A Multi-factor Jump-Diffusion Model for Commodities 1

Transcription

1 A Muli-facor Jup-Diffusion Model for Coodiies JOHN CROSBY Lloyds SB Financial Markes, nd floor, Gresha Sree, London ECV 7AE Eail address: h July 5, revised 7 h Ocober 6 Acknowledgeens: he auhor wishes o hank Sion Babbs, Michael Depser, Farshid Jashidian and an anonyous referee for valuable coens, which have significanly enhanced his paper since is firs draf. In addiion, he hanks eer Carr, Mark Davis, Darrell Duffie, Sewar Hodges, Andrew Johnson, Daryl Mcarhur, Anhony Neuberger, Olga avlova, Chris Rogers, Nick Webber and seinar paricipans a Cabridge Universiy, a Iperial College, London and a he Universiy of Warwick. However, any errors are he auhor s responsibiliy alone. Absrac In his paper, we develop an arbirage-free odel for he pricing of coodiy derivaives. he odel generaes fuures (or forward) coodiy prices consisen wih any iniial er srucure. he odel is consisen wih ean reversion in coodiy prices and also generaes sochasic convenience yields. Our odel is a uli-facor jup-diffusion odel, one specificaion of which allows he prices of long-daed fuures conracs o jup by saller agniudes han shor-daed fuures conracs, which, o our knowledge, is a feaure ha has no previously appeared in he lieraure, in spie of i being in line wih sylised epirical observaions (especially for energy-relaed coodiies). Our odel also allows for sochasic ineres-raes. he odel produces sei-analyic soluions for sandard European opions, which enable opion prices o be evaluaed in ypically abou /5 h of a second (depending upon paraeer values and he required accuracy). his opens he possibiliy o calibrae he odel paraeers by deriving iplied paraeers fro he arke prices of opions. We perfor such a calibraion on crude oil opions and show ha, allowing long-daed fuures conracs o jup by saller agniudes han shor-daed conracs, gives a grealy enhanced fi. Keywords : Coodiy opions, coodiy derivaives, jup-diffusion, ean reversion. Inroducion he ai of his paper is o develop an arbirage-free uli-facor jup-diffusion odel for pricing coodiy opions. he coodiy could be, for exaple, crude oil, anoher peroleu produc, gold, a base eal, naural gas or elecriciy. Before urning our aenion o coodiies, i is worh reflecing on he developen of ineresrae odels. he paper by Vasicek (977) inroduced an equilibriu ean revering ineres-rae odel ino he lieraure. By inroducing a ie-dependen ean reversion level, his becae he exended Vasicek odel (Babbs (99), Hull and Whie (993)) which could auoaically fi any iniial er srucure of ineres-raes. hese odels focused principally on insananeous shor raes. Furher research (Babbs (99), Heah e al. (99)) developed no-arbirage odels (including uli-facor versions) evolving he enire yield curve, consisen wih is iniial values. here are any parallels beween he above ineres-rae odels and odelling fuures (or forward) coodiy prices. Soe of he coodiies lieraure (Gibson and Schwarz (99)) has focused on equilibriu odels wih he firs facor being he spo coodiy price and he second facor being he insananeous convenience yield whils Schwarz (997) inroduced a hird facor, wih sochasic ineres-raes. However hese odels leave he arke price of convenience yield risk o be deerined in equilibriu and are no necessarily consisen wih any iniial er srucure of fuures (or forward) prices. Subsequen odels (by analogous echniques o ineres-rae odelling) have been consisen wih any iniial er srucure. See for exaple, Corazar and Schwarz (994), Carr and Jarrow (995), Beaglehole and Chebanier (), Milersen and Schwarz (998), Milersen (3), Clewlow and Srickland (),(999) wih he laer paricularly focussing on evolving he forward price curve. In his respec, his is a condensed and enhanced version of a previous working paper (Crosby (5)) eniled Coodiies: A siple uli-facor jup-diffusion odel.

2 our paper is closes in spiri o Clewlow and Srickland (999) hough we also incorporae sochasic ineres-raes and uliple jup processes wih differen specificaions of jups. As a general rule, aenion has osly focused on pure diffusion odels. Jups were incorporaed ino ineres-rae odels in Babbs and Webber (994),(997), Bjork e al. (997) and Jarrow and Madan (995). See also Meron (976),(99), Hoogland e al. (), Duffie e al. () and Runggaldier (). aralleling hese odels, jups have also been inroduced ino odels for coodiy prices in Hilliard and Reis (998), Deng (998), Clewlow and Srickland (), Benh e al. (3) and Casassus and Collin-Dufresne (5). We will inroduce a uli-facor jup-diffusion odel which significanly exends exising odels in he lieraure. Firsly, we ouline soe feaures of he coodiies and coodiy opions arkes. I is an epirical fac (Bessebinder e al. (995), Casassus and Collin-Dufresne (5)) ha he (spo) prices of os coodiies see o exhibi ean reversion. Furherore, i is also epirically observed ha price jups are usually boh ore frequen 3 and larger in agniude in he coodiies arkes han in, for exaple, he equiy or foreign exchange (fx) arkes. Suppose here is a price jup in he spo fx rae, and we assue ha here are no siulaneous jups in he bond arkes of he doesic and foreign currencies, hen, clearly, he forward fx rae o all enors us jup by he sae proporional aoun. When here is a jup in he price of gold, sylised observaions sugges ha he forward (or fuures) gold price o all enors do jup by he sae proporional aoun. In his respec, gold rades like a currency. However, in os oher coodiies arkes, especially in he case of crude oil, naural gas and elecriciy, sylised observaions sugges a very differen behaviour. he prices of shor-daed (close o delivery) fuures (or forward) conracs exhibi large jups bu he agniude of hese jups is uch lower for conracs wih a greaer ie o delivery indeed very long-daed conracs are ofen observed o scarcely jup a all, even when he very shor-daed conracs have juped by hundreds of per cen. Sylised observaions also sugges ha, afer a large jup in he prices of hese very shor-daed conracs, prices ofen see o rever o a ore usual level raher quickly. he epirically observed feaure ha he prices of shor-daed fuures (or forward) conracs jup by ore han long-daed conracs is easily seen wih wo illusraions. he firs illusraion is fro he hisorical prices of crude oil fuures. he price of crude oil fuures for nex onh delivery was approxiaely 7 dollars per barrel iediaely before Iraq s invasion of Kuwai in Augus 99 whils he price of crude oil fuures for 8 onh delivery was approxiaely 9 dollars per barrel. Over he nex 5 / onhs (up o he sar of he Gulf War), he (rolling) price for nex onh delivery, on wo separae occasions, ouched 4 dollars per barrel. In conras, he (rolling) price of crude oil fuures for 8 onhs delivery never wen above 7 dollars per barrel during he whole period. Qualiaively siilar behaviour has been observed (Gean (5)) in he arke for naural gas. Our second illusraion, which is even ore sriking, is he case of elecriciy and can be found in Villaplana (3). A he beginning of he second half of 998, he prevailing price for elecriciy in he ennsylvania-new Jersey-Maryland area of he U.S. was around 5 dollars per MWh for boh spo (nex day delivery) and one onh forward delivery. During he course of he following six onhs, here were large jups in he price of elecriciy which caused he spo price o rise above 35 dollars per MWh on hree separae occasions. On each occasion ha here was a jup in he spo price, a jup was also observed in he forward price of elecriciy. However, during his enire period, he forward price of elecriciy for one onh delivery never exceeded 98 dollars per MWh. I is also sriking o Casassus and Collin-Dufresne (5) show epirically ha (spo) coodiy prices exhibi a high degree of ean reversion in he real-world physical easure and also, albei perhaps o a lesser degree, usually in he riskneural easure. hey show how his possible difference in behaviour under he wo differen easures can be explained by sae-dependen risk-preia. Sae-dependen risk-preia could also be used in precisely he sae way in our odel. However, since our ai is o develop a odel for pricing coodiy derivaives, we focus only on he risk-neural easure and do no pursue he approach of Casassus and Collin-Dufresne (5) here. Bessebinder e al. (995) also find srong epirical evidence for ean reversion in he risk-neural easure. In secion 3, we will show ha our odel is consisen wih ean reversion in he risk-neural easure and ha jups ay (depending upon heir precise specificaion) also conribue o his effec. 3 Of course, i is possible ha he values of he inensiy raes of he jups and (if he jup apliudes are rando) he ean jup apliudes are differen (perhaps very differen) in he real-world physical easure (in which he price jups are observed) and in he risk-neural easure (relevan for derivaives pricing), iplying ha jups ay be iporan in one easure bu no he oher. Casassus and Collin-Dufresne (5) do no find any srong evidence for his and conclude (p38) ha jups would have a significan ipac for he cross secion of opion prices. Of course, i would be grossly preaure o conclude ha he values under he wo differen easures would be idenical or even approxiaely equal. In any even, since our ai is o price coodiy derivaives, we will only be concerned wih he values of hese paraeers in he risk-neural easure.

3 observe ha, on each of he hree occasions he spo price juped above 35 dollars per MWh, he spo price quickly (wihin wo or hree weeks) revered back o a level below 4 dollars per MWh. hese illusraions deonsrae he iporance of allowing prices of shor-daed fuures conracs (or forward prices) o jup by ore hen long-daed conracs wihin he conex of hisorical price oveens. One quesion ha igh be asked is: Is his iporan in he conex of opion pricing? We answer his quesion affiraively in wo ways. Firsly, in secion 5, we will perfor a calibraion of our odel o he arke prices of opions on crude oil fuures which deonsraes ha allowing for he effec of prices of shor-daed fuures conracs juping by ore han long-daed conracs gives a grealy iproved fi. For he second way, suppose now we have our calibraed odel paraeers and we wish o price a European opion, whose payoff is he greaer of zero or he raio of he price of a fuures conrac wih a furher (a opion auriy) one onh o delivery divided by he price of a fuures conrac wih a furher (a opion auriy) wo years o delivery inus a fixed srike. his is a siple ype of exoic opion on he slope of he er srucure of fuures coodiy prices. If we assue (as he exising lieraure does) ha, when here are jups, fuures conracs of all auriies jup by he sae proporional aoun hen, i is easy o show ha, he price of his exoic opion, given he odel paraeers, is indifferen o jups. However, if we assue ha he prices of shordaed fuures conracs jup by ore han long-daed conracs, hen he price of he opion will be influenced by jups, which is wha one would expec o be he case given he epirically observed behaviour described above. his illusraes ha he epirically observed hisorical price oveens are iporan for pricing derivaives (we will ouline in secion 4 how our odel can be used wih Mone Carlo siulaion o price exoic opions for a Fourier ransfor based algorih for pricing he ype of exoic opion jus described, see Crosby (6b)). Despie he epirical evidence and is iporance in pricing opions, o our bes knowledge, no previous papers have considered odels for coodiies which allow he prices of long-daed fuures conracs o jup by saller agniudes han hose of shor-daed fuures conracs. We also observe ha he arke prices of opions on any coodiies iply Black (976) volailiies which vary wih he srike of he opion ie arke prices iply a volailiy skew (or sile). Furherore, he volailiy skew is uch, uch ore pronounced for opions wih shorer auriies (Gean (5) and Gean and Nguyen (3) provide epirical daa which deonsraes how arked his behaviour is, paricularly for energy-relaed coodiies) and i is well known ha such behaviour could be accouned for hrough a jup-diffusion odel. Addiionally, we noe (see Gean (5)) ha he iplied volailiies of coodiy opions ypically decrease wih increasing opion auriy (his is very arked for energy-relaed coodiies). We would like our odel o incorporae all of he above sylised observaions of he coodiies and coodiy opions arkes. Exainaion of our odel will show ha indeed i does. We will deliberaely develop our odel wih considerable generaliy so ha i is flexible enough o be applied o opions on alos any coodiy. For exaple, he iplied volailiies of opions on soe coodiies (for exaple, naural gas) exhibi seasonaliy 4. he volailiy specificaion we will use is flexible enough o odel his when appropriae (for exaple, by uilising he specific for in Milersen (3)). Siilarly, our odel will be general enough o caer for jups ore appropriae for gold as well as for jups ore appropriae for crude oil, naural gas and elecriciy (as oulined earlier). We will also briefly discuss convenience yields in our odel. In he lieraure, convenience yields are ofen relaed o cos of sorage. We will no explicily do his because i does no appear helpful for odelling, for exaple, he price of an opion on elecriciy which is exreely expensive o sore. Insead, we will posulae he dynaics of fuures coodiy prices and hen infer fro he he dynaics of convenience yields. We will show ha, in our odel, convenience yields (in general) can exhibi jups and furherore ha jups in convenience yields are inrinsically linked wih he abiliy o capure he effec of shor-daed fuures conracs juping by ore han long-daed fuures conracs. We will assue ha ineres-raes are sochasic. When ineres-raes are sochasic, fuures coodiy prices and forward coodiy prices are no longer he sae. In his paper, we will work wih boh fuures and forward prices bu osly wih fuures coodiy prices. We will assue ha arkes are fricionless. ha is, coninuous rading is possible and we assue ha here are no bid-offer spreads in he coodiies arkes or in he bond arkes. Of course, we do no assue ha he coodiy can be sored or insured wihou cos since i is precisely hese coss which give rise o he noion of convenience yield. We will assue ha arkes are free of arbirage. 4 Noe ha our odel is auoaically consisen wih any seasonaliy in he er srucure of fuures (or forward) coodiy prices since i is consisen wih any given iniial er srucure. 3

4 I is well known (Harrison and liska (98), Duffie (996)) ha, under hese assupions, here exiss an equivalen aringale easure under which fuures prices are aringales. In he case of a diffusion odel, if here are sufficien fuures (or forward) conracs (and risk-free bonds) raded, hen any derivaive can be insananeously hedged or replicaed by a dynaic self-financing porfolio of fuures conracs (and risk-free bonds). he arke would hus be coplee. In his case, he equivalen aringale easure is unique. However, in he case of a jup-diffusion odel, he arke ay be eiher coplee or incoplee. If he arke is incoplee hen he equivalen aringale easure would no be unique. In he case of incopleeness, we will assue ha an equivalen aringale easure is fixed by he arke hrough he arke prices of opions and we will call his (by an abuse of language bu for he sake of breviy) he (raher han an) equivalen aringale easure. I is also possible for our jup-diffusion odel o lead o a arke which is coplee. he circusances in which our jup-diffusion odel gives rise o a coplee arke are specified in secion. he reainder of his paper is srucured as follows. In secion, we will provide noaion and inroduce he odel. In secion 3, we will relae i o sochasic convenience yields and o ean reversion. In secion 4, we will show how he odel can be used in connecion wih Mone Carlo siulaion o price coplex (exoic) coodiy derivaives. In secion 5, we will derive he prices of sandard opions, in sei-analyical for, illusrae our odel wih five exaples and calibrae i o he arke prices of opions on crude oil fuures. Secion 6 is a shor conclusion.. he odel of fuures coodiy prices Noaion: Le us explain soe noaion. All jup-diffusion processes are assued righ coninuous. he value, H, = li H u,. When, for he sake of of H( ) jus before a jup a ie is ( ) u ( ) dh (, ) dh (, ) breviy, we wrie in a SDE, we sricly ean. H(, ) H(, ) and we denoe calendar ie by, ( We define oday o be ie ). In his and subsequen secions, we will work exclusively in he equivalen aringale easure which, as already indicaed, ay, in fac, no be unique. If i is no unique, we will assue ha one has been fixed by he arke and we will call his he (raher han an) equivalen aringale easure. We ΩI,,Q and an denoe he equivalen aringale easure by Q. We fix a probabiliy space ( ) I which we assue saisfies he usual condiions. We denoe inforaion filraion ( ) expecaions, a ie, wih respec o he equivalen aringale easure Q by E [ ]. Sochasic evoluion of ineres-raes: We assue ha ineres-raes in our odel are sochasic. Le us inroduce soe noaion. We denoe he (coninuously copounded) risk-free shor rae, a ie, by r ( ) and we denoe he price, a ie, of a (credi risk free) zero coupon bond auring a ie by (, ). We assue ha (under he equivalen aringale easure Q ) he shor rae follows he exended Vasicek (one facor Gaussian) process, (Babbs (99), Hull and Whie (99),(993)) naely, dr () = α ( γ () r( ) ) d dz ( ), where dz ( ) r σ r denoes sandard Brownian increens, or equivalenly (Babbs (99), Heah e al. (99)) he dynaics of bond prices are (, ) σ = r () d + σ (, ) dz(), where σ (, ) exp( ( )) (, ) r αr α d r ( ), (.) 4

5 where σ r and r γ is defined so as o be consisen wih he iniial er srucure (ie he er srucure of ineres raes oday, ie ), which we ake as given. Coodiies: α are posiive, finie consans and ( ) We denoe he value of he coodiy, a ie, by C. he value of he coodiy is usually ered he spo price. However, in his paper, we shall ofen use he expression value of he coodiy because, in soe coodiies arkes, he spo price is no always exacly easy o define. We denoe he forward coodiy price, a ie, o (ie for delivery a) ie, by F (, ). We denoe he fuures coodiy price, a ie, o (ie he fuures conrac aures a) ie, by H (, ). I can be shown (Cox e al. (98), Duffie (996)), ha in he absence of arbirage, ha (, ) F E exp r( s) dsc = and H( ) E [ C ], (, ) (.) =. (.3) A key o odelling coodiy prices when ineres-raes are sochasic is o recognise ha, in his case, fuures coodiy prices and forward coodiy prices are no he sae. Indeed equaions. and.3 show ha fuures prices are aringales wih respec o he equivalen aringale easure whereas, when ineres-raes are sochasic, forward prices are no. Noe ha equaions. and.3 are consisen wih (, ) = C = H ( ) and F(, ) C = H (, ) F, = (.4) We ake as given our iniial er srucure (ie he er srucure oday, ie ) of fuures coodiy prices. ha is, we know H (, ) for all of ineres, ( ) (perhaps, in pracice, by inerpolaion of he fuures prices of a finie nuber of fuures conracs). In soe odels, he dynaics of he value of he coodiy are posied and hen equaions. and.3 would be used o derive he dynaics of forward coodiy prices and fuures coodiy prices. By conras, our odel will posi he dynaics of fuures coodiy prices. In oher words, fuures conracs are no derivaives bu, insead, are he priiive asses of our odel. he dynaics of fuures coodiy prices H(, ), under he equivalen aringale easure, which we will posi, will be consisen wih he aringale propery of equaion.3. Now we inroduce he insananeous fuures convenience yield forward rae ε (, ), a ie, o ie via he relaion ( ) C = H, exp ( ) ε,. (.5), s= ( s) ds We inroduce K sandard Brownian increens denoed by ( ) We denoe he correlaion beween dz ( ) and dz ( ) by ρ beween () k dz and () dz, for each k, k =,,..., K., for each k, and he correlaion dz Hj by ρ Hj for each j and k, j, =,,..., K. We also inroduce M independen oisson processes denoed by N, for each, =,..., M, wih λ under he equivalen aringale N, whose inensiy raes are ( ) 5

6 easure Q. We assue ha, for each,,..., M funcions of a os. We inroduce ( ) b, for each,,..., M =, ( ) λ are posiive, bounded deerinisic =, which are non-negaive deerinisic funcions, which we call jup decay coefficien funcions. hey conrol, when here are jups, how uch less long-daed fuures conracs jup by han shor-daed fuures conracs, in a way which we ake precise in assupion.5 and reark.. We inroduce γ, for each, =,..., M, which we call spo jup apliudes (in view of reark.), which are paraeers, which deerine he size of he jup, condiional on a jup in N. A risk of coplicaion, bu for he sake of breviy, we will consider wo possible specificaions for he spo jup apliudes, and in urn, hese are linked o wo possible specificaions of he jup decay coefficien funcions. For each,,..., M Assupion. : he spo jup apliudes coefficien funcions () =, we assue ha eiher: γ are assued o be finie consans. In his case, he jup decay b are assued o be any non-negaive deerinisic funcions. Or: Assupion. : he spo jup apliudes γ are assued o be independen and idenically disribued rando variables, whose disribuion is defined wih respec o he equivalen aringale easure Q, saisfying < EN ( exp( γ )) <, each of which is independen of each of he Brownian oions and of each of he oisson processes. In his case, he jup decay coefficien funcions b () are assued o be idenically equal o zero ie b ( ) for all. where, for each, E N denoes he expecaion operaor, a ie, condiional on a jup occurring in N. If, for a given, he spo jup apliude is consan (assupion.), he expecaion operaor is se equal o is arguen (see equaion.6). Reark.3 : Noe ha for each, we assue eiher assupion. or assupion. is saisfied. For differen i could be a differen assupion (ie if M >, we can ix he assupions). Noe also ha alhough we index he spo jup apliudes γ wih, boh assupions iply ha heir oucoes do no depend on, ie he index siply refers o he ie a which a jup ay occur. Reark.4 : he oivaion for hese assupions is as follows: Crosby (5) and (in he conex of an ineres-rae odel) Bjork e al. (997) show ha i ay no be possible, in general, unless a very specific and non-rivial condiion is saisfied, whils being consisen wih he absence of arbirage, o have boh jups whose apliudes are rando variables and siulaneously have jup decay coefficien funcions ( b () ) which are no idenically zero. Hence we assue ha all he oisson processes saisfy eiher assupion. or assupion.. We will develop he odel wih assupions. and. in parallel since he choice of hese assupions scarcely alers he developen., in he denoinaor of equaion.5 (see also C coens in secion 3), he effec of applying Io s lea o exp ε ( sds, ) (, ) and by s= he knowledge ha fuures coodiy prices are aringales in he equivalen aringale easure. We are oivaed by he presence of ( ) Assupion.5 : We assue ha he dynaics of fuures prices in he equivalen aringale easure Q are: 6

7 dh H (, ) (, ) K k = (, ) dz ( ) ( ) dz ( ) = σ σ, M + exp γ exp b ( u) du dn = M λ () EN exp γ exp b ( u) du d = (.6) σ,, for each k, k =,,..., K, are bounded (ie saisfying he Novikov condiion) deerinisic funcions of and, of he for: where ( ) σ (, ) = η () + χ () exp a ( u) du, (.7) where η (), χ ( ) and a ( u ) are deerinisic funcions 5. Reark.6 : Fuures coodiy prices are aringales in he equivalen aringale easure Q. Reark.7 : In he absence of jups, he dynaics of fuures coodiy prices in he equivalen aringale easure are very siilar o hose of forward prices in Clewlow and Srickland (999) (alhough we also incorporae sochasic ineres-raes). When K = (and in he absence of jups), equaion.6 gives dynaics for fuures coodiy prices which are essenially idenical o hose in Milersen and Schwarz (998) alhough hey ake he saring poin of heir odel, he dynaics of spo coodiy prices and convenience yields. hus, equaion.6 generalises well known diffusion odels in he lieraure. I also generalises Casassus and Collin-Dufresne (5), Clewlow and Srickland () and Hilliard and Reis (998) in wo ain ways. Firsly, our odel is auoaically consisen wih any iniial er srucure of fuures coodiy prices. Secondly, our odel has a uch ore general specificaion of jups. For exaple, he forer odels only consider jups of he ype in assupion.. We explain in reark 3.3 why hese laer odels canno generae jups of he ype in assupion.. Reark.8 : If he spo jup apliudes for all, =,..., M, are consans (ie hey all saisfy assupion.), and if he he nuber of fuures conracs in he arke is greaer han or equal o K + M +, hen he resuls of Babbs and Webber (994),(997), Bjork e al. (997), Jarrow and Madan (995), Hoogland e al. () and Crosby (5) show ha our arke is coplee. In his case, here exiss a unique equivalen aringale easure. All derivaives can be replicaed or hedged (see, for exaple, Hoogland e al. () and Crosby (5) for ore deails) by a self-financing porfolio of fuures conracs and risk-free bonds. his will iply he unique pricing of all derivaives. Reark.9 : In pracise, os coodiies arkes have fuures conracs of any differen auriies. For exaple, here are fuures conracs on WI grade crude oil for ore han differen auriies. Even for base eals, which are less acively raded han crude oil, he London Meal Exchange rades fuures conracs for 7 differen auriies on a wide variey of differen base eals. Hence, in he case ha all spo jup apliudes are consans (ie he spo jup apliudes for all saisfy assupion.), hen, for exaple, if K were se equal o hree, hen he nuber of oisson processes M could be se o or ore and, in his case, our arke would sill be coplee. In 5 We noe ha i will becoe clear laer ha in order o avoid a poenial degeneracy we ay pu () all k excep one, (or cobine ers of he for η ( ) dz ( ) o ease noaion. In addiion, we noe ha he funcions a ( ) ean reversion rae paraeers and hence we expec he o be posiive. η for ) bu we will wrie ou equaions below in full u will be seen o have an obvious inerpreaion as 7

8 pracise, seing M o equal jus one or wo, say, would be ore realisic o allow for an easier calibraion whils sill allowing considerable flexibiliy in he odel. Reark. : If any of he spo jup apliudes, for any, =,..., M, saisfy assupion. (ie if any are rando variables), or if here are insufficien fuures conracs in he arke, hen he sae references, cied in reark.8, iply ha our arke will no be coplee. he absence of arbirage iplies (Harrison and liska (98)) ha an equivalen aringale easure exiss bu he incopleeness of our arke iplies i will no be unique and hence we canno uniquely deerine he price of a derivaive since ebedded wihin he values of he inensiy raes λ ( ) and (in he case of assupion.) he paraeers of he disribuion of he spo jup apliudes, under an equivalen aringale easure, are arke prices of risk. Corresponding o each possible equivalen aringale easure, here will be differen values of hese paraeers leading o differen derivaive prices. In secion 5, we show ha sandard opions have prices of a siple for. We can esiae he paraeers of our odel, by invering he arke prices of such opions (provided here are sufficien opions in he arke). Ebedded wihin hose paraeers, specifically he inensiy raes and (for assupion.) he paraeers of he disribuion of he spo jup apliudes, are arke prices of risk which are fixed by he arke and which herefore also fix he equivalen aringale easure Q. his is a sandard echnique in incoplee arkes. For noaional convenience, we define, for each, he deerinisic funcion e (, ) via: e(, ) λ( ) EN exp γ exp b( u) du (.8) By Io s lea for jup-diffusions, applied o equaion.6, and using equaion.8, K K d( ln H (, )) = σ (, ) + σ (, ) ρ σ (, ) σ (, ) d k = k = K k K ρ Hj σ (, ) σ Hj (, ) d + σ (, ) dz ( ) σ (, ) dz ( ) k = j= k = M M + γ exp b ( u) du dn e (, ) d (.9) = = and where we have used he usual convenion ha if he upper index is sricly less han he lower index in a suaion, hen he su is se o zero. Reark. : Equaion.9 enables us o beer describe he size of he jup when one happens. When here is a jup in N, ln H(, ) changes by γ exp b ( u) du. Le us briefly consider he iplicaions of his. When here is a jup, he log of he fuures coodiy prices infiniesially close o auriy (ie he spo price C ln H(, ) ) jup by γ. However, he log of he fuures coodiy prices for delivery ( ) years ahead jup by γ exp b ( u) du. Considering he lii, as ( ), (and provided exp b ( u) du ), hen very longdaed fuures coodiy prices do no jup a all. he effec of he jup decay coefficien funcion b (), (which is assued always non-negaive), is o exponenially dapen he effec of he jup hrough fuures coodiy price enor. his sees o be in line wih epirical observaions in os 8

9 coodiies arkes (his is paricularly a feaure in he case of oil, naural gas and elecriciy). In he case of assupion., b () and jups cause parallel shifs in he log of he fuures coodiy prices across differen enors (sylised observaions sugges his is ore appropriae for gold). Le us reurn o he odel: Define he sae variables: () σ exp( α ( )) ( ), Y() σ rdz( s) X s dz s r r χ exp s, and, for each k, k =,,..., K, ( ) ( ) ( ) ( ), Y ( ) ( ) ( ) η s dz s X s a u du dz s Define, for each, =,..., M, he jup sae variables: X () γ exp b ( u) dudn N s s s. (.). (.) roposiion. : he evoluion, fro ie o ie, of he fuures coodiy price o ie, can be expressed in ers of he sae variables as: K K ( ) ( ) H, = H ( ) ( ) ( ) ( ), exp σ s, + σ s, ρ σ s, σ s, ds k = k = K k exp ( ) ( ) ρ Hjσ s, σ Hj s, ds k = j= K K exp( α r ( ) ) exp Y () + exp a ( u) du X () + X () Y () k= k= α r α r M M exp exp b( u) du XN( ) e( s, ) ds = = (.) roof : Rewrie equaion.9 in inegral for fro o, hen use equaions. and.. his shows ha H(, ) and H ( ) C, are Markov in a finie nuber of sae variables 6. Reark.3 : Wih he help of resuls in secion 4 (specifically equaion 4.5), i is sraighforward o verify by direc calculaion, using equaion., ha [ ] (, ) = (, ) E C E H H which confirs consisency wih equaion.3. roposiion.4 : he forward coodiy price F(, ), a ie, o (ie for delivery a) ie is relaed o he fuures coodiy price H(, ) via: 6 We noe ha i is sraighforward o cobine he Y ( ) and Y ( ) ino a single sae variable. We could do his, bu prefer no o, in order o axiise he inuiion behind he odel. However, i shows ha H (, ) C and H(, ) are, in fac, Markovian in K + + M sae variables. 9

10 F(, ) = H(, ) exp ρ σ ( s, ) σ ( s, ) σ ( s, ) ds K. (.3) k = roof : We can change he probabiliy easure in equaion. o he forward risk-adjused easure under which forward coodiy prices are aringales and hen use Girsanov s heore. Since Jashidian (993) (in a diffusion seing) proves a siilar resul, we oi he full proof here. Reark.5 : Noe ha forward coodiy prices and fuures coodiy prices differ only by a deerinisic er. Our odel has, hus far, been expressed in ers of fuures coodiy prices, bu, clearly, i would have been sraighforward o have worked wih forward coodiy prices insead. 3. Sochasic convenience yields and ean revering coodiy prices Our ai in his secion is o give greaer insigh ino our odel and, in paricular, o give resuls abou sochasic convenience yields and ean reversion 7 which show ha our odel is able o capure he sylised observaions of he coodiies arkes ha were ade in secion. Firsly, we derive he dynaics of he value of he coodiy. roposiion 3. : he dynaics of he value of he coodiy are as follows. If we define ( s) ( s) K σ, σ, εr() ε(, ) σ ( s, ) + 4 σ( s, ) ds k = ( ) ( ) K k σ Hj s, σ s, ( ) ( ) ρ + s Hj σ s, σ Hj, ds k = j= K ( ) K ( ) ( ) ( ) σ s, σ s, ρ σ s, ds ρ σ s, ds k = k = K σ ( s, ) dz ( s) k = M M e ( s, ) + γ sb () exp b ( u) du dns + ds = s = hen (3.) K dc = + C k = M M + ( exp( γ ) ) dn e (, ) d = = ( r() εr() ) d σ (, ) dz ( ). (3.) roof : We rewrie equaion.9, our SDE for ln H(, ), for ln (, ) H s insead, and hen rewrie in inegral for fro o. hen by differeniaing wih respec o, we ge he SDE for he dynaics 7 Casassus and Collin-Dufresne (5) show, by using sae-dependen risk-preia, ha here can be a differen degree of ean revering behaviour beween he real-world physical easure and he risk-neural easure. Saedependen risk-preia could also be used in precisely he sae way in our odel. However, for he sake of breviy and since our presen ai is o develop a odel for pricing coodiy derivaives, we focus only on he riskneural equivalen aringale easure. We will show ha our odel produces ean reversion in he risk-neural equivalen aringale easure (excep in he (degenerae) case ha all he ean reversion rae paraeers and all he jup decay coefficien funcions are idenically equal o zero).

11 of he log of he value of he coodiy, ln C ln H(, ). We can siplify his SDE and hen subsiue fro equaion 3., whence Io s lea gives equaion 3.. Noe ha he SDE in equaion 3. has a drif er which (by consrucion) is of a failiar for. We can also use Io s lea applied o ε (, ) = ln ( H(, ) (, ) C ) o obain he SDE for he dynaics of he insananeous fuures convenience yield forward rae ε (, ). Furherore, if rewrie his SDE for ε (, ) for ε ( s,) insead, and hen rewrie his SDE in inegral for fro o, hen we can, wih a lile algebra, show ha ε (, ) = ε r ( ) where ε r () is defined as in equaion 3.. his jusifies our noaion for ε r ( ) and ε (, ) (ie i jusifies our choice of ε r () in equaion 3. and shows is consisency wih equaion.5). We will call ε (, ) (using erinology analogous o ineres-raes) he fuures convenience yield shor rae. Reark 3. : We have sared he developen of our odel by assuing ha he dynaics of fuures coodiy prices are as in equaion.6 and hen shown ha our odel iplies ha he dynaics of he value of he coodiy and he fuures convenience yield shor rae are given by equaions 3. and 3. respecively. One could, of course, go in he opposie direcion and sar by assuing he dynaics of equaions 3. and 3. and hen showing ha he dynaics of fuures coodiy prices us be given by.6. However, i sees o us ha saring wih he dynaics of fuures coodiy prices is a uch ore naural saring poin, firsly, because, one can direcly observe he prices of, and rade in, fuures conracs (which one cerainly canno do direcly wih convenience yields), and, secondly, he for of equaion.6 sees uch ore inuiive han he for of equaion 3.. Reark 3.3 : Noe ha he fuures convenience yield shor rae ε (,) follows a ean revering sochasic process and ha i also (excep in he special case ha b ( ) for all ) exhibis jups. his leads o an iporan conclusion. We know, fro equaion.6 ha o capure he effec ha long-daed fuures conracs jup by saller agniudes han shor-daed fuures conracs, a leas one of he M spo jup apliudes us saisfy assupion. wih b ( ) >. If his is he case, hen fro equaion 3., he fuures convenience yield shor rae ε (,) us exhibi jups. We now see why (see also reark.7) exising odels in he lieraure such as Casassus and Collin-Dufresne (5), Clewlow and Srickland () and Hilliard and Reis (998) canno capure he effec ha long-daed fuures conracs jup by saller agniudes han shor-daed fuures conracs. he exising odels in he lieraure only consider jups in spo coodiy prices hey do no allow jups in he dynaics of convenience yields. In our noaion, his is equivalen o assuing b () for all. Hence, exising odels iply fuures coodiy prices across differen enors jup by he sae proporional aouns which is conrary o he epirical evidence, for os coodiies arkes, presened in secion. Noe ha (observing equaion 3.) he volailiy of he value of he coodiy does no depend on he volailiy of bond prices or ineres-raes. If we exaine he SDE for fuures coodiy prices σ, dz? In view of (equaion.6), he quesion igh be asked: Why have he er ( ) ( ) equaion.5, Io s lea iplies ha he dynaics of (, ) (, ) = exp ε (, ) H C s ds s= σ, dz in do no depend on he er σ (, ) dz ( ). If we did no have he er ( ) ( ) equaion.6, hen he dynaics of he value of he coodiy and hose of he fuures convenience yield shor rae would depend on he Brownian oion driving ineres-raes and bond prices. Alhough here would be nohing wrong wih his, i jus sees less inuiively appealing. Of course, in a sense, wriing he dynaics of fuures coodiy prices in he for of equaion.6 is a non-assupion in ha given he diffusion ers in equaion.6 for any K, we can rewrie he in he for

12 K σ (, ) dz () σ (, ) dz () = σ (, ) dz (), where ' K k= k= r r dz () dz () HK ' and σ ( ) σ( ) αr( ) HK αr αr Hence, he volailiy er ( ) K ' K +, ', σ σ, = + exp( ). σ HK ', is sill of he for of equaion.7 and hence he odel is sill of essenially he sae srucure. he following proposiion provides furher insigh ino our odel because i shows ha he log of he value of he coodiy exhibis ean reversion. roposiion 3.4 : he SDE for he log of he value of he coodiy can be wrien: d K H, k = ( ln C ) = a ( ) ( Λ(,,ln H (, ) ) ( ln C )) d + σ ( ) dz ( ) M M + = = γ dn e (, ) d (3.3) where (,ln H (, ) ) Λ, is a funcion of he sae variables (whose exac for is easily obained a he expense of soe algebra). roof: We obain an expression for C H(, ) by seing equal o in equaion. and hen we ake logs. We hen use his wih he SDE for ln C ln H(, ), we obained in he proof of proposiion 3., o eliinae one of he sae variables X ( ). he choice is arbirary bu o be definie, we eliinae X H (). We obain equaion 3.3. Reark 3.5 : his shows ha ln C follows a ean revering jup-diffusion process wih a long run ean reversion level of Λ (,,ln H(, ) ). In fac, we can show ha Λ (,,ln H (, ) ) is iself also a ean revering jup-diffusion process. Noe also ha if he jup decay coefficien funcions b () are sricly posiive, hen hey also conribue o he effec of ean reversion. his can be seen even ore clearly if we follow he proof of proposiion 3.4 bu, insead of eliinaing he sae variable, say. We hen obain an equaion soewha X H (), we eliinae he jup sae variable X N ( ) siilar o equaion 3.3 in for, bu wih he (ean revering) drif er conaining b () insead of a H (). Indeed we see ha he jup decay coefficien funcions b ( ) play an analogous role for he jup processes as he ean reversion rae paraeers ( ) a play for he Brownian oions. We noed, in secion, he sylised epirical observaion ha, afer a large jup in he price of a shordaed fuures conrac, he price ofen sees o rever o a ore usual level very quickly. Our odel is poenially able o capure his effec hrough large values of he jup decay coefficien funcions. How can we suarise his secion? We have seen ha he drif er in he SDE for he value of he coodiy (equaion 3.) is equal o he risk-free rae inus he fuures convenience yield shor rae. he drif er (equaion 3.3) can also siulaneously be viewed as ha for a ean revering sochasic process. he fuures convenience yield shor rae is a ean revering sochasic process and b for all ) i also exhibi jups. Mos iporanly, we have (excep in he special case ha () shown ha, in order o capure he effec ha he prices of long-daed fuures conracs jup by saller agniudes han shor-daed conracs, i is necessary o have jups in he dynaics of he fuures convenience yield shor rae his is a feaure which is issing fro exising odels in he lieraure.

13 4. Mone Carlo siulaion In his secion, we show how we can siulae fuures coodiy prices. he key o his will be o siulae he sae variables since hen we can use equaion.. Mone Carlo siulaion of he diffusion sae variables is sraighforward (see Babbs (99), Depser and Huon (997) or Glasseran (4)). So now we exaine how we can siulae he jup sae variables, X N (). Firsly, for fuure noaional convenience, we define, for each, φ b u du (, ) exp ( ). (4.) =, Recall, ha for each,,..., M λ. he process sars a zero ie N and every ie a jup occurs, he process increens by one. Now, by he definiion of a non-hoogenous oisson process, he probabiliy Q (,; n ) ha here are n jups in he oisson process N has a oisson disribuion wih inensiy rae () N in he ie period o is: Q n N n u du (,; ) r ( ) exp = = λ ( ) (4.) n! We now sae a very useful aheaical proposiion. λ u du roposiion 4. : Suppose ha we know ha here have been n jups beween ie and ie. Wrie he arrival ies of he jups as S, S,..., S n. he condiional join densiy funcion of he arrival ies, when he arrival ies are viewed as unordered rando variables, condiional on N = n is: ( S s S s Sn sn N n) r = & = &...& = = = n λ( s ) λ( s )... λ( sn ) ( ) λ u du (4.3) roof : he above resul is proved in, for exaple, Karlin and aylor (975) in he case ha he inensiy rae is consan and he exension o a ie-dependen deerinisic inensiy rae is relaively sraighforward (and herefore he proof is oied). his is an iporan resul because now i is sraighforward o siulae X N (). Firsly, we siulae he nuber of jups n up o ie. here are several ways o siulae he nuber of jups, in a given ie inerval, of a non-hoogenous oisson process (for exaple, see Glasseran (4)). Using equaion 4.3, we can siulae he arrival ies S, S,..., S n of he n jups beween ie and ie. (his is paricularly sraighforward if λ ( ) is consan since hen he arrival ies, condiional on n, are unifor on (, ). If λ ( ) is no consan, we can use he inverse ransforaion ehod (Glasseran (4))). X, iplies ha Now noe ha equaion., he definiion of ( ) N ( ) n 3

14 n n X N () = γ S exp b ( ) (, ) i u du = γ S φ i Si. (4.4) i= S i= i If n =, hen XN ( ) =. We include his case in equaion 4.4 by using he usual convenion ha a suaion is zero if he upper index is sricly less han he lower index. I only reains o siulae γ (in he case of assupion., he jup sizes are known consans, so, in fac, no furher siulaion is required, and, in he case of assupion., hey are independen and idenically disribued which eans hey do no depend on he arrival ies, and, in fac, equaion 4.4 siplifies since () b ) and hen we obain ( ) X fro equaion 4.4. N In order o siulae fuures coodiy prices, we also need he final deerinisic er in equaion.. his involves an inegral which would, in general, have o be done nuerically (see also Crosby (6a)), bu i is a siple one diensional deerinisic inegral which can be pre-copued before enering he Mone Carlo siulaion. We will use he following proposiion in secion 5. roposiion 4. : M M E exp exp b ( ) ( ) (, ) u du XN e s ds = = = (4.5) roof : Use equaions 4. and 4.3 and sandard resuls abou condiional expecaions. Reark 4.3 : Noe (leaving aside he issue of any errors in he evaluaion of he final deerinisic inegral in equaion.), ha here are no discreisaion error biases in he siulaion of fuures coodiy prices in our odel as here igh be in soe odels involving he siulaion of non- Gaussian sochasic processes (for discussions on his opic, see Babbs () or Glasseran (4)). 5. Opion pricing Our ai in his secion is o derive he prices of sandard opions and o do so (in spie of he obvious fac ha, in our odel, fuures coodiy prices are no log-norally disribued) in a for suiable for rapid copuaion. he key o his will be he observaion ha, condiional on he nuber of jups and heir arrival ies (and wih a suiable assupion abou he spo jup apliudes), fuures coodiy prices are log-norally disribued, a which poin failiar resuls coe ino play (see also Meron (976) and Jarrow and Madan (995)). We will derive he prices of sandard European opions on fuures, fuures-syle opions on fuures and sandard European opions on forward coodiy prices. Laer in his secion, we will provide soe nuerical exaples which illusrae our odel. We will also show ha we can rapidly (ypically of he order of /5 h of a second per opion depending upon he required accuracy) copue he prices of sandard opions. o achieve our goals, we will have o ake an assupion abou he disribuion of he spo jup apliudes γ in he case of assupion.. We assue he spo jup apliudes γ are norally disribued. For each : In he case of assupion., he spo jup apliudes are assued o be equal o β, a consan. In he case of assupion., he spo jup apliudes are assued o be norally disribued wih ean β and sandard deviaion υ, which hen iplies exp γ exp b ( ) s b ) u du is log- s norally disribued and using sandard resuls, we have (puing ( ) 4

15 ENs exp γ s exp b ( u) du = exp β + υ s (5.) A generic opion pricing forula: Our ai is o value, a ie, a European (non-pah-dependen) opion, auring a ie, wrien on he fuures coodiy price, where he fuures conrac aures a ie, ( ). Firsly, define he indicaor funcions, for each, =,..., M, (.) = if assupion. is saisfied, for his, and (.) = oherwise and (.) = if assupion. is saisfied, for his, and (.) = oherwise. Condiional on he nuber of jups n, =,..., M, in he ie period o, and he arrival ies s, s,..., s, =,..., M of hese jups, hen (using equaions 4. and 4.4): n n exp exp b( u) du XN( ) = exp βφ ( si, ) if ( ) = (5.). i= or: exp exp b( u) du XN( ) is log-norally disribued wih ean n exp β + υ= exp n β + υ i= if ( ) = (5.3). roposiion 5. : he fuures coodiy price (, ) fuures price (, ) H a ie o ie, condiional on he H a ie (where ) and condiional on he nuber of jups n, =,..., M, in he ie period o, and he arrival ies s, s,..., s n, =,..., M of hese jups, is log-norally disribued wih ean n M M H(, ) exp (.) βφ ( si, ) + (.) n β + υ e( s, ) ds = i= = H, V, ; n ;, M (5.4) ( ) ( ) where V (, ; n ;, ) M is defined by equaion 5.4 and where: exp, = exp exp, and e( s ) ds λ( s) ( ( φ( s ) β) ) ds if exp e( s, ) ds = exp exp β + υ λ( u) du if (.) = (5.5) (.) = (5.6) 5

16 roof: Equaion 5.4 follows iediaely fro equaion., aken ogeher wih equaions 5. and 5.3. Equaions 5.5 and 5.6 use he definiions in equaions 4. and.8. Noe ha, in general, i would be necessary o copue he inegral in equaion 5.5 nuerically. Now, define Σ (,,, M), via: K K k= k= K k M ρhjσ ( s, ) σhj ( s, ) ds ( nυ (.) ) k= j= = Σ (,,, M) σ ( s, ) + σ ( s, ) ρ σ ( s, ) σ ( s, ) ds + + (5.7) Consider a non-pah-dependen European opion wrien on he fuures coodiy price. he opion aures a ie and he fuures conrac aures a ie. Le he payoff of he opion a ie be D( H(, ) ) for soe I -easurable funcion D ( ). Condiional on he nuber of jups n, =,..., M and he arrival ies s, s,..., s n, =,..., M of hese jups, he value of he opion a ie is (where ): E exp r( u) du D( H(, ) ) n, s, s,..., sn ;,..., = M (5.8) Reark 5. : In view of proposiion 5., given he payoff ( (, ) ) D H of he opion a ie, we will be able o use sandard resuls (for log-norally disribued prices), ogeher wih equaions 5.4, 5.5, 5.6 and 5.7, o calculae he expecaion in equaion 5.8. Noe ha he probabiliy ha here are, for each,,..., M n jups in he oisson process =, is Q (, ; n ), where (, ; ) roposiion 5.3 : he price of he opion a ie is: n= n= nm = n= n= nm = ( ) ( ) ( )... Q, ; n Q, ; n... Q, ; n M n N in he ie period o Q n is defined as in equaion E exp r( u) du D( H(, )) n, s, s,..., s ; =,..., M λ( s) λ( s)... λ( s ) n dsds... ds dsds... ds... ds ds... ds n λ ( u) du M = M n n M M nm M (5.9) roof: I follows iediaely fro he resuls of secion 4 (in paricular equaions 4. and 4.3) and sandard resuls abou condiional expecaions. Reark 5.4 : Furher, in he special case ha assupion. is saisfied for all, =,..., M, he V, ; n ;, M is siplified and he opion price a ie also siplifies o: for of ( ) 6

17 n= n= nm = n= n= nm = ( ) ( ) ( )... Q, ; n Q, ; n... Q, ; n M E exp r( u) du D( H(, ) ) n; =,..., M (5.) M Using equaions 5.4, 5.7, 5.8 and 5.9, we are now in a posiion o wrie down he prices of various sandard opions. Our specific opion pricing forulae will coe fro subsiuing a specific for for equaion 5.8 ino 8 equaion 5.9. We sae he resuls wihou proof bu for full deails and ehodologies, see Meron (973),(976), Babbs (99), Ain and Jarrow (99), Duffie and Sanon (99), Jashidian (993), Jarrow and Madan (995) and especially Milersen and Schwarz (998). In each of he following opion pricing forulae (equaions 5. o 5.4), we denoe he srike of he opion by K and we wrie η if he opion is a call and η if he opion is a pu. Sandard European Opions on Fuures: = Suppose ha we wish o value a ie a sandard European (call or pu) opion on he fuures coodiy price. he opion aures a ie and he fuures conrac aures a ie, where. he payoff of he opion a ie is ax( η ( H (, ) K ),). he price of he opion a ie is given by 8 equaion 5.9 wih E exp r( u) du D( H(, ) ) n, s, s,..., sn ;,..., = M replaced by = η H V n M A s ds N ηd KN ηd where A d (, ) (, ) (, ; ;, ) exp (,, ) ( ) ( ) K ( s,, ) ρ σ ( s, ) σ ( s, ) σ ( s, ) σ ( s ), k = ln,, ; ;,,,,,, ( H( ) V( n M) K) + A( s ) ds+ Σ ( M) Σ(,,, M) Σ (,,, ) and (, ; ;, ) d d M V n M is as in equaion 5.4.,, (5.) Fuures-syle Opions on Fuures: Suppose ha we wish o value, a ie, a fuures-syle opion (call or pu) on he fuures coodiy price. Fuures-syle opions are raded on soe exchanges and he key poin abou he is ha hey are siilar o fuures conracs in ha he gains and losses of he fuures-syle opion are reseled coninuously (in pracice, daily), wih a ark-o-arke procedure, and, as wih fuures conracs, here is no iniial cos in buying a fuures-syle opion. We assue ha he fuures-syle opion aures a ie and he fuures conrac aures a ie, where. he fuuressyle opion price (ie is delivery value) a ie is ax( η ( H (, ) K ),). I can be shown (see Meron (99), Duffie (996) or Duffie and Sanon (99)) ha he fuures-syle opion price, a ie, is he price of a sandard (ie non-fuures-syle) opion, a ie, which has a payoff of 7

18 exp r( u) du ax ( η ( H(, ) K),) a ie. Hence he fuures-syle opion price, a ie, is: E exp r( u) du exp r( u) du ax ( η ( H(, ) K),) = E ax ( η ( H(, ) K),) Hence, we can show ha he fuures-syle opion price a ie is given by 8 equaion 5.9 wih equaion 5.8 replaced by ( H(, ) V (, ; n;, M) N( d) KN( d) ) η η η (5.) where ln,, ; ;,,,, d ( H( ) V( n M) K) + Σ ( M) Σ(,,, M) and d d Σ (,,, M) Reark 5.5 : I can be shown (using he ehods of Meron (973), Duffie and Sanon (99) and Jashidian (993)) ha i is never opial o exercise Aerican fuures-syle opions on fuures prices before auriy (his applies o boh calls and pus). Hence equaion 5., cobined wih 8 equaion 5.9, is equally valid for boh European and Aerican fuures-syle opions on fuures prices. Sandard European Opions on Forwards: Suppose ha we wish o value a ie a sandard European (call or pu) opion on he forward coodiy price. he opion aures a ie and he forward price is o ie, where. here are wo possible payoffs: is ( ( ( ) ) ) We consider firs he case where he payoff of he opion a ie ax η F, K,. he price of he opion a ie is given by 8 equaion 5.9 wih equaion 5.8 replaced by η F V n M B s ds N ηd KN ηd (, ) (, ) (, ; ;, ) exp (,, ) ( ) ( ) where K B( s,, ) ρ σ ( s, ) σ ( s, ) σ s, σ s, σ s, σ s,, d ( ) ( ) k = ( ( ) ( ) ) ( ) ln,, ; ;,,,,,, Σ ( F( ) V( n M) K) + B( s ) ds+ Σ ( M) and d d Σ (,,, M) (,,, M) (5.3) 8 or equaion 5., in he special case noed in reark

19 We consider secondly he case where he payoff of he opion is also ( ( F(, ) ),) ax η K bu now he payoff occurs a ie. his eans he payoff is he sae as a payoff of ax( η (, )( F(, ) K ),) a ie. he price of he opion a ie is given by 8 equaion 5.9 wih equaion 5.8 replaced by (, ) ( (, ) (, ; ;, ) ( ) ( ) ) η F V n M N ηd KN ηd (5.4) where ln,, ; ;,,,, d ( F( ) V( n M) K) + Σ ( M) Σ(,,, M) and d d Σ (,,, M) As an iediae corollary, we can obain he price of a sandard European opion on he spo coodiy price, by seing equal o in equaion 5.3 or 5.4 (subsiued ino 8 equaion 5.9). Nuerical exaples and copuaional issues: he above resuls are very useful as hey also allow he possibiliy o calibrae he odel hrough deriving iplied paraeers fro he arke prices of opions (for which purpose rapid copuaion is iporan). We will, laer in his secion, illusrae our odel wih a oal of five nuerical exaples, which we spli ino wo caegories, labelled exaples o 3 and exaples 4 and 5. Firsly, we discuss copuaional issues surrounding he rapid copuaion of opion prices using equaions 5.9 o 5.4. he probabiliies in he oisson ass funcions will rapidly end o zero once he nuber of jups is greaer han he ean nuber of jups. herefore, copuaion ies in he case when all he oisson processes saisfy assupion. will ypically be very sall (a leas when M is no oo large). When all or soe of he oisson processes saisfy assupion., i is necessary o copue he inegrals over he arrival ies. he os appropriae ehod would see o be o use Mone Carlo siulaion of he arrival ies (we sress only of he arrival ies no of he nuber of oisson jups nor he diffusion processes which can be done analyically). his is he ehod we use in he nuerical exaples below. Alhough his igh sound copuaionally inensive, he siulaion is jus of he arrival ies of he jups. In any cases, he variaion of he inegral wih differen arrival ies will be quie sall leading o sall sandard errors. his igh ypically be he case for b ) opions which are deep in or ou of he oney or when he jup decay coefficien funcions ( () are close o zero. In addiion o iniise sandard errors, we used he ehod of aniheic variaes and we also used equaion 4.5 as a conrol variae using he opial-weighing/linear-regression ehodology described, for exaple, in Glasseran (4). he opion prices in exaple and (when appropriae) in exaples 4 and 5 were all copued using 5 Mone Carlo siulaions. he deerinisic inegral in equaion 5.5 was copued using he rapeziu rule wih 5 poins. Using a uch larger nuber of poins confired ha he poenial errors in he opion prices in exaples, 4 and 5 due o he approxiaion inheren in copuing his inegral were, in all cases, uch less han. which is negligible copared o he sandard errors repored. In all he exaples, he suaion over he oisson ass funcions was runcaed when boh he proporional and absolue convergence of he opion price were less han.. Copuaions were perfored on a desk-op p.c., running a.8 GHz, wih Gb of RAM wih a progra wrien in Microsof C++. We now illusrae our odel wih our firs hree exaples, labelled exaples o 3, he resuls of which are in ables o 3 respecively. In all hree exaples o 3, we assue ha he fuures coodiy prices o all auriies are 95. We assue ha ineres-raes are sochasic and ha σ r =.96 and α r =.. We assue he ineres-rae yield curve is fla wih a coninuously copounded risk-free rae of.5 (as in Milersen and Schwarz (998)). Alhough any of he paraeers in our odel can be ie-dependen (and indeed i ay be useful o allow for his o capure, for exaple, seasonaliy (see Milersen (3))), we will illusrae he odel wih consan paraeers. In order o ach he paraeers of Milersen and Schwarz (998), whose se-up is slighly differen o ours bu enirely equivalen in he wo facor pure-diffusion case, 9

20 we choose o have wo Brownian oions (in addiion o he Brownian oion driving ineres-raes) ie K = and η =.66, η =.49 / , χ =., H H χ H =.49 /.45, a H =.45, ρ HH =.85, ρ H =.964, ρ H =.43. Noe he negaive value of χ H is arificial in order o ach he Milersen and Schwarz (998) daa and could be ade posiive by cobining η H and η H ino one er and aking consisen adjusens o he correlaions in he obvious anner. We now consider exaples o 3. Exaple is pure-diffusion and exaples and 3 are wih jups. he pure-diffusion exaple is effecively idenical o ha used in Milersen and Schwarz (998). We value sandard European call opions (using equaions 5.9 and 5.) on fuures conracs whose auriies are.5 years afer he auriy of he opion. We price opions wih srikes 75, 8, 95,, 5 and auriies equal o.5,.5,.75,,, 3 years (here are 3 opions in oal). Exaple : In exaple, we price opions in he pure-diffusion case (using equaion 5.). he resuls are in able. Clearly he resuls are exacly as in able of Milersen and Schwarz (998) (we have exra opion auriies and exra srikes) since we have (albei in a slighly differen for) he sae diffusion paraeers. Now we inroduce jup processes for exaples and 3 bu keep he diffusion paraeers as in exaple. he paraeers of our processes are purely for illusraion. Exaple : In exaple, we assue ha here is one oisson process, i has consan paraeers: λ =.75, β =., b =. H M = and i saisfies assupion. and he paraeers are only for illusraion. he value of b is roughly equivalen o he effec of a jup being dapened o approxiaely 37.8 % of he jup size over half a year which sees plausible. Now we price opions, using equaions 5.9 and 5., wih all he oher paraeers he sae as in exaple. he resuls are in able. Also in he able are he corresponding sandard errors (all are less han.8) and he corresponding iplied Black (976) volailiies wih 9 a price of 95. he oal copuaion ie for all 3 opions in his exaple was less han.5 seconds or an average of less han.7 seconds per opion. Exaple 3 : In exaple 3, we assue ha here are wo oisson processes, M = and hey boh saisfy assupion., wih paraeers: λ =.75, β =., υ =. (and b =. which is required for assupion.) λ =.75, β =.5, υ =. (and b =. dio) Again, he paraeers are only for illusraion. he inuiion of he paraeer values, loosely speaking, is o ry o capure a coodiy which can have upward jups and also have downward jups of slighly saller size, he inensiy raes of he wo oisson processes being equal. Now we again price opions, using equaions 5. and 5., wih he oher paraeers he sae as in exaple. he resuls are in able 3. Also in he able are he corresponding iplied Black (976) volailiies wih 9 a price of 95. Since b =. and b =., here is no inegraion over he arrival ies and hence copuaion ies were negligible copared o hose in exaple. I can be seen ha in boh exaples wih jups (ha is, exaples and 3), he odel produces a volailiy skew. he agniude of he skew decreases wih increasing opion auriy which is ypical for jup-diffusion processes (and is also in line wih epirical observaions in he coodiy opions 9 We use he fuures price of 95 when calculaing he iplied Black (976) volailiies for all our exaples because his sees o be in line wih he arke convenion even hough his convenion appears o effecively ignore he ipac of sochasic ineres-raes and hence he difference beween forward prices and fuures prices.

21 arkes (Gean (5)). In all hree exaples, we see ha iplied volailiies decrease wih increasing opion auriy (again in line wih ypical epirical observaions (Gean (5)). Calibraion o arke daa: We will now calibrae wo differen specificaions of our odel o arke daa. As in oher opions arkes, he coodiy opions arke quoes iplied Black (976) volailiies. We obained fro iner-dealer brokers live arke daa, as of 5 h January 5, for he iplied Black (976) volailiies of OC opions on crude oil fuures. he opions had eleven differen auriies, naely, onh, onhs, 3 onhs, 6 onhs, onhs, 8 onhs, years, 3 years, 4 years, 5 years and 6 years. he opions, a each auriy, were of seven differen srikes. he srikes were differen for each auriy because he arke convenion is o quoe opions a srikes given by specific Black (976) delas. In oher words, he srike is deerined by solving for he srike ha gives a specified dela when subsiued ino he forula for he dela in he Black (976) odel. he opions are eiher pus or calls according o which is ou-of-he-oney (excep, obviously, for a-he-oney-forward opions when pus and calls have he sae price). Our opions were a he srikes corresponding o (in order of increasing srike price) delas of -. (pu), -.5 (pu), -.35 (pu), a-he-oney-forward,.35 (call),.5 (call) and. (call). hus we have seven differen srikes a each of eleven opion auriies ie a oal of 77 opions. For all eleven differen opion auriies, he fuures conrac auriies are beween and 5 days afer opion auriy (he average ie beween fuures conrac auriy and he opion auriy was 3 days bu he exac ie depends on holidays and he daes of weekends). We firs exraced he US dollar ineres-rae yield curve fro LIBOR deposi raes and by boosrapping swap raes. We hen deerined he exended Vasicek odel paraeers by calibraing he exended Vasicek (Babbs (99), Hull and Whie (993)) odel o he arke prices of liquid European swapions. We hen calibraed wo differen specificaions of our odel o he arke iplied Black (976) volailiies. he specificaions were as follows: In each specificaion, we had wo Brownian oions (in addiion o he one driving ineres-raes). We se η H (in order o avoid a degeneracy) and we assued η H, χ H, χ H, a H and a H were all consans. In each specificaion, we had wo oisson processes ie M =. In he firs specificaion (we will call i Specificaion ), he jups apliudes were boh of he ype of assupion., wih λ, λ, b, b all assued consans and b >, b >. he paraeers o be deerined fro he calibraion were hus η H, χ H, χ H, a H, a H, ρ HH, ρ H, ρ H, λ, λ, β, β, b, b. herefore, here were 4 paraeers (in addiion o σ r and α r which were calibraed independenly o he arke prices of European swapions). However, in he second specificaion (we will call i Specificaion ), he jup apliudes were boh of he ype of assupion., wih λ, λ assued consans. he paraeers o be deerined fro he calibraion were hus η H, χ H, χ H, a H, a H, ρ HH, ρ H, ρ H, λ, λ, β, β, υ, υ. herefore, here were 4 paraeers he sae as for specificaion. o be precise, in our calibraion, we solved for he paraeers which iniised he sus of squares of proporional differences beween he odel and arke prices of he 77 opions. he resuls of he calibraion were ha we obained odel paraeers as follows: Exended Vasicek paraeers: σ r =. 9, α r =. 43. Specificaion paraeers: η =.646 H, η H (by design), χ =. H 93, χ =. H 795, a =. H 647, a =.63 H, ρ HH =. 434, ρ H =. 3485, ρ H =. 356, λ =.74, λ =. 6, β. 47 =, β =. 59, b =. 789, b =. 8 Specificaion paraeers: η =.34 H, η H (by design), χ =. H 37, χ =. H 577, a =. H 578, a =.88 H, ρ HH =. 3743, ρ H =. 38, ρ H =. 345, λ =.677, λ =. 588, β. 58 =, β =. 743, υ =.759, υ =.99

22 We have ploed he resuls of he calibraion, graphically, in ers of iplied Black (976) volailiies, in figures and (only wo auriies, naely onh and 4 years, are shown for breviy bu he res of he daa is qualiaively siilar). Boh specificaions were able o give a good fi o he arke iplied Black (976) volailiies. However, he fi for specificaion is uch beer han ha for specificaion. he residual value of he su of he squares of he proporional differences beween he odel and arke prices was.8 for specificaion whereas i was.3778 for specificaion. In addiion, he su of he squares of he differences beween he odel and arke iplied volailiies was.3 for specificaion whereas i was.467 for specificaion. In addiion, he axiu difference (in absolue value), across all 77 opions beween he odel and arke iplied volailiies was.68 percenage poins for specificaion whereas i was.98 percenage poins for specificaion. hus, by hree differen easures, specificaion gave a uch beer fi o he arke daa han specificaion. In addiion, we noe ha if we exclude he opions wih -. dela and. dela (ie in he wings), he differences (in absolue value) beween he odel and he arke iplied Black (976) volailiies, for specificaion, for he reaining 55 opions, were all under one percenage poin. his is indicaive of an excellen fi because % is he approxiae bid-offer spread (in volailiy ers) for hese opions (he bid-offer spread would ypically be soewha wider for he opions wih -. dela and. dela). o suarise, we have seen he fi o arke daa is uch iproved when he jups boh saisfy assupion. (as in specificaion ) which is in line wih he epirical feaures of he coodiies arkes presened in secion. We will now provide a furher wo exaples, labelled exaples 4 and 5, which illusrae our odel using he odel paraeers we have calibraed above. In boh exaples 4 and 5, we price sandard European call opions wih an opion auriy of years. he (coninuously copounded) risk free spo ineres-rae o wo years is Hence = and (, ) In each exaple, we price he opions wih he calibraed daa fro above for each of he wo specificaions of jups processes. Exaple 4 : In exaple 4, we price sandard European call opions on fuures conracs auring 3 days afer he opion auriy. Opions wih an opion auriy of wo years and a fuures conrac auriy 3 days laer fored a subse of he opions we used in he calibraion o he arke prices of opions on crude oil fuures above (bu now we will consider slighly differen srikes). he fuures price is 4.. Hence =, ( 3 365) = + and H (, ) = 4.. We priced sandard European calls whose srikes are 37., 4. and 45. ie wih srikes which are he curren fuures price and 4 dollars eiher side. he resuls are in able 4 where we display he opion price, he iplied Black (976) volailiy and (for specificaion ) he sandard error of he opion price. Exaple 5 : In exaple 5, we price sandard European call opions on fuures conracs auring 3 years and 3 days afer he opion auriy (he opion auriy is, again, wo years). In he calibraion o he arke prices of opions on crude oil fuures above, 5 year opions ino hese sae fuures conracs fored a subse of he opions o which we calibraed. By conras, in his exaple, we are pricing year opions ino hese fuures conracs ie we are now pricing year opions ino fuures conracs which aure 3 years and 3 days afer he opion auriy. he fuures price is 8.4. Hence =, 3 ( 3 365) = + +, H (, ) = We priced sandard European calls whose srikes are 4.4, 8.4 and 3.4 ie wih srikes which are he curren fuures price and 4 dollars eiher side. he resuls are in able 5 where we display he opion price, he iplied Black (976) volailiy and (for specificaion ) he sandard error of he opion price. Exaining able 4 closely we see ha he prices of he hree opions are very, very close in specificaion and specificaion. his is no surprising since opions wih he sae opion auriy and he sae fuures conrac auriy (albei wih slighly differen srikes) were used in he calibraion. he opion prices in specificaion are jus very slighly higher han for specificaion, for each of he hree srikes, bu he axiu difference beween he iplied volailiies of he corresponding hree opions in he wo differen specificaions is.3 %. Exaining able 5 closely we see a differen picure. For each of he hree opions, he opion price is now raher lower for specificaion han for specificaion. his sees inuiive because in specificaion, we have jups whose agniudes are exponenially dapened wih fuures conrac enor. We also see ha he iplied volailiy of he hree opions for specificaion iply a uch

23 flaer skew han in specificaion. Furherore, whils he slope of he skew in specificaion is very slighly upwards (ie he iplied volailiy of he hree call opions slighly increases wih increasing srike), he slope of he skew in specificaion is downwards (ie he iplied volailiy of he hree call opions decreases wih increasing srike). I is well-known ha i is possible o have wo differen sochasic processes which give rise o he sae prices for sandard European opions bu give rise o differen prices for exoic opions. In exaples 4 and 5, we are seeing he sae issue a work. We canno rejec or accep eiher specificaion or specificaion on he basis of he opion prices in exaples 4 and 5 bu we noe again ha specificaion is in line wih he epirical observaions we presened in secion and i also gives a beer fi o he arke prices of opions han specificaion. Milersen and Schwarz (998) reark (and we concur) on he iporance of fuures auriy (no jus opion auriy) on opion prices. Our experience is ha, a he oen, os sandard (plain vanilla) opions currenly raded in he coodiies arkes have auriies which are usually less han wo onhs (and ofen jus a few weeks) before he auriy of he underlying fuures conracs. As he coodiy derivaive arkes expand, his ay well change (copare he developen of he ineres-rae derivaives arkes: a one ie, caps were uch ore coon han swapions bu now he siuaion is alos reversed), in which case, as exaples 4 and 5 show, our odel will be paricularly useful. Our odel can also price coplex (exoic) coodiy derivaives uilising he Mone Carlo siulaion ehod described in secion 4. A possible area for fuure research would be o furher invesigae variance reducion echniques or o use quasi-rando nuber ehods (see for exaple, Glasseran (4)) o speed up he evaluaion of opion prices. A furher possibiliy is o price opions using he Fourier ransfor ehodology of, for exaple, Carr and Madan (999) and his is he subjec of furher research, on which we repor in Crosby (6a). Finally, exending our odel o allow for a uli-facor (raher han one facor) Gaussian ineres-rae odel is very sraighforward. 6. Conclusions We have considered a racable arbirage-free uli-facor jup-diffusion odel for he evoluion of fuures coodiy prices consisen wih any iniial er srucure. he odel allows long-daed fuures conracs o jup by saller agniudes han shor-daed fuures conracs, which is in line wih sylised epirical observaions (especially for energy-relaed coodiies). his is a feaure which, o our bes knowledge has never appeared in he lieraure before. We have relaed his odel o forward coodiy prices and o he value of he coodiy. We have shown ha he value of he coodiy exhibis ean reversion in he risk-neural equivalen aringale easure. We have relaed our odel o sochasic convenience yields which heselves exhibi ean reversion and also, in general (bu depending on he precise specificaion), exhibi jups. Whils soe of he expressions appear quie long, he odel described in his paper is concepually very inuiive. he odel is highly aenable o Mone Carlo siulaion (wihou discreisaion error bias), uilising equaions derived in secion 4. Hence, Mone Carlo siulaion can be used o price coplex (exoic) coodiy derivaives wihin our odel. We have deonsraed ha our odel can produce volailiy skews which, in line wih hose observed in he coodiy opions arkes, are uch ore pronounced for opions wih shorer auriies. We have shown ha he prices of sandard opions have sei-analyical soluions which can be rapidly evaluaed (in ypically of he order of /5 h of a second). his opens he possibiliy of calibraing he odel hrough deriving iplied paraeers fro he arke prices of opions. We have calibraed our odel o he arke prices of opions on crude oil fuures and shown ha, by allowing he feaure ha he prices of long-daed fuures conracs can jup by saller agniudes han shor-daed fuures conracs, we can ge a significanly beer fi. 3

24 Figure : 44% 43% 4% 4% 4% 39% 38% 37% 36% Fig : Iplied volailiies for onh opions on crude oil fuures -. pu -.5 pu -.35 pu AMF.35 call.5 call. call Opion Dela Marke Specificaion Specificaion Figure : Fig : Iplied volailiies for 4 year opions on crude oil fuures 4% 3% % % % 9% 8% -. pu -.5 pu -.35 pu AMF.35 call.5 call. call Opion Dela Marke Specificaion Specificaion 4

25 able (Exaple ) No oisson processes = +.5 All opions are sandard European calls on fuures. he values of are down he firs colun. Below are he opion prices Srikes -> Below are he iplied Black (976) volailiies (expressed as percenages), using a price of %.5.77%.75.67% 9.47% 7.789% % he above opion prices are in he pure-diffusion case and are priced using equaion 5.. In all cases, ( ) H, = 95 for all and (, ) exp(.5( ) ) = for all. In all cases, we have wo Brownian oions (in addiion o he Brownian oion driving ineres-raes) and η H =.66, η H =.49 / , χ H =., χ H =.49 /.45 a H =.45, σ r =.96, α r =., ρ HH =.85, ρ H =.964, ρ H =.43 5

26 able (Exaple ) One oisson process = +.5 All opions are sandard European calls on fuures. he values of are down he firs colun. Below are he opion prices Srikes-> Below are he sandard errors for he opion prices above Srikes-> <. <. <. <. <..5 <. < Below are he iplied Black (976) volailiies (expressed as percenages) of he opion prices above Srikes-> % 4.37% 5.393% 6.769% 7.58%.5.734%.885% 3.355% 3.88% 4.63%.75.49%.59%.874%.67%.7%.55%.598%.798%.988%.49% 8.44% 8.489% 8.575% 8.65% 8.676% % 7.633% 7.7% 7.756% 7.764% For he above opion prices, we have used equaions 5.9 and 5. wih M = and λ =.75, β =., b =.. In all cases, ( ) H, = 95 for all and (, ) exp(.5( ) ) = for all. 6

27 able 3 (Exaple 3) wo oisson processes = +.5 All opions are sandard European calls on fuures. he values of are down he firs colun. Below are he opion prices Srikes-> Below are he iplied Black (976) volailiies (expressed as percenages) of he opion prices above Srikes-> % 3.8% 3.685% 34.33% 35.95%.5 3.7% 3.373% 3.8% 3.53% 3.947% % 3.46% 3.785% 3.6% 3.897% 9.588% 9.769% 3.38% 3.% 3.7% 9.5% 9.43% 9.59% 9.847% 9.953% 3 8.8% 8.9% 9.68% 9.43% 9.476% For he above opion prices, we have used equaions 5. and 5. wih M = and λ =.75, β =., υ =., b =., λ =.75, β =.5, υ =., b =. In all cases, ( ) H, = 95 for all and (, ) exp(.5( ) ) = for all. 7

28 able 4 (Exaple 4) =, (, ) = + ( 3 365) , H (, ) = 4.. All opions are sandard European calls. Specificaion Specificaion Opion ype call call call call call call Srike rice Iplied vol 4.87% 4.85% 4.88% 4.84% 4.665% 4.56% San. error N/A N/A N/A able 5 (Exaple 5) =, (, ) = ( 3 365) , H (, ) = All opions are sandard European calls. Specificaion Specificaion Opion ype call call call call call call Srike rice Iplied vol 7.3% 7.333% 7.46% 9.6% 8.4% 7.86% San. error <. <. <. N/A N/A N/A References Ain K. and R. Jarrow (99) ricing foreign currency opions under sochasic ineres raes Journal of Inernaional Money and Finance (3) p3-39 Babbs S. (99) he er srucure of ineres-raes: Sochasic processes and Coningen Clais h. D. hesis, Iperial College London Babbs S. and N. Webber (994) A heory of he er srucure wih an official shor rae FORC Working paper 94/49 Universiy of Warwick, Unied Kingdo Babbs S. and N. Webber (997) "er Srucure Modelling under Alernaive Official Regies" in Maheaics of Derivaive Securiies M Depser and S liska (eds) Cabrdge, CU Babbs S. () Condiional Gaussian odels of he er srucure of ineres raes Finance and Sochasics vol 6 p Beaglehole D. and A. Chebanier (July ) A wo-facor ean-revering odel Risk p65-69 Benh F., L. Ekeland, R. Hauge and B. Nielsen (3) A noe on arbirage-free pricing of forward conracs on energy arkes Applied Maheaical Finance p

29 Bessebinder H., J. Coughenour,. Seguin and M. Soller (995) Mean reversion in Equilibriu Asse rices: Evidence fro he Fuures er Srucure Journal of Finance 6 () p Bjork., A. Kabanov and W. Runggaldier (April 997) Bond arke srucure in he presence of arked poin processes Maheaical Finance vol 7 no p-39 Black F. (March 976) he ricing of Coodiy Conracs Journal of Financial Econoics 3 p67-79 Carr. and D. Madan (Suer 999) Opion valuaion using he fas Fourier ransfor Journal of Copuaional Finance Vol No. 4 p6-73 Carr. and R. Jarrow (995) A Discree ie Synhesis of Derivaive Securiy Valuaion using a er Srucure of Fuures rices in Finance, volue 9 of Handbooks in Operaions Research and Manageen Science, chaper 7 p5-49 Aserda, he Neherlands Norh-Holland ublishing Copany Casassus J. and. Collin-Dufresne (5) Sochasic Convenience Yield iplied fro coodiy Fuures and Ineres Raes Journal of Finance 6 (5) p83-33 Clewlow L. and C. Srickland (Aug 999) A uli-facor Model for Energy Derivaives School of Finance and Econoics, Universiy of echnology, Sydney, Ausralia Clewlow L. and C. Srickland () Energy Derivaives: ricing and Risk Manageen Lacia Group Corazar G. and E. Schwarz (994) he valuaion of Coodiy-Coningen clais he Journal of Derivaives vol no 4 p7-39 Cox J., J. Ingersoll and S. Ross (98) he relaion beween Forward prices and Fuures rices Journal of Financial Econoics 9 p3-346 Crosby J. (5) Coodiies: A siple uli-facor jup-diffusion odel Working paper available fro he auhor on reques. Crosby J. (6a) Coodiy opions opiised Risk Magazine, May 6 Vol 9 p7-77 Crosby J. (6b) ricing a class of exoic coodiy opions in a uli-facor jup-diffusion odel Subied for publicaion Depser M. and J. Huon (997) Nuerical valuaion of cross-currency swaps and swapions in Maheaics of Derivaive Securiies M Depser and S liska (eds) Cabridge, CU Deng S. (Oc 998) Sochasic Models of Energy Coodiy rices and heir Applicaions; Mean-reversion wih Jups and Spikes IE & OR Dep, Universiy of California a Berkeley, Berkeley, CA 947 Duffie D. (996) Dynaic Asse ricing heory rinceon, rinceon Universiy ress Duffie D. and R. Sanon (99) ricing Coninuously Reseled Coningen Clais'' Journal of Econoic Dynaics and Conrol vol. 6 p Duffie D., J. an and K. Singleon () ransfor Analysis and Asse ricing for Affine Jup Diffusions Econoerica vol 68 p Gean H. (5) Coodiies and coodiy derivaives Chicheser, UK, John Wiley Gean H. and V.N. Nguyen (3) Analysing volailiy surfaces for energy coodiies Working paper ESSEC, France Gibson R. and E. Schwarz (99) Sochasic convenience yield and he pricing of Oil Coningen Clais Journal of Finance, vol 45 no 3 p Glasseran. (4) Mone Carlo ehods in Financial Engineering New York, Springer-Verlag Harrison M. and S. liska (98) Maringales and Sochasic Inegrals in he heory of Coninuous rading Sochasic rocesses and heir Applicaions vol p5-6 Heah D., R. Jarrow and A. Moron (99) Bond ricing and he er Srucure of Ineres Raes: A new ehodology for coningen clai valuaion Econoerica Vol 6 p77-5 9

30 Hilliard J. and J. Reis (998) Valuaion of coodiy fuures and opions under sochasic convenience yields, ineres raes and jup diffusions in he spo Journal of Financial and Quaniaive Analysis 33() p6-86 Hoogland J., C. Neuann and M. Vellekoop (Dec ) Syeries in jup-diffusion odels wih applicaions in opion pricing and credi risk Deparen of Applied Maheaics, Universiy of wene,.o. Box 7, 75 AE, Enschende, he Neherlands Hull J. and A. Whie (99) ricing Ineres-Rae-Derivaive securiies Review of Financial Sudies 3 (4) p Hull J. and A. Whie (993) One-facor Ineres-Rae Models and he Valuaion of Ineres-Rae Derivaive securiies Journal of Financial and Quaniaive Analysis 8 () p35-54 Jashidian F. (April 993) Opion and Fuures Evaluaion wih deerinisic volailiies Maheaical Finance vol 3 no p49-59 Jarrow R. and D. Madan (Oc 995) Opion ricing using he er Srucure of Ineres Raes o Hedge Syseaic Disconinuiies in Asse Reurns Maheaical Finance 5 p3-336 Karlin S. and H. aylor (975) A Firs Course in Sochasic rocesses nd ediion New York, Acadeic ress Meron R. (973) heory of Raional Opion ricing Bell Journal of Econoics and Manageen Science p4-83 (reprined in Meron (99), chaper 8) Meron R. (July 976) Opion pricing when underlying sock reurns are disconinuous Journal of Financial Econoics 3 p5-44 (reprined in Meron (99), chaper 9) Meron R. (99) Coninuous-ie Finance adsow, Unied Kingdo, Basil Blackwell Inc Milersen K. and E. Schwarz (March 998) ricing of opions on coodiy fuures wih sochasic er srucures of convenience yields and ineres raes Journal of Financial & Quaniaive Analysis vol 33 p33-59 Milersen K. (Feb 3) Coodiy price odelling ha aches curren observables: A new approach Quaniiaive Finance vol 3 p5-58 (also available a Runggaldier W. (Nov ) Jup-Diffusion odels Deparen of ure and Applied Maheaics, Universiy of adova, Ialy Schwarz E. (July 997) he sochasic Behaviour of Coodiy rices: Iplicaions for valuaion and hedging Journal of Finance p Vasicek O. (977) An Equilibriu Characerisaion of he er Srucure Journal of Financial Econoics 5 p77-88 Villaplana. (3) ricing power derivaives: A wo-facor jup-diffusion approach Working paper 3-8, Universidad Carlos de Madrid, Spain he auhor, John Crosby, can be conaced a : [email protected] 3