, (A) whr h n x n n Z M d Y R R R,,,, h is h numbr of sa variabls, or facors, in h modl, and n R η is a vcor of srially uncorrlad Gaussian disurbancs wih zro man and covarianc marix H. In h simaion procdur, a discr im vrsion of his quaion is ncssary; in h cas of h join modl wih a common long-rm rnd for h hr commodiis, his quaion is givn by h following xprssions: 3 3 ln ln ln ln ln ln Tn T Tn T Tn T F F F F F F Y M M M, ) ( ) ( ) ( ) ( ) ( ) ( 3 3 n n n T A T A T A T A T A T A d M M M, n n n T T T T T T M 3 3 M M M M M M M M M M M M and i T F is h pric of a fuurs conrac for h commodiy i (i,,3) wih mauriy a im T + radd a im. In principl, i would b possibl o us a diffrn numbr of fuurs conracs for ach commodiy; howvr, in his wor, w considr i mor suiabl o us h sam numbr ( n ) of fuurs conracs for all commodiis. Th ransiion quaion accouns for h voluion of h sa variabls: Z T c Z ψ + +,, N, (A) whr h h x h h and T c R R R ψ, is a vcor of srially uncorrlad Gaussian disurbancs wih zro man and covarianc marix Q.

112 In h cas of h join modl wih a common long-rm rnd for h hr commodiis, h discr im vrsion of his quaion, which is ndd in h simaion procdur, is givn by h following xprssions: Z 3 ξ, c µ ξ, T 3 and ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ) ) /( ( ) / ( ) / ( ) / ( ) / ( ) ) /( ( ) / ( ) / ( ) / ( ) / ( ) ) /( ( ) / ( ) / ( ) / ( ) ( Var ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ψ Hr, Y is h condiional xpcaion of Y, and Ξ is h covarianc marix of Y condiional on all informaion availabl a im. Afr omiing unssnial consans, h log-lilihood funcion can b xprssd as Ξ Ξ Y Y Y Y l ) ( )' ( ln. (A3)

113 APPENDIX B. STOCHASTIC DIFERENTIAL EQUATIONS (SDE) INTEGRATION Mos of h modls proposd in h liraur assum ha h ris-nural dynamics of a commodiy pric (or is log) is givn by a linar sochasic diffrnial sysm: dx Y ( b+ AX ) cx d+ RdW whr Y is h commodiy pric (or is log), b, A, R and c ar drminisic paramrs 4 n n x n n indpndn of ( b R, A, R R, c R ) and W is a n-dimnsional canonical Brownian moion (i.. all componns uncorrlad and is varianc qual o uniy) undr h ris-nural masur. L us s ha h soluion of ha problm is 4 : X A A s A s X + + bds RdW s (B) In ordr o proof i, w shall apply h gnral rul for h drivaion of h produc of sochasic componns (Osndal, 99): dx A ( d ) As X + bds + + As RdW s + As d X + bds + ( ) + + A As As d d X bds RdWs A As RdW s + I is asy o show ha: As X bds+ d + As RdW s A bd+ A RdW 4 Again no ha R dos no nd o b compud. 4 Evn in h cas ha b, A and R wr funcion of, if A and A ds commu, h soluion of ha problm is (B). s 3

114 Th firs diffrnial only has lmns of yp d, hnc h produc of h firs diffrnial ims h scond diffrnial is zro. Thus: + Consqunly w obain xprssion (B): A A [ bd+ RdW] A X d+ bd RdW A As As A dx A d X + bds RdWs + + X A A s X + + bds A s RdW. s I is asy o prov ha h soluion is uniqu (Osndal, 99). 4

115 APPENDIX C. CANONICAL REPRESENTATION Inroducion In his appndix, w shall s how our modls can b rlad o Dai-Singlon A (n) class, wih h imporan disincion of allowing complx ignvalus. Afrwards, w shall show global idnificaion propris. Gnral sup L Z log( ) b h log of h spo pric. If w assum a linar modl, is ral S dynamics is givn by: ( m+ AX ) d ( + CX ) whras is ris nural dynamics is givn by: dx + RdW S xpφ (F) X givn dx S xp X givn ( mλ+ AX ) ( φ + CX ) d+ RdW (FN) whr R is full ran lowr riangular (w shall xamin his assumpion lar). W would li o now how his gnral sup can b rducd o a modl which is maximal, i.. canno b rducd o an quivaln modl wih lss sas and paramrs (anohr way o s his is saying ha has h maximum numbr of idnificabl paramrs). W shall concnra firs in (F). Firs of all, (s for xampl Sonag 99), a modl has h minimal numbr of sas if and only if is obsrvabl and conrolabl, i.. C CA ran CA n n (obsrvabiliy n condiion) and ran( R AR A R A R) n (conrolabiliy condiion). As h lar is always saisfid if R is full ran, w jus impos h formr. Morovr, in h conx of 5

116 sochasic sysms, conrolabiliy plays a small rol as i mans ha som sas ar unaffcd by nois so whhr hy ar obsrvaionally quivaln o ohr sysm dpnds only on iniial sas. Invarian ransformaions Following Dai and Singlon, w allow for h following ransformaions ~. Affin ransformaions of sas: X v+ GX whr G is nonsingular and v is an arbirary vcor. No h imporan rol of consans φ and φ. If hy whr no prsn and oupu quaion wr would hav o accomplish Cv. CX, hn v could no b arbirary bu insad ~. Roaions of brownian moions. W UW whr UU T I as Brownian moion is unobsrvd. No ha hs ransformaions prsrv obsrvabiliy and ran of R. Rlaionship wih A (n) W shall firs show now how o rla our modl o Dai-Singlon A (n) class, i.. a ~ dy KY d+σdw sysm li: (DS) ~ S xp( δ + CY) whr R I, C ( ) and K is lowr riangular wih all hir diagonal lmns sricly posiiv, i.. K >. This mans svral rsricions wihin h sysm:. Th dynamics marix -K is full ran and all hir ignvalus ar ral and ngaiv.. Nois marix is also full ran. All hs propris ar prsrvd hrough invarian ransformaions, so w would hav o impos hm on our sysm. Bu w hav complx ignvalus, so w hav o us a diffrn, alhough similar, canonical form. To sum up, w rplac Dai-Singlon rsricions wih ohrs, so our approachs ar similar bu no dircly comparabl. ii 6

117 Firs canonical form If all ignvalus ar diffrn hn h pair (F) can b rducd o: (F) ~ dx S xp ( m~ ~ ~ + AX ) ~ ~ ( CX ) ~ ~ d+ RdW, whr. A ~ is diagonal (ral only if hr ar no complx ignvalus) C.. ( ) 3. R ~ is lowr riangular and all is diagonal lmns ar sricly possiiv. 4. ~ m m wih m R Morovr, if w sar wih a canonical form (F) h sysm is obsrvabl and conrolabl (hrfor has h minimal numbr possibl of sas). Proof If all h ingnvalus ar diffrn, hn A is diagonalizabl. Thrfor, changing h bas, w hav a rprsnaion whr A ~ is diagonal. W shall s now ha all lmns in C ar no null. ~ L A diag( d d ). By h obsrvabiliy condiion, h marix n ~ C ~~ CA ~~ CA n is full ran. c cd Bu his marix quals n cd c c c d d n n n n c c c n d n d n. Should any of h c i b null, hn is full column would b null and hrfor h sysm would no b obsrvabl. This also provs ha, saring from canonical form (F), h sysm is obsrvabl. 7

118 As a rsul, w can dfin h ransformaion L diag,,. Undr his chang c c n of variabl, ( ) ~ ~ C and A is diagonal. Using a suiabl orogonal ransformaion of h nois, w can also impos h condiions on R ~ via a Cholsi dcomposiion (hus proving also ha h sysm is conrolabl, du o h fac ha nois marix is full ran). Now for h form of m ~. W dfin h nw sa as φ + µ / d µ / d µ n / d n ~ ~ X X + + µ / d n n. Clarly i vrifis h condiions. Complx ignvalus I is now im o considr complx ignvalus. Th rsuls ar ssnially h sam, bu h canonical form is slighly diffrn. Boh ar, howvr, prfcly quivaln. W nd a fw prvious lmmas. Lmma If A is a x ral marix wih complx ignvalus such ha h pair ( A, C) is obsrvabl hn ± iϕ and C ia a x ral marix + iϕ. A is diagonalizabl and, if Λ iϕ and H i i hn H ΛH ϕ ϕ ϕ. Thr xis a ral marix T such ha T AT ϕ and CT ( ) 8

119 9 Proof A has wo all ignvalus disinc hrfor is diagonalizabl. As i is ral, is ignvalus ar conjuga. W jus hav o do h produc. + Λ i i i i i i H H ϕ ϕ. I quals i i i i i i ϕ ϕ ϕ ϕ ϕ ϕ In ordr o proof par, l us g bac o h original A. I has wo ignvcors, bu is a ral marix. Thrfor, if v is an ingnvcor associad o an ignvalu λ, hn Av λv. Taing conjugas, v v A λ. Bu A is ral, hrfor A A so v v A λ. I mans ha v is h ingnvcor associad o h ohr ignvalu. L v v v v T b h marix of ignvcors. Thn AT T i i + ϕ ϕ. L H T T. W now hn, T AT ϕ ϕ. W shall proof now ha T is ral. [ ] [ ] [ ] [ ] x Im R Im R R v v v v v iv v v iv iv v v i i v v v v Finally, l ( ) c,c C. As ( ) C A, is obsrvabl, > + c c.w dfin + c c c c c c H W now ( ) CH + c c c c c c c c c c H H ϕ ϕ ϕ ϕ c c c c c c c c c c c c c c ϕ ϕ ϕ ϕ ϕ ϕ So, dfining T H T w g h rsul.

120 Lmma L C x n n x n R, A R b ral marics whr all ignvalus of A ar diffrn. Thr xis a ral marix T such ha T A AT whr ig( ) λ R or ig( A ) { + iϕ, + iϕ } Proof A j j ` j j A A r CT ( C C r ) j j j L λ,,λp b h ral ignvalus and µ, µ,, µ q, µ q b h complx ons. L v,,v p and w, w,., w, w q q b h corrsponding ignvcors. W dfin h subspacs V Sp( ) and W Sp( w, w ) i v i i i i. i V is dfind by a ral vcor (and hus has wi wi a ral basis) and Wi Sp wi + wi, hrfor has also a ral basis. L T b h i basis of all h subspacs oghr, which is a ral marix. Clarly V W p q R + p V W q and all subspacs ar A invarian. Using h abov ral basis, w can pariion T AT A.. A A r whr A A or i V i A A hus vrifying h hsis. i W i W ar now rady o sa h complx canonical form. Scond canonical form If all ignvalus ar diffrn hn (F) can b rducd o ( ) ~ dx m~ ~ + AX (F) ~ S xp( CX ) ~ ~ d+ RdW, whr all marics ar ral and:

121 . A A.. A ~ and ihr A r ϕ A i λ i R or A i ϕ. C ( C ) C r ach corrsponding o i A and C if R i A λ or C ( ) ohrwis. 3. R ~ is lowr riangular and all is diagonal lmns ar sricly possiiv. 4. ~ m m wih m R Proof Combining h wo prvious lmmas, i is obvious ha hr is a ral marix ha ransforms A and C ino h prvious forms. By procding as in h ohr hird rducd form, w obain h rs of h rsul. i i i Maximaliy In ordr o show ha h modl s is maximal w s ha h modl is globally idnificabl, as in gnral h lar implis h formr if all paramrs ar admisibl. To s his, rmmbr ha in a globally idnifiabl modl, diffrn paramrs giv diffrn ralizaions. Suppos ha a modl has n paramrs and is no maximal bu admis a rprsnaion wih <n paramrs. By rdfining h paramr spac (undr som condiions) i mans ha h las paramrs ar funcions of h firs, formally ( φ ϕ( φ) ) θ,. Bu, for a valu ( ) * * * * * φ, w can a a diffrn valu (, ϕ ) φ, ϕ( φ ) φ hus obaining a diffrn admisibl valu. Th only way o avoid conradicion would b ha

122 * * * * (, ϕ ) ( φ, ϕ( φ ) φ achiv h sam ralizaion, bu his is imposibl sinc h modl is globally idnificabl. W hus hav o conclud ha h modl is no maximal. W shall firs proof h vrsion whr spo prics ar obsrvabl and hn xplain why ris prmia can also b idnifid. Proposiion If S is obsrvabl, modl (F3) is globally idnificabl (incluiding h iniial sa X ) Proof L Z log( ). W assum ha w can obsrv h man and varianc of Z a any S momn in im. If h modl has complx ignvalus, w prform h ransformaion i i for ach A hus convring C ino ( ) i and maing A diagonal. If afr his ransformaion h modl is globally idnificabl, so is h original modl. W now ha ( Z ) var Cvc ( xp( Au) xp( Au) ) vc( RR' ) d C' (s García al., ). I is h sum of xponncials of ignvalus of A and in all sums appars ii d T d ii ( RR' ) ii. As ( RR ' ) ii is no null and d ii is h doubl of an ingnvalu all ignvalus ar idnifid and so is A. No ha his argumn os vn valid if is an ngnvalu, as w would only b abl o idnify n valus, which mans ha h ohr is. Thrfor, no rsricions xiss in h ignvalus of A so any maximal modl nds all. Bu, as Cvc ( xp( Au) xp( Au) ) vc( RR' ) d C', if A is idnifid, so is RR ' (in h complx cas is HRR ' H ' whr H is h chang of variabl, bu w can g h

123 original by muliplying by boh invrss). W jus hav o xrac from h ingrals (as all ingrals ar posiiv). Thrfor RR ' and A ar idnifid. W hav now wo cass. L us firs assum A is NOT full ran. Thn, d A As m m [ Y ] C X + ds ( ) X + E Y. dn d dn So w hav h qualiy [ ] n X + m + X + X. As all his funcions ar linarly indpndn, i mans ha all hir cofficins ar univocaly dfind. Now, w shall assum ha A is full ran. W dfin A φ, A X X and ( C ) C. Th sysm is sill obsrvabl, by consrucion and w ar bac o h C r prvious cas. Ris prmia I is now im o considr whhr ris prmia can b idnifid. If w sar wih modl ( ) ~ dx m~ ~ + AX (F) ~ S xp( CX ) ~ dx m~ ~ λ+ AX (FN) ~ S xp( CX ) ~ ~ d+ RdW ( ) ~ ~ d+ RdW is ris nural vrsion is givn by: W shall now assum ha all fuurs ar obsrvabl and show ha h sysm, wih h ris nural dynamics is also globally idnificabl. Proposiion Q In h abov condiions, if F E [ S / I ] globally idnificabl. is obsrvabl, hn modl (F3N) is, T + T 3

124 Proof Firs, if F, T is obsrvabl, maing T i mans ha S is obsrvabl. So all paramrs apar from (possibly) ris prmia ar idnifid. Howvr, Z + T T AT As φ + C X + λds+ T As RdWs+. If w a xpcaions [ ] Q wih rspc firs o h firs masur and afr o h scond E E [ / I ] Q ingral disapars and [ ] Q Thrfor w ar lf wih E E [ Z I ], h Io E X / I X only dpnds on idnifiabl paramrs. d dn [ / ] φ+ ( m λ) λ λn in h singular A cas and wihou h rm in h nonsingular cas. Anyway, indpndn funcions which mans idnifiabl paramrs. REFERENCES Borovova, S. & Gman, H. (6), Sasonal and sochasic ffcs in commodiy forward curvs Rviw of Drivaivs Rsarch, 9, Brnnan, Michal J., 958, Th supply of sorag, Amrican Economic Rviw 48, 5 7. Brnnan, Michal J., and Eduardo S. Schwarz, 985, Evaluaing naural rsourc invsmns, Journal of Businss 58, Casassus, J. and Collin-Dufrsn, P., 5, Sochasic Convninc Yild Implid from Commodiy Fuurs and Inrs Ras, Th Journal of Financ, Vol. LX, No. 5, Dai, Qiang, and Knnh J. Singlon,, Spcificaion analysis of affin rm srucur modls, Journal of Financ 55,

125 Fama, Eugn F., and Knnh R. Frnch, 987, Commodiy fuurs prics: Som vidnc on forcas powr, prmiums and h hory of sorag, Journal of Businss 6, García A., Población J., Srna, G.,. Th sochasic sasonal bhavior of naural gas prics. Europan Financial Managmn 8, Gibson, Rajna, and Eduardo S. Schwarz, 99, Sochasic convninc yild and h pricing of oil coningn claims, Journal of Financ 45, Hilliard, Jimmy E., and Jorg Ris, 998, Valuaion of commodiy fuurs and opions undr sochasic convninc yilds, inrs ras, and jump diffusions in h spo, Journal of Financial and Quaniaiv Analysis 33, Hull, John, 3, Opions, Opions, Fuurs and Ohr Drivaivs, Fifh Ediion (Prnic Hall, Nw Jrsy). Lucia, J. & Schwarz, E.S. (), Elcriciy Prics and Powr drivaivs: Evidnc from h Nordic Powr Exchang Rviw of Drivaiv Rsarch, 5, 5-5. Manoliu, M. & Tompaidis, S. (), Enrgy Fuurs Prics: Trm Srucur Modls wih Kalman Filr Esimaion, Applid Mahmaical Financ, 9,.43. Milrsn, K. and Schwarz, E., (998), Pricing of Opions on Commodiy Fuurs wih Sochasic Trm Srucurs of Convninc Yilds and Inrs Ras. Th Journal of Financial and Quaniaiv Analysis, Vol. 33, No., pp Osndal B. 99. Sochasic Diffrnial Equaions. An Inroducion wih Applicaions, 3rd d. Springr-Vrlag: Brlin Hidlbrg. 5

126 Schwarz, E.S. Th sochasic bhavior of commodiy prics: Implicaion for valuaion and hdging. Th Journal of Financ, 997, 5, Schwarz, E.S., Smih, J.E. Shor-rm variaions and long-rm dynamics in commodiy prics. Managmn Scinc,, 46(7), Sonag, E. D. (99). Mahmaical Conrol Thory: Drminisic Fini Dimnsional Sysms. Scond Ediion, Springr, Nw Yor, 998. Sornsn, C. Modling sasonaliy in agriculural commodiy fuurs. Th Journal of Fuurs Mars,,, Todorova, M.I. (4), Modling Enrgy Commodiy Fuurs: Is Sasonaliy Par of i?, Journal of Alrnaiv Invsmns, 7, -3. Wi, S. Z. C., and Z. Zhu. (6). Commodiy convninc yild and ris prmium drminaion: Th cas of h U.S. naural gas mar. Enrgy Economics, 8,

127 TABLES AND FIGURES TABLE DESCRIPTIVE STATISTICS. FUTURES PRICES Th abl shows h man and volailiy of h four commodiy fuurs prics sris. Th sampl priod is 9/7/999 o 7/4/ (65 wly obsrvaions). F is h fuurs conrac closs o mauriy, F is h conrac scond-closs o mauriy and so on. WTI Crud Oil Haing Oil Gasolin Hnry Hub Man Volailiy Man Volailiy Man Volailiy Man Volailiy F % F % F % F % F % F % F % F % F % F % F % F % F % F % F % F % F % F % F % F % F % F % F % F % F % F % F % F % F % F % F % F % F % TABLE DESCRIPTIVE STATISTICS. CONVENIENCE YIELD Th abl shows h man and volailiy of h commodiy convninc yild simad prics sris for h four commodiis undr sudy. Th sampl priod is 9/7/999 o 7/4/ (65 wly obsrvaions). δ x_x+ dnos h implici convninc yild from x monh o x+ monhs. WTI Crud Oil Haing Oil Gasolin Hnry Hub Man Sand. Dv. Man Sand. Dv. Man Sand.Dv. Man Sand. Dv. δ _ -..9 δ _..8 δ _.8.4 δ _ δ 4 _ δ 3 _ δ 3 _ δ 5 _ δ 7 _ 8.8. δ 5 _ 6.6. δ 5 _ δ 9 _.3.5 δ _.8.9 δ 7 _ δ 7 _ δ 4 _ δ 4 _ δ _.7.8 δ 9 _.7.35 δ 8 _ δ 7 _ δ _ δ _ δ _ δ _.6.6 δ 4 _ δ 7 _ δ 4 _ δ 6 _ δ 3 _ δ 7 _

128 TABLE 3 UNIT ROOT TEST Th abl shows h saisic of h Augmnd Dicy-Fullr (ADF) s. Th MacKinnon criical valus for h rjcion of h null hypohsis of a uni roo ss ar (%), (5%) and (%). δ _ δ _ 3 δ 3 _ 4 δ 5 _ 6 δ 8 _ 9 δ 9_ δ 3 _ 4 δ 5 _ 6 δ 9 _ WTI Haing Oil RBOB Hnry Hub TABLE 4 SEASONALITY TEST Th abl shows h saisic of h Krusal-Wallis s for h prsnc of sasonal ffcs in h simad convninc yild sris. Th s saisic is disribud, undr h null hypohsis of no sasonal ffcs, as a wih dgrs of frdom. Th criical valu for h rjcion of h null hypohsis a 99% is δ _ δ _ 3 δ 3 _ 4 δ 4 _ 5 δ 5 _ 6 δ 6 _ 7 δ 7 _ 8 δ 8 _ 9 δ 9 _ WTI H. Oil RBOB H. Hub

129 TABLE 5 ESTIMATION RESULTS. FOUR-FACTOR MODEL Th abl prsns h rsuls for h four-facor modl applid o h four commodiis undr sudy: WTI crud oil, haing oil, RBOB gasolin and Hnry Hub naural gas. Sandard rrors ar in parnhss. Th simad valus ar rpord wih * dnoing significanc a %, ** dnoing significanc a 5%, and *** dnoing significanc a %. WTI Haing Oil RBOB Hnry Hub µ ξ.3 *** (.49).58 *** (.395).84 ** (.474).655 ** (.3) κ.5 *** (.).3 *** (.43).9649 *** (.69).33 *** (.6) ϕ.9566 *** (.5).9978 *** (.).9 *** (.9).88 *** (.) ξ.66 *** (.45).573 *** (.44).885 *** (.58). *** (.49).75 *** (.9).458 *** (.7).35 *** (.9).4367 *** (.65) α.6 *** (.5).8 *** (.6).45 *** (.).385 *** (.) ρ ξ.58 (.49).3 *** (.49).573 (.7).7 (.63) ρ ξα *** (.79) -.6 ** (.65) -.5 * (.556) -.89 (.845) ρ ξα* *** (.695) * (.693).353 *** (.6) -.67 (.797) ρ α.373 *** (.759).994 (.685).76 *** (.655).58 *** (.8) ρ α*.366 *** (.77).957 *** (.7) *** (.549).345 *** (.74) λ ξ.37 *** (.49).53 *** (.396).55 *** (.49).5 *** (.33) λ.53 (.69) -. (.69) -.65 (.85) (.) λ α -.7 (.9) -.4 (.3) -.6 (.). (.5) λ α* -.5 * (.9) -.77 ** (.3).7 (.3) -.38 (.6) η. *** (.).94 *** (.).7 *** (.).376 *** (.3) Log-lilihood AIC SIC

130 TABLE 6 ESTIMATION RESULTS. THREE-FACTOR MODEL FOR THE CONVENIENCE YIELD Th abl prsns h rsuls for h hr-facor modl applid o h four commodiy convninc yild sris undr sudy: WTI crud oil, haing oil, RBOB gasolin and Hnry Hub naural gas. Sandard rrors ar in parnhss. Th simad valus ar rpord wih * dnoing significanc a %, ** dnoing significanc a 5%, and *** dnoing significanc a %. WTI Haing Oil RBOB Hnry Hub µ ξ.788 *** (.83) -.69 (.8).34 (.99) *** (.747) κ.75 *** (.).9639 *** (.94).8 *** (.55) *** (.3) ϕ.796 *** (.).4 *** (.3).8 *** (.3). *** (.).65 *** (.).77 *** (.68).3 *** (.483).847 *** (.3) α.53 *** (.).75 *** (.54).4773 *** (.47).377 *** (.3) ρ α.777 *** (.).6338 *** (.86).438 *** (.63) -.34 *** (.3) ρ α*.375 *** (.).87 (.55).49 *** (.6).639 *** (.3) λ.887 *** (.89) -.65 (.798).94 (.983) -.5 (.6875) λ α.6 *** (.63).66 *** (.4) -.66 (.5) -.38 (.86) λ α* -.9 (.64) -.47 ** (.35).3 (.437).375 (.9) η.57 *** (.).779 *** (.8).7 *** (.3).377 *** (.3) Log-lilihood AIC SIC

131 TABLE 7 WEIGHTS OF SEASONAL COMPONENTS Th abl prsns h avrag wighs of h simad sasonal facors in h spo pric (fourfacor modl for spo commodiy prics) and in h convninc yild (hr-facor for convninc yilds), for h four commodiis undr sudy: WTI crud oil, haing oil, RBOB gasolin and Hnry Hub. PANEL A: FOUR FACTOR MODEL, COMMODITY SPOT PRICES α /log(s) ( α + α* )/log(s) Hnry Hub.8866% 5.799% Haing Oil.57%.65% RBOB.895%.689% WTI.86%.439% PANEL B: THREE-FACTOR MODEL, CONVENIENCE YIELDS α /log(s) ( α + α* )/log(s) Hnry Hub % 94.58% Haing Oil 46.66% 8.67% RBOB % 8.68% WTI 9.859%.83% 3

132 FIGURE TIME SERIES EVOLUTION OF ESTIMATED CONVENIENCE YIELDS WTI, CY WTI CY_9_ HO CY HO CY_9_ 3

133 FIGURE TIME SERIES EVOLUTION OF ESTMATED CONVENIENCE YIELDS (CONT.) RBOB CY RBOB CY_9_ HH CY HH CY_9_ 33

134 FIGURE RBOBO GASOLINE CONVENIENCE YIELD SPECTRUM 34

135 FIGURE 3 FORWARD CURVES FUTURES AND CONVENIENCE YIELD Forward Curvs Fuurs and Convninc Yild HH 7/4/ ,8,6,4, -, -,4 -,6 Fuurs Convninc Yild 35

136 FIGURE 4 COMMODITY SEASONAL COMPONENTS AND CONVENIENCE YIELD..5 Sasonal componns of HH 4 Facor Convninc yild 4 Facor complmnary Sasonal componns of N 4 Facor Convninc yild 4 Facor complmnary Sasonal componns of RBOB 4 Facor Convninc yild 4 Facor complmnary x -3 5 Sasonal componns of WTI 4 Facor Convninc yild 4 Facor complmnary