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