Long Term Spread Option Valuation and Hedging


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1 Long Term Spread Opion Valuaion and Hedging M.A.H. Demper, Elena Medova and Ke Tang Cenre for Financial Reearch, Judge Buine School, Univeriy of Cambridge, Trumpingon Sree, Cambridge CB 1AG & Cambridge Syem Aociae Limied 57 Porugal Place, Cambridge CB5 8AF Abrac Thi paper inveigae he valuaion and hedging of pread opion on wo commodiy price which in he long run are coinegraed. For long erm opion pricing he pread beween he wo price hould herefore be modelled direcly. Thi approach offer ignifican advanage relaive o he radiional mulifacor pread opion pricing model ince he correlaion beween wo ae reurn i nooriouly hard o model. In hi paper, we propoe one and wo facor model for po pread procee under boh he rineural and mare meaure. We develop pricing and hedging formulae for opion on po and fuure pread. Two example of pread opion in energy mare he crac pread beween heaing oil and WTI crude oil and he locaion pread beween Bren blend and WTI crude oil are analyzed o illurae he reul. JEL claificaion: G1 Key word: pread opion, coinegraion, meanreverion, opion pricing, energy mare Correponding auhor, addre: Judge Buine School, Univeriy of Cambridge, Trumpingon Sree, Cambridge CB 1AG, Tel: +44 (0) , Fax: +44 (0) , 1
2 1. Inroducion A pread opion i an opion wrien on he difference (pread) of wo underlying ae price S 1 and S repecively. We conider European opion wih payoff he greaer or leer of S (T)S 1 (T)K and 0 a mauriy T and rie price K and focu on pread in he commodiy (epecially energy) mare (boh for po and fuure). In hee mare pread opion are uually baed on difference beween price of he ame commodiy a wo differen locaion (locaion pread) or ime (calendar pread), beween he price of inpu and oupu (producion pread) or beween he price of differen grade of he ame commodiy (qualiy pread). The New Yor Mercanile Exchange (NYMEX) alo offer radable opion on he heaing oil/ crude oil or gaoline/crude oil pread (crac pread). I i naural o model he pread by modelling each ae eparaely. Margrabe (1978) wa he fir o rea pread opion and gave an analyical oluion for rie price zero (he exchange opion). Wilcox (1990) and Carmona and Durrleman (003) ue Bachelier (1900) formula o analyically price pread opion auming he underlying price follow arihmeic Brownian moion. I i more difficul o value a pread opion if he wo underlying price follow geomeric Brownian moion. Variou numerical echnique have been propoed o price uch an opion. Rubinein (1991) value he pread opion in erm of a double inegral. Demper and Hong (000) ue he fa Fourier ranform o evaluae hi inegral numerically. Carmona and Durrleman (003) offer a good review of pread opion pricing. Many reearcher have modelled a pread opion by modelling he wo underlying ae price in he ri neural meaure a 1 ds = rs d+ σ S dw ds = rs d+ σ S dw W1 W = ρ. Ed d d (1) The correlaion ρ play a ubanial rôle in valuing a pread opion; rading a pread 1 Boldface i ued hroughou o denoe random eniie here condiional on S 1 and S having realized value S 1 and S a ime which i uppreed for impliciy of noaion.
3 opion i equivalen o rading he correlaion beween he wo ae reurn. However, Kir (1995), Mbanefo (1997) and Alexander (1999) have uggeed ha reurn correlaion i very volaile in energy mare. Thu auming a conan correlaion beween wo ae a in (1) i inappropriae for modelling. Bu here i anoher longer erm relaionhip beween wo ae price ermed coinegraion which ha been lile udied by ae pricing reearcher. If a coinegraion relaionhip exi beween wo ae price he pread hould be modelled direcly for long erm opion pricing. Thi i he opic of hi paper which i organized a follow. Secion give a brief review of price coinegraion and he principal aiical e for coinegraion and for he mean reverion of pread. Secion 3 propoe one and wo facor model of he underlying pread proce in he rineural and mare meaure and how how o calibrae hee model. Secion 4 preen opion pricing formulae for opion on po and fuure pread. Secion 5 provide wo example in energy mare which illurae he heoreical wor and Secion 6 conclude.. Coinegraed price and mean reverion of he pread The value of a pread opion i deermined by he dynamic relaionhip beween wo underlying ae price and he correlaion of he correponding reurn ime erie i commonly underood and widely ued. Coinegraion i a mehod for reaing he longrun equilibrium relaionhip beween wo ae price generaed by mare force and behavioural rule. Engle and Granger (1987) formalized he idea of inegraed variable haring an equilibrium relaion which urn ou o be eiher aionary or o have a lower degree of inegraion han he original erie. They ued he erm coinegraion o ignify comovemen among rending variable which could be exploied o e for he exience of equilibrium relaionhip wihin he framewor of fully dynamic mare. 3
4 The parameer in equaion (1) can be calibraed uing reurn daa bu he coinegraion relaionhip mu be inveigaed wih price daa (Hamilon, 1994). In general, he reurn correlaion i imporan for hor erm price relaionhip and he price coinegraion for heir long run counerpar. If wo ae price are coinegraed (1) i only ueful for hor erm valuaion even when he correlaion beween heir reurn i nown exacly. Since we wih o rea long erm pread opion pricing we hall inveigae he coinegraion (long erm equilibrium) relaionhip beween ae price. Fir we briefly explain he economic reaon why uch a longrun equilibrium exi beween price of he ame commodiy a wo differen locaion, price of inpu and oupu and price of differen grade of he ame commodiy. The law of one price (or purchaing power pariy) implie ha coinegraion exi for price of he ame commodiy a differen locaion. Due o mare fricion (rading co, hipping co, ec.) he ame good may have differen price bu he mipricing canno go beyond a hrehold wihou allowing mare arbirage (Samuelon, 1964). Inpu (raw maerial) and oupu (produc) price hould alo be coinegraed becaue hey direcly deermine upply and demand for manufacuring firm. There alo exi an equilibrium involving a hrehold beween he price of a commodiy of differen grade ince hey are ubiue for each oher. If uch longerm equilibria hold for hee hree pair of price coinegraion relaionhip hould be deeced in he empirical daa. Duan and Plia (003) model he logprice raher han he price coinegraion beween wo US mare indice. In equiy mare inveor are concerned wih index reurn raher han level o hi may be a good choice. However from he economic argumen i follow ha he pread beween wo po commodiy price reflec deviaion from a general (poibly growh) equilibrium, e.g. he profi of producing (producion pread), hipping (locaion pread) or wiching (qualiy pread). Conien wih Jure and Yang (006) we model here price coinegraion. Alhough he model of hi paper are applicable o calendar pread we will no rea hem here. 4
5 In empirical analyi economi uually ue equaion () and (3) o decribe he coinegraion relaionhip: S = c + ds + ε () 1 ε  ε = ωε + u, (3) 11 where S 1 and S are he wo ae price and u i a Gauian diurbance. Engle and Granger (1987) demonrae ha a coinegraion model i he ame a an error correcion model, i.e. he error erm ε in () mu be meanrevering (3). Thu a imple way o e he coinegraion relaionhip i o e wheher ω i a ignificanly negaive number in equaion (3), i.e. wheher he pread proce i meanrevering (DicyFuller, 1979). Equaion () can be een a he dynamic equilibrium of an economic yem. When S 1 and S deviae from he longrun equilibrium relaionhip hey rever bac o i in he fuure. For boh locaion and qualiy pread S 1 and S hould ideally follow he ame rend, i.e. d hould be equal o 1. However for producion pread uch a he par pread (he pread beween he elecriciy price and he ga price) d may no be exacly 1. Uually 3/4 of a ga conrac i equivalen o 1 elecriciy conrac o ha inveor rade a 1 elecriciy / 3/4 ga pread which repreen he profi of elecriciy plan (Carmona and Durrleman, 003). Since gaoline and heaing oil are coinegraed ubiue, he d value could be 1 for boh he heaing oil/crude oil pread and he heaing oil/gaoline pread (Girma and Paulon, 1999). For our hree pread of inere locaion, producion and qualiy d hould be 1. Leing x denoe he pread beween wo coinegraed po price S 1 and S i follow from () and (3) in hi cae ha x x 1 =c  c 1 ω(c 1 x 1 )+ u, (4) i.e., he pread of he wo underlying ae i meanrevering. No maer wha he naure of he underlying S 1 and S procee he pread beween hem can behave quie differenly from heir individual behaviour. Thi ugge modelling he pread 5
6 direcly uing an OrneinUhlenbec proce for longerm opion evaluaion becaue he coinegraion relaionhip ha ubanial influence in he long run. Such an approach give a lea hree advanage over alernaive a i: 1) avoid modelling he correlaion beween he wo ae reurn, ) cache he longrun equilibrium relaionhip beween he wo ae price and 3) yield an analyical oluion for pread opion. For example, Jure and Yang (006) employ an OrneinUhlenbec proce o model he pread beween Siamee win equiie 3 and preen an opimal aeallocaion raegy for pread holder..1 Coinegraion e To e he coinegraion of wo ae price, we fir need o e wheher each generae a uni roo ime erie. In an efficien mare ae price ingly will uually generae uni roo (independen incremen) ime erie becaue he curren price hould no provide forecaing power for fuure price. If wo ae price procee are uni roo bu he pread proce i no, here exi a coinegraion relaionhip beween he price and he pread will no deviae ouide economically deermined bound. The augmened DiceyFuller (ADF) e may be ued o chec for uni roo in ae price ime erie. The ADF aiic ue an ordinary lea quare (OLS) auo regreion S S = δ + δ S + δ ( S S ) + η (5) i+ 1 i i 1 i= 1 p o e for uni roo, where S i he ae price a ime, δ i, i=0,,p, are conan and η i a Gauian diurbance. If he coefficien δ 1 i negaive and exceed he criical value in Fuller (1976) hen he nullhypohei ha he erie ha no uni roo i rejeced. We can ue an exenion of (4) correponding o (5) o e he coinegraion 3 For deail on Siamee win ee Froo and Dabora (1999). 6
7 relaionhip: S = c + d S + ε 1 ε  ε = χ + χ ε + χ ( ε ε ) + u i+ 1 i i 1 i= 1 p (6) When χ 1 i ignificanly negaive he hypohei ha coinegraion exi beween he wo underlying ae price procee S 1 and S i acceped (Hamilon, 1994).. Mare meaure mean reverion e In Secion 3 we will ee ha he meanrevering propery of he po pread can be deeced by examining he meanreverion of fuure pread wih a conan ime o mauriy. Empirically we eimae F ( + Δ, + Δ + τ ) F(, + τ ) = α + β F(, + τ ) + ε, (7) where F(,+τ) i he fuure pread of mauriy +τ oberved a, Δ i he ampling ime inerval and ε i a random diurbance. If β i ignificanly negaive hen he po pread i deemed o be meanrevering and a coinegraion relaionhip i aen o exi beween he wo underlying ae price. Thi mehod examine he evidence for meanreverion in he mare meaure uing hiorical fuure price daa..3 Rineural meaure meanreverion e We can ue (ex ane) mare daa analyi o e wheher inveor expec he fuure pread o rever in he rineural meaure. Thi mehodology focue on relaion beween pread level and he pread erm rucure lope defined a he change acro he mauriie of fuure pread. A negaive relaionhip beween he po pread level (or horerm fuure pread level) and he fuure pread erm rucure lope how ha ri neural inveor expec meanreverion in he po pread. Indeed, ince each fuure price equal he rading dae expecaion of he delivery dae po price in he rineural meaure he curren erm rucure of he fuure pread reveal where inveor expec he po pread o be in fuure. Deecing an invere relaionhip beween curren pread level and fuure lope uppor a negaive relaionhip in he ri neural meaure beween he curren pread level and i fuure 7
8 movemen. Beembinder e al (1995) aemp o dicover ex ane mean reverion in commodiy po price. To dicover he negaive relaionhip in our cae we eimae x x = ς + γ x + ε, (8) L S S where x L and x S are repecively longend and horend pread level in he fuure pread erm rucure and ε i a noie erm. If γ i ignificanly negaive here i evidence ha he po pread i ex ane meanrevering in he rineural meaure..4 Coninuou ime conequence Regreion model (7) and (8) allow he empirical examinaion of he meanrevering properie of he po pread in repecively he mare and rineural meaure. Moreover by eing wheher he pread proce wih d:=1 i meanrevering we can examine wheher hi ideal coinegraion relaionhip hold in (). If eing indicae meanreverion hen a coinegraion relaionhip may be uppoed and he pread proce can be modelled direcly. The underlying coninuou ime po pread proce x hould hen follow he coninuou ime verion of equaion (4) in he mare meaure: dx = ( ψ ( ) x ) d + σdw, (9) where i he mean reverion peed, ψ() i a funcion of phyical ime, σ i a conan volailiy and W i a Wiener proce. The oluion of (9) ha he propery ha he variance of he pread x will no blow up aympoically in ime and i uncondiional variance i aionary. The radiional wo price pread proce model doe no poe hee properie. By differencing he wo equaion in (1) we obain he pread of he wo conrac price a where d( S S ) = r( S S ) d+σ ( S, S, ρ) dw, (10) σ( S, S, ρ) = σ S + σ S + σ σ S S ρ i he inananeou volailiy of he pread a ime. Thu he price pread proce 8
9 doe no meanrever in he ri neural meaure and he andard deviaion of he pread increae over ime and blow up aympoically. We do no ee hi for pread beween coinegraed commodiy price in hiorical mare daa (Villar and Jouz, 006). Commodiy and equiy pread procee are differen however. In he ri neural meaure any radable equiy porfolio wihou dividend paymen  including pread  hould grow a he rifree rae. Thu nondividend paying oc pread will no be meanrevering in hi meaure. However in he cae of phyical commodiie, epecially hoe commodiie which canno be ored, he rineural drif mu encompa ome form of ochaic convenience yield. We ee hi empirically in he meanrevering properie of ome commodiy price (Schwarz, 1997). Becaue of convenience yield pread procee for commodiie can be quie differen from hoe for equiie. 3. Modelling he Spread Proce We have een ha we can model he pread proce direcly uing an OrneinUhlenbec proce if a coinegraion relaionhip exi beween he wo underlying ae price. If we wih o price coningen claim on he pread we mu repecify (9) in he ri neural meaure. We now preen direc model of he po and fuure pread procee wih one and wo facor. 3.1 One facor model Fir conider a onefacor model of he po pread in he ri neural meaure pecified by dx = ( θ x ) d+ σ dw, (11) where θ i a conan which repreen he longrun mean of he pread proce. For impliciy we aume ha θ doe no depend on ime. Solving (11) given he aring ime v and pread poiion x v we obain 9
10 ( v) ( v) x = xve + θ[1 e ] + e σe dw (1) which follow a normal diribuion a ime wih mean a x e e ( v) ( v) = v + θ[1 ] (13) v and andard deviaion b ( v) 1 e = σ. (14) Thu b σ a o ha he pread andard deviaion end o a conan aympoically. Define F(,T,x ) a he fuure pread ( he pread of wo fuure price) of mauriy T oberved in he mare a ime when he po pread i x. In he ri neural meaure he po pread proce x mu aify he no arbirage condiion E[ x x ] = F(,T,x ), (15) T i.e. in he abence of arbirage he condiional expecaion wih repec o he po pread x of he ouurn po pread a T in he rineural meaure i he fuure pread oberved a ime <T. Thi mu hold becaue i i cole o ener a fuure pread (long one fuure and hor he oher). From (1) and he no arbirage conrain (15) we have FT x = xe + e (16) T ( ) T ( ) (, ; ) θ[1 ] From Io lemma i follow ha he fuure pread F(,T;x ) wih fixed mauriy dae T aifie ( ) d F F F T (,T;x ) = d + d = e σ d x x W, (17) i.e. he fuure pread i a maringale and i volailiy decay exponenially in ime o mauriy (T). 3. Two facor model 10
11 The mean revering po pread in (11) mainly reflec he hor o medium erm properie of he fuure pread (Gabillon, 1995) wih he volailiy of he fuure pread decaying exponenially wih ime o mauriy. Thi i a erm rucure which i no flexible enough o mach he volailiy erm rucure oberved in he mare. The po pread uually ha an eimaed rong meanreverion peed o ha he one facor model eimaed volailiie of fuure pread wih mauriie longer han or 3 year are quie cloe o zero while he oberved pread normally have quie noiceable volailiy. Thu anoher facor y i needed o reflec he longend movemen of he fuure pread erm rucure which relae o movemen of fundamenal uch a orage or hipping co change beween he wo commodiie. For impliciy and conien wih mare equilibrium we will aume ha he longrun mean of he y proce i zero in he ri neural meaure. In he ri neural meaure he underlying po pread proce x and he longrun facor y follow dx = ( θ + y x ) d+ σ dw dy = yd+ σ dw EdWW d = ρd, (18) where he laen facor y i a 0 meanrevering proce (if i poiive) repreening he long end of he pread erm rucure. We can inerpre he dynamic of he po pread pecified by (18) a revering o a ochaic long run mean θ+y. Since he volailiy of he long end pread i uually much maller han ha of he hor end σ hould be quie mall. Since fundamenal (e.g. orage co) have lower peed of adjumen, we can expec o be much maller han. Obviouly he longend movemen of he pread hould no be a priori conrained o be meanrevering and i could be Wiener proce lie in ome circumance. Given σ, he maller he value he cloer he y proce i o a Wiener proce. Gabillon (1995) conidered a imilar model o (17) for oil fuure conrac price. Appendix 1 how ha he oluion of (18) in he ri neural meaure i 11
12 x ( v) ( v) y v ( v) ( v) = xe v + θ[1 e ] + [ e e ] σ ( u) ( u) ( ) + + W v v [ e e ] d ( u) e σ dw( ). (19) Define: σ ( v) A1 : = [1 e ] 1 1 A e e e ( v) ( ) ( )( ) : ( [1 ] [1 v + v = + ] [1 ]) ( + ) ( ) σσ 1 1 A e e ( + )( v) ( v) 3 : = ( [1 ] [1 ]). + σ (0) Then he andard deviaion of x a ime i and var(x) i b = A + A + ρ A (1) 1 3 σ ρσσ (1 + / ) ( + ) σ + + () aympoically if i no zero. Thi i again becaue boh he x and y procee are meanrevering. If i zero, A become ' 1 ( v) ( v) A = {( v) + [1 e ] [1 e ]} σ. (3) In hi cae he andard deviaion of he pread will grow wih ime and blow up aympoically. However, ince σ i uually quie mall, he peed of growh of he andard deviaion i alo quie mall wheher or no i zero. Thi i conien wih he noion menioned in Mbanefo (1997) ha he pread andard deviaion grow much more lowly han i underlying wo leg. In ummary he meanrevering wo facor model cover wo cae of longend movemen: 1) a aionary y facor and ) a Brownian moion y facor. From he no arbirage conrain (15) and (19), we have y FTx xe e e e (, ; ) T ( ) [1 T ( ) ] [ ( T ) T ( ) = + θ + ] 1. (4) From Io lemma i follow ha he ri neural fuure pread F(,T) proce wih
13 fixed mauriy dae T aifie (, ) T ( ) [ ( T ) d T e d e e T ( F = σ W+ ) ] σdw (5) and i hu a maringale. I volailiy i compoed of wo par: he fir relaing o he one facor model and he econd o he long run facor y. A a reul, if i much maller han he volailiy of he longerm fuure pread will end o decay only lowly o zero over ime. 3.3 Spread proce in he mare meaure We will need a riadjued verion in order o calibrae he model preened above o mare daa. If we pecify a ri premium proce for x and y hen he drif par of our model can incorporae hee ri premia in he mare meaure (Duffie, 1988). Previou udie aume conan ri premia when modelling OrneinUhlenbec procee (ee, e.g. Hull & Whie (1990) and Schwarz (1997)) and o will we. Thu, he inglefacor model for he po pread proce in he mare meaure follow dx = [ ( θ x ) + λ] d+ σdw, (6) where λ i he ri premium. The wofacor model in he mare meaure follow dx = [ ( θ + y x ) + λ] d+ σdw dy = ( y + λ ) d+ σ dw EdWW d = ρd, (7) where λ and λ are he ri premia of he x and y procee repecively. Again uing Io lemma on he ri adjued verion of (16) and (4) wih (6) and (7) we obain he fuure pread proce in he mare meaure for boh model. For he one facor model he fuure pread wih a fixed mauriy dae T follow T ( ) T ( ) d (, T) = λe d+ e σd F W. (8) For he wo facor model (8) become 13
14 λ F( ) = λ + [ ] + T ( ) ( T ) T ( ) T ( ) d,t e d e e d e d + ( T ) ( T ) [ e e ] σ dw. σ W (9) From (8) and (9) he fuure pread wih fixed mauriy dae are no meanrevering in he mare meaure. Alhough he model preened above are for po pread, i i no eay o oberve direcly he po price of a commodiy and inveor ypically ue he neare mauriy fuure price o repreen he po price (Clewlow and Sricland, 1999). Bu ince he fuure pread wih fixed mauriy dae i no meanrevering for our model hi mae i difficul o eimae he meanreverion parameer of he po pread. However we will now how ha he fuure pread wih conan ime o mauriy τ : = T i meanrevering in our model which can be ued o deermine he meanreverion peed of he po pread. Uing Io Lemma we find (ee Appendix ) ha he proce for he fuure pread wih a conan ime o mauriy can be pecified in he mare meaure a follow. One facor model: τ λe τ df(, + τ) = [ θ + F(, + τ)] d+ e σdw. (30) Two facor model: τ τ λe df(, + τ) = [ θ + ye + + φλ F(, + τ)] d+ σ3dw, (31) where e φ : = τ e τ and σ : = e σ + θ σ + ρθe σσ are conan. τ τ 3 From (30) and (31) a fuure pread wih conan ime o mauriy i meanrevering in boh he one and wo facor model wih he ame meanreverion peed a he po proce. Noe ha when τ 0, (30) and (31) converge o (8) and (9) repecively. 3.4 Calibraion 14
15 In energy mare conrac mauriie are uually ordered by monhly dae. For example, crude oil fuure conrac raded on he NYMEX are ordered by 30 conecuive fuure monh and hen by quarer up o 7 year. In order o chec he meanrevering propery of he po pread we can ue he 1monh (ime o mauriy) fuure pread wih a monhly obervaion inerval for daa. We ue maximumlielihood eimaion (MLE) on he panel daa of fuure pread curve in he piri of Chen and Sco (1993) and Pearon and Sun (1994). Thi mehod i commonly ued in fixed income yield curve modelling (e.g. Duffie and Singleon, 1997; Dai, Singleon and Yang, 006). Recenly hi echnique ha alo been ued in he eimaion of convenience yield curve model (Caau and CollinDufrene, 005). Since he ae variable are no direcly oberved in our daa e, he Chen and Sco approach pecifie hee laen variable by olving expreion for ome ecuriie which are arbirarily aumed o be priced wihou error in he mare. The remaining ecuriie are aumed o be priced wih meauremen error. To illurae he mehod le F(,T 1 ) o F(,T 5 ) repreen 5 fuure pread available o deermine he model parameer. For he one facor model we uppoe he fir fuure pread a ime i oberved wihou pricing error bu F(,T ) o F(,T 5 ) are priced wih error. Thu he model eimaion equaion are FT (, ) = C+ Dx FT (, ) = C+ Dx+ u FT (, ) = C+ Dx+ u FT (, ) = C+ Dx+ u FT (, ) = C+ Dx+ u, (3) where C i := θ ( i ) [1 T e ], D i := T ( i ) e and u o u 5 are join normally diribued pricing error. The loglielihood funcion for all fuure pread a ime i given by L : = lnd + lnl + lnl, (33) e 1 where ln L i he log lielihood of he ae variable x (aen a he one monh fuure pread F(,T 1 )) a ime and ln L e i he log lielihood of he oher ecuriie F(,T ) o F(,T 5 ) wih 15
16 1 1 1 ( x xm) ln L : = ln( π ) ln( V) V Δ 1 e V : = σ Δ λ Δ xm : = x 1e + ( θ + )(1 e ) e ' 1 L : = ln( π ) ln( Ω ) uω u and denoe he (1 monh) obervaion inerval. In (33) D 1 i he coefficien Ti ( ) e in he affine ranformaion (16) from x o F(,T i ) and hu he Jacobian of hi ranformaion i 1/D 1. Since he fir one monh fuure pread i priced wihou error i loglielihood i deermined by he loglielihood of he ae variable ln L adjued by he Jacobian muliplier 1/D 1. In (34) V i he variance of he ae variable condiional on x 1, x m i he mean of x condiional on x 1 and Ω i he covariance marix for u. The oal log lielihood i deermine he parameer of he one facor model. L (34), which i maximized o For he wo facor model he correponding expreion are FT (, ) = C+ Dx+ Ey FT (, ) = C+ Dx+ Ey FT (, ) = C+ Dx+ Ey+ u FT (, ) = C+ Dx+ Ey+ u FT (, ) = C+ Dx+ Ey+ u, (35) where C i := ( i ) [1 T θ e ], D i := T ( i ) e and E i := ( Ti ) ( Ti ) [ e e ]. Defining J : D E 1 1 = D E he loglielihood for he wo facor model i e L : = ln J + lnl + lnl, (36) where 16
17 1 1 1 ( x x ) 1 ( y y ) ln L : ln( ) ln( V ) ln( V ) m m = π x y Vx Vy e V : = + { [1 e ] + [1 e ] [1 e } V x y Δ 1 1 Δ 1 Δ ( + ) Δ σ + ( ) 1 e : = σ ( ρσ σ 1 1 e + ( + ) Δ Δ + [ (1 ) (1 )] Δ e σ : Δ Δ 1 1 ( Δ ) (1 Δ Δ ) ( Δ m ) ( ) λ y : = e y + (1 e ) m Δ Δ 1 ) λ λ x = x e + y e e + e e e λ Δ + ( θ + )(1 e ) e ' 1 L : = ln( π ) ln( Ω ) uω u. (37) In preliminary udy we found ha he eimaed correlaion ρ beween he long end and he hor end wa inignifican. Thi mae economic ene in ha he long end movemen are low and driven by fundamenal while he horerm movemen which are random, fa and driven by mare rading aciviie. Tha innovaion in he long and hor run hould be uncorrelaed ha been ued o analyze he longrun and horrun componen of oc price (Fama and French, 1988). Rouledge, Seppi and Spa (000) aer ha he long run movemen of commodiy fuure price hould have zero correlaion wih he horrun movemen becaue he phyical invenory can regenerae or renew in oc ou period (Corollary 1., p.1304). Thu if he ime o mauriy of he fuure daa are long enough hee correlaion eimae hould be zero. Bu due o daa availabiliy hee eimae may no be eimaed a inignificanly differen from zero and hen he fir ri facor doe no aborb enough of he horerm price movemen (Schwarz and Smih, 000). In our wo facor model we will aume he correlaion ρ o be zero. 4. Spread opion pricing and hedging If he underlying ae price follow a Gauian proce he European call and pu 17
18 price wih mauriy T on hi ae can be calculaed repecively a c = B b ( a exp[ π K) b ] + B( a a K) Φ( K ) b (38) p = B b ( a exp[ π K) b ] B( a a K) Φ( K ), (39) b where B i he price of a dicoun bond, a and b are repecively he mean and andard deviaion of he underlying a mauriy, and K i he rie price of he opion (ee Appendix 3). We have een ha he pread diribuion a ime T follow a normal diribuion in boh one facor and wo facor model o ha equaion (38) and (39) can be ued o price he pread opion. 4.1 Pricing wihin fuure price mauriie Opion on he po pread Since he fuure pread i he expecaion of he fuure po pread in he rineural meaure he mean of he underlying ae a opion mauriy T can be obained a curren ime a a = F(, T ). (40) Equaion (14) and (1) how b for he one and wo facor model repecively a conan given an iniial ime and a fixed mauriy dae T. A pread opion value wih mauriy T depend on F(,T) hrough (40) inveor can uilize fuure of he ame mauriy dae o hedge. The dela of call and pu on he pread are given repecively by c a K Δ c = = BΦ( ) (41) a b p a K Δ p = = BΦ( ). (4) a b Since a pread can be een a long one ae and hor he oher imulaneouly he dela hedge yield an equal volume hedge, i.e. long and hor he ame value of 18
19 commodiy fuure conrac. No maer how many facor are deemed o drive he fuure price, only he correponding mauriy fuure conrac are uilized o hedge he pread opion providing hey are available in he mare. Opion on he fuure pread Define he opion mauriy a R, he fuure mauriy a T>R and he curren ime a. The fuure pread i a maringale o ha i mean in he ri neural meaure i a [ (, )] (, ) = EQ F R T = F T. (43) I andard deviaion in he one facor model i b ( T R) ( T ) e e = σ (44) and in he wo facor model i b = A + A + ρ A, (45) F F F 1 3 where F σ ( T R) ( T ) A1 = [ e e ] σ 1 1 A e e e e F ( T R) ( T ) T ( R) T ( ) = { [ ] + [ ] ( ) e ( + ) e ( + )( T R) ( + )( T ) [ ]} σσ 1 1 A e e e e F ( + )( T R) ( + )( T ) T ( R) T ( 3 = { [ ] [ ) ]}. + (46) Since R and T are nown uing (38) and (39) b i a conan in boh model and (38) and (39) can be ued o price call and pu opion on he pread a. Again inveor can ue fuure conac wih mauriy T o hedge hee opion poiion wih equal volume hedge given by (41) and (4) repecively. The hedgeraio difference beween hedging a po and a fuure pread opion arie from he difference in he b 19
20 value (compare (14) and (1) wih (44) and (45)). If an opion (on he po pread) ha a mauriy longer han he correponding fuure raded in he mare (which i common for many real opion), inveor canno ue hee mehod o price and hedge he opion. We udy hi iuaion nex. 4. Pricing beyond fuure price mauriie Suppoe a ime we now he value of he ae variable x in he one facor model or x and y in he wo facor model, hen we can foreca he mean pread a ime T> uing hee ae variable. For he one facor model a xe e T ( ) T ( ) = + θ[1 ] (47) and for he wo facor model T ( ) T ( y ) [1 ] [ ( T ) T ( a ) = xe + θ e + e e ]. (48) The andard deviaion b will be he ame a hoe in previou ecion. If no fuure conrac of long enough mauriy are available, inveor mu ue everal hor erm fuure o hedge he long erm opion, i.e. hey need o hedge he individual facor underlying he fuure conrac. There i quie a large lieraure on how o ue horerm fuure o hedge longerm one, e.g. Brennan and Crew (1997), Neuberger (1998) and Hilliard (1999). For he one facor model he call dela on he laen po pread i c a Δ x = =Δc a x e T ( ), (49) where c i he price of a call opion, Δ c i given by (41) and T i he opion mauriy. If he fuure pread F(,T 1 ) i uilized o hedge hi opion poiion he hedge raio i c x Δ = =Δ e e =Δe x F T ( ) T ( 1 ) T ( T1) F c c. (50) 0
21 Similarly he dela for pu on he po pread i p x Δ = =Δ e e =Δ e x F T ( ) T ( 1 ) T ( T1) F p p, (51) where Δ p i given by (4). Applying he wo facor model, inveor mu ue wo horer erm fuure F(,T 1 ) and F(,T ) o hedge long erm opion. Ideally T 1 hould be hor (e.g. 1 monh) and T he longe fuure mauriy available in he mare. The call dela on he laen wo facor x and y are repecively c a Δ x = =Δc a x e T ( ) (5) and c a ( T [ ) T ( Δ ) y = =Δ c e e ] a y. (53) Suppoe he fuure pread F(,T 1 ) and F(,T ) are uilized o hedge he opion poiion. Then o obain he dela neural hedge raio n 1 and n one mu olve Δ + ne + ne = 0 x T ( 1 ) T ( ) 1 Δ + n [ e e ] + n [ e e ] = 0. y ( T1 ) ( T1 ) ( T ) ( T ) 1 (54) Since hi paper focue on long erm opion valuaion and hedging we hould dicu he opion mauriie appropriae for he ue of our valuaion model. The anwer i relaed o he meanreverion peed. The decay half life ln/ of he meanrevering pread proce can be ued o repreen i meanreverion rengh. We propoe ha if an opion ime o mauriy i longer han hi half decay ime, our mehodology (boh one facor and wo facor model) i appropriae o evaluae he pread opion. Alo, we expec ha he radiional pread opion model (1) will overvalue hee longer erm opion becaue of he variance blowup phenomenon previouly dicued. Mbanefo (1997) noed ha longerm pread opion (longer 1
22 han 90 day) will be overvalued if meanreverion of he pread i no conidered. We noe ha jump and ochaic volailiy are no imporan in deermining heoreical or empirical long erm pread opion price (Bae, 1996; Pan, 003) which allow model of hi paper o remain parimoniou. 5. Example 5.1 Crac pread: Heaing oil/ WTI crude oil (CSHC) 4 The crac pread beween heaing oil and WTI crude oil (heaing oil crac pread) repreen he profi from refining heaing oil from crude oil, i.e. he price of heaing oil minu he price of crude oil. We have een in Secion ha a relaive deviaion beween he (equilibrium) inpu and oupu price relaionhip could exi for hor period of ime, bu a prolonged large deviaion will lead o he producion of more end produc unil he oupu and inpu price are nearer he longerm equilibrium relaionhip. Thu we expec he heaing oil crac pread o be meanrevering. Daa The daa for modelling crac pread coni of NYMEX daily fuure price of WTI crude oil (CL) and heaing oil (HO) from January 1984 o January 005. The ime o mauriy of hee fuure range from 1 monh o more han year. In order o e for uni roo, a ingle monhly daa poin i colleced on he fir day of each monh by aing he price of he fuure conrac wih one monh ime o mauriy. For example, if he rading day i 1 February 000 hen he fuure conrac aen for he ime erie i he 1 March 000 conrac. We alo creae a longend crac pread wih ime o mauriy 1 year. The mehodology i exacly he ame a wih he 1 monh ime erie, bu due o daa unavailabiliy we only ue daa from January 1989 o January 005 o conruc he longend crac pread. 4 Abbreviaion ued in hi ecion are NYMEX rading code.
23 To calibrae he onefacor and wofacor pread model uing (3) and (35), we calculae he monhly fuure pread wih 5 fuure conrac from January 1989 o January 005. The ime ep Δ i choen o be 1 monh and he conrac choen are 1 monh, 6 monh, 9 monh, 1 monh and 15 monh ime o mauriy fuure pread. Uni roo and coinegraion e Fir, we conduc he ADF e on he heaing oil and crude oil price uing he longer ime erie. A noed in Secion 3.4 we can e he meanreverion of he po pread by examining he 1 monh fuure pread. Figure 1 how he 1 monh fuure price of crude oil and heaing oil. Iner Figure 1 abou here Table 1 how he reul of he ADF e of Secion.1 on 1 monh fuure of crude oil and heaing oil ariing from eimaing (5). In order o rejec he hypohei ha a ime erie ha a uni roo he coefficien δ 1 mu be ignificanly negaive. Iner Table 1 abou here From Table 1, we canno rejec he hypohei ha boh crude oil and heaing oil are uniroo ime erie (cf. Girma and Paulon (1999) and Alexander (1999)). Nex we e he pread ime erie direcly by eimaing (5) o find a very rong meanrevering peed ignifican a he 1% level which ugge ha coinegraion doe exi in he daa. In oher word, he meanreverion of he pread doe no appear o be caued by he eparae meanreverion of he heaing and crude oil price bu by he longrun equilibrium (coinegraion) beween hem which mu be conidered in longrun derivaive pricing of he crac pread. Thi agree wih Girma and Paulon (1999). Hiorical pread only offer u heir maremeaure characeriic bu a regreion relaing he hor erm and long erm pread can give u he ri neural pread. We eimae he regreion equaion (8) uing he 1 year crac pread a he longend 3
24 fuure pread and he 1 monh crac pread a he horend fuure pread. The reul are given in Table and he 1 year and 1 monh crac pread depiced in Figure. Iner Table abou here In Table he eimae of γ i ignificanly le han zero. Since hi e inveigae he meanrevering propery of pread in he rineural meaure, we conclude ha an OrneinUhlenbec proce i appropriae o model he po proce. Thu boh he ex ane and ex po e give evidence ha he po crac pread i meanrevering in boh he mare and ri neural meaure. Iner Figure abou here Model calibraion In order o calibrae he (onefacor and wofacor) model we ue he full daa e and employ equaion (3) and (35). From preliminary udy we noiced ha he heaing oil price how a eaonal paern which i inheried in he crac pread. To eliminae he influence of eaonaliy, we ued an equally weighed porfolio of 1 monh and 6 monh fuure (oppoie eaon) o deermine he x facor and a imilar porfolio of 9 monh and 15 monh fuure o deermine he y facor. We hen performed a MLE opimizaion o calibrae he one facor model. Table 3 li he reul. One can ee ha he andard deviaion of he θ, σ and eimae are quie mall, i.e. θ, and σ can be deermined quie preciely, unlie he λ eimae. However we do no need he mare price of ri λ a inpu o opion pricing. The aympoic eimae of he andard deviaion of he pread (when goe o infiniy in (14) i $.03. Figure 3 how he one monh pread and he laen po pread, which are very cloe o each oher. Iner Table 3 abou here Iner Figure 3 abou here 4
25 Iner Table 4 abou here Similar o he iuaion in he one facor model, he mare price of ri λ and λ canno be preciely eimaed in he wo facor model, bu eimae of all he oher parameer (σ, σ,, and θ) can. The aympoic eimae of he andard deviaion of he pread i $.65, which i higher han ha for he one facor model. Thi i eay o underand, ince he one facor model only coun he horerm variance of he pread, while he wo facor model ae accoun of boh he long end (fundamenal) and he hor end (rading aciviie). Alo, auming zero correlaion beween he wo facor, we can examine he raio of he longend and he horend variance A 1 :A in (0) a ime goe o infiniy, which i abou 1:1 in hi example. Thu he aympoic variance of he crac pread i nearly equally conribued by horend (fir facor) and longend (econd facor) movemen. Since hi raio i quie high he econd facor i obviouly imporan in derivaive pricing and hould no be omied. Since he one facor model i need in he wo facor model by aing σ, λ and y v (he aring value of he y facor) o be zero, we can compare he difference in loglielihood core for each daa e o ee wheher he addiional parameer of he wofacor model provide a aiically ignifican improvemen in ha model abiliy o explain he oberved daa. The relevan e aiic for hi comparion i he chiquared lielihood raio e (Hamilon, 1994) wih 3 degree of freedom and he 99 h percenile of hi diribuion i Given ha he loglielihood core increae by abou 1, he improvemen provided by he wo facor model are quie ignifican. Iner Figure 4 abou here Figure 4 how he ime evoluion of he long and hor facor wih correlaion coefficien , which i no ignifican and hu conien wih he aumpion ha he correlaion beween he wo facor i zero. Opion valuaion on he po pread 5
26 The decay half life i abou half a year for he one facor model o ha when valuing an opion longer han half a year he mehodology (uing one facor or wo facor model) in hi paper i appropriae. On 3 January 005 he HO06N (heaing oil fuure wih mauriy July 006) conrac had a value 44. ($/Barrel); on he ame day he CL06N (crude oil fuure wih mauriy July 006) raded a ($/Barrel). By uing he parameer in Table 3 and 4, Table 5 give he European opion value on he po crac pread wih mauriy July 006 on hi dae. For comparion we alo calculae opion value from a model which ignore he coinegraion effec. We can ue he Blac (1976) drifle GBM model o imulae boh crude and heaing oil fuure price pah, and hu calculae he pread opion value 5. The average correlaion coefficien (over a 0 year period) i 0.89 beween heaing and crude oil. We ue call opion value o compare he differen model; i i hen eay o obain he pu value by pucall pariy. Iner Table 5 abou here Iner Table 6 abou here From Table 5 we can ee ha he opion value from he one facor model i ypically maller han ha from he wofacor model and he laer i much maller han ha from he Blac model. Since he Blac model doe no conider meanreverion (he coinegraion) of he pread, i pread diribuion a mauriy i wider han ha of a coinegraed model and hu yield a larger opion value. Pu imply, a noncoinegraed model ignore he longrun equilibrium beween crude and heaing oil price and hu overprice he opion. Since he wo facor model accoun for longerm pread movemen, i hould yield a wider pread diribuion a mauriy and hu ha a larger opion value han he one facor model. In Table 6, boh he one facor and wo facor model yield an equal volume hedge bu he Blac model doe no. A i well nown, he le dipere he underlying erminal diribuion, he more 5 Since here i no analyical oluion when he rie i no zero one convenien way o calculae he opion value i by Mone Carlo pah imulaion. 6
27 eniive he opion dela are o he rie price 6. Thu he onefacor model yield he mo eniive dela and he Blac model ha he lea eniive dela among he hree model. 5. Locaion pread: Bren / WTI crude oil (LSBW) We define he LSBW locaion pread a he price of WTI crude oil (CL) minu he price of he Bren blend crude oil (ITCO). WTI i delivered in he USA and Bren in he UK. Daa NYMEX daily fuure price of WTI crude oil were decribed in he previou example. The daily Bren fuure price are from January 1993 o January 005. The ime o mauriy of he Bren fuure conrac range from 1 monh o abou 3 year. A in he previou example, monhly daa i ued o e for he uni roo in Bren oil price. We alo creae a monhly longend LSBW pread wih ime o mauriy of 1 year ranging from January 1993 o January 005. In order o calibrae he onefacor and wofacor model, we calculae he monhly fuure pread wih 5 mauriie from January 1993 o January 005 a monhly inerval, i.e. he ime ep Δ in (30) and (31) i 1 monh. The 5 conrac involved are 1 monh, 3 monh, 6 monh, 9 monh and 1 monh fuure pread. Uni roo and coinegraion e A from he previou example we now ha he WTI crude oil price follow a uni roo proce, in hi example we need only conduc he ADF e on Bren crude oil price. Figure 5 how he 1 monh fuure price of WTI crude oil and Bren blend. We again ae he 1 monh fuure price a repreenaive of he po price. Iner Figure 5 abou here 6 The eniiviy i defined a he raio of he change of he dela in he change of he rie price. 7
28 Table 7 how he reul of ADF e eimaing (5) on he 1 monh fuure of Bren blend. Similar o WTI crude oil, he Bren blend price i alo a uni roo proce (ince δ 1 i poiive), bu he LSBW locaion pread appear o be a meanrevering proce. Thi again ugge he exience of a longrun equilibrium in he daa. To eimae fuure expecaion of he pread, we eimae he regreion equaion (8). A in he previou example, we ue he 1 year and 1 monh LSBW fuure pread. The reul are lied in Table 8. The 1 year and 1 monh LSBW pread evoluion i depiced in Figure 6. Iner Table 7 abou here Iner Figure 6 abou here Iner Table 8 abou here From Table 8 we ee ha he eimae of γ i rongly negaive, o ha he mare appear o expec he po LSBW pread o be meanrevering in he rineural meaure. Hence boh he ex ane and ex po analye uppor ha he po LSBW pread i meanrevering o ha meanreverion hould be accouned for in opion pricing. Model Calibraion We did no find evidence of eaonaliy in he LSBW pread. Hence we ue he 1 monh fuure pread o bac ou he laen po pread facor for he one facor model. For he wo facor model we ue he 1 monh fuure pread o eimae he hor erm x facor and an equally weighed porfolio coniing of he 9 monh and 1 monh fuure pread o eimae he long erm y facor. Table 9 and 10 li he calibraion reul for he one facor and wo facor model. We ee ha he aympoic andard deviaion of he pread are eimaed o be $1.60 and $3.90 repecively for he one and wo facor model. The raio of longend o he horend variance (A 1 :A in (9)) i 5:1 in he wo facor model, i.e. he long end (econd facor) movemen of he pread accoun for much more variance han (fir 8
29 facor) hor end variaion. Similar o he previou example he eimae of he aympoic andard deviaion from he wo facor model i higher han ha from he one facor model. From he Chiquared (lielihood raio) e he wo facor model i very ignificanly beer han he one facor model in explaining he oberved LSBW pread daa. The laen po pread facor and he wo facor (x and y) are hown in Figure 7 and 8 repecively. The correlaion beween he wo facor i 0.04 which i again conien wih our zero correlaion aumpion. Iner Table 9 abou here Iner Table 10 abou here Iner Figure 7 abou here Iner Figure 8 abou here Opion valuaion on he po pread The decay half life i abou ix monh from he one facor model. Thu o model an opion longer han ix monh he mehod (one facor or wo facor model) in hi paper hould be ued. On he day of 1 December 003 he ITCO06Z (Bren blend crude oil fuure wih mauriy December 006) conrac had a value 4.6 ($/Barrel); on he ame day he CL06Z (WTI crude oil fuure wih mauriy December 006) raded a 5.69 ($/Barrel). By uing he eimaed parameer in Table 9 and 10, Table 11 how he European opion value on he po pread wih mauriy December 006. Since he opion i on he po pread he hedging fuure mauriie hould be he ame a he opion mauriy December 006. The Bren and WTI crude oil conrac price boh follow uni roo procee o we may imulae boh price o calculae he noncoinegraed Blac model pread opion value. Noe ha he average correlaion beween he Bren blend and WTI crude oil i Iner Table 11 abou here 9
30 Table 11 how ha, imilar o he previou example, he opion value of he one facor model i ypically maller han ha of he wofacor model and he laer i much maller han he Blac model. A before by ignoring coinegraion he Blac model end o overvalue he long erm opion. We obain a imilar paern of dela a in he previou example (ee Table 1) and he explanaion for hi remain he ame. Iner Table 1 abou here 6. Concluion In hi paper we have developed pread opion pricing model in which he wo price leg of he pread are coinegraed. Since he coinegraion relaionhip i imporan for he longrun relaionhip beween he wo price pread opion evaluaion hould ae accoun of hi relaionhip if he opion mauriy i long. Auming a coinegraion relaionhip beween he wo underlying ae, we model he pread proce direcly uing he OrneinUhlenbec proce, i.e. we model direcly he dynamic deviaion from he longrun equilibrium which canno be pecified correcly by modelling he wo underlying ae eparaely. We fir pecify rineural procee for he pread and hen deermine he mare procee by auming conan ripremia. We alo propoe wo mehod (ex ane and ex po) o e for meanreverion of he pread proce. Finally we give analyical oluion for he pread opion price and dela. In order o illurae he heory, we udy wo example opion on crac and locaion pread repecively. Boh mare pread procee are found o be meanrevering, which implie ha heir wo price leg are coinegraed. From lielihood raio e he wo facor model i found o be ignificanly beer han he one facor model in explaining he crac and locaion pread daa. The opion value and Gree from our coinegraion model are quie differen from hoe of andard model bu are conien wih he pracical obervaion of Mbanefo (1997). We are currenly woring on Lévy proce verion of our model which may be more appropriae o commodiy mare uch a ga or 30
31 elecriciy and for horer erm opion mauriie generally. Reference Alexander, C., Correlaion and Coinegraion in Energy Mare, in: Kaminy, V. (Ed.), Managing Energy Price Ri, Second Ediion. Ri Publicaion, London, pp Amin, K., Ng, V., Pirrong, S., Valuing Energy Derivaive, in: Managing Energy Price Ri. Ri Publicaion, London, pp Bae, D., Jump and Sochaic Volailiy: Exchange Rae Procee Implici in Deuche Mar Opion. Review of Financial Sudie 9, Beembinder, H., Coughenour, J., Seguin, P., Smoller, M., MeanReverion in Equilibrium Ae Price: Evidence from he Fuure Term Srucure. Journal of Finance 50, Blac, F., The Pricing of Commodiy Conrac. Journal of Financial Economic 3, Blac, F., Schole, M., The Pricing of Opion and Corporae Liabiliie. Journal of Poliical Economy 81, Brennan, M., The Pricing of Coningen Claim in Dicree Time Model. Journal of Finance 34, Brennan, M., Crew N., Hedging Long Mauriy Commodiy Commimen wih ShorDaed Fuure Conrac, in: Demper, M.A.H., Plia, S.R. (Ed.), Mahemaic of Derivaive Securiie. Cambridge Univeriy Pre, Cambridge, pp Carmona, R., Durrleman, V., 003. Pricing and Hedging Spread Opion. SIAM Review 45, Caau, J., CollinDufrene, P., 005. Sochaic Convenience Yield Implied from Commodiy Fuure and Inere Rae. Journal of Finance 60, Chen, R., Sco, L., Maximum Lielihood Eimaion for a Mulifacor Equilibrium Model of he Term Srucure of Inere Rae. Journal of Fixed 31
32 Income 3, Clewlow, L., Sricland, C., Valuing Energy Opion in a One Facor Model Fied o Forward Price. Woring Paper, Univeriy of Technology, Sydney, Auralia. Dai, Q., Singleon, K., Yang, W., 006. Regime Shif in a Dynamic Term Srucure Model of US Treaury Bond Yield. Woring Paper, Sanford Univeriy. Demper, M., Hong, S., 000. Pricing Spread Opion wih he Fa Fourier Tranform, in: Geman, H., Madan, D., Plia, S.R., Vor, T. (Ed.), Mahemaical Finance, Bachelier Congre. SpringerVerlag, Berlin, pp Dicy, D., Fuller, W., Diribuion of he Eimae for Auoregreive Time Serie wih a Uni Roo. Journal of he American Saiical Aociaion 74, Duan, J., Plia, S., 004. Opion Valuaion wih Coinegraed Ae Price. Journal of Dynamic and Economic Conrol 8, Duffie, D., Securiy Mare: Sochaic Model. Academic, Boon, MA. Duffie, D., Singleon, K., An Economeric Model of he Term Srucure of Inere Rae Swap Yield. Journal of Finance 5, Engle, R., Granger, C., Coinegraion and Error Correcion: Repreenaion, Eimaion and Teing. Economeric 55, Fama, E., French, K., Permanen and Temporary Componen of Soc Price. Journal of Poliical Economy 96, Froo, K., Dabora, E., How are Soc Price Affeced by he Locaion of Trade. Journal of Financial Economic 53, Fuller, W., Inroducion o Saiical Time Serie. John Wiley & Son, New Yor. Gabillon, J., Analyzing he Forward Curve, in: Managing Energy Price Ri. Ri Publicaion, London, pp Gibon, R., Schwarz, E., Sochaic Convenience Yield and he Pricing of Oil Coningen Claim. Journal of Finance 45,
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