John Crosby Coodes: A sple ul-facor Jup-Dffuson odel John Crosby Lloyds SB Fnancal arkes Faryners House 5 onuen Sree London EC3R 8BQ Eal : johnc5@yahoo.co el: 7 63 9755 Conens Coodes: A sple ul-facor Jup-Dffuson odel... Absrac.... Inroducon.... he odel of fuures coody prces...4 3. Sochasc convenence yelds and ean reverng coody prces...3 4. one Carlo sulaon...9 5. Opon prcng... 6. No-arbrage arke copleeness and ncopleeness...34 6. Consan spo jup apludes...35 6. Spo jup apludes are dscree rando varables...4 6.3 Spo jup apludes are connuous rando varables...4 6.4 Spo jup apludes are of xed for...45 6.5 he dynacs of forward coody prces...46 7. Conclusons...49 hs revsed verson h July 5 revsed h Augus 5 All coens welcoe. Acknowledgeens: he auhor wshes o hank Son Babbs and Farshd Jashdan for valuable coens whch have sgnfcanly enhanced hs paper snce s frs draf. In addon he hanks eer Carr Sewar Hodges Andrew Johnson Daryl carhur Anhony Neuberger Nck Webber and senar parcpans a he Unversy of Warwck. However any errors are he auhor s responsbly alone.
Coodes: A sple ul-facor Jup-Dffuson odel John Crosby h July 5 revsed h Augus 5 Absrac In hs paper we develop an arbrage-free odel for he prcng and rsk anageen of coody dervaves. he odel generaes fuures (or forward) coody prces conssen wh any nal er srucure. he odel s conssen wh ean reverson n coody prces whch s an eprcally observed sylsed fac abou coody arkes and also generaes sochasc convenence yelds. Our odel s a ul-facor jup-dffuson odel one verson of whch allows for long-daed fuures conracs o jup by saller aouns han shor-daed fuures conracs whch s n lne wh sylsed eprcal observaons. Fnally our odel also allows for sochasc neres-raes. he odel produces se-analyc soluons for sandard European opons. hs opens he possbly o calbrae he odel paraeers by dervng pled paraeers fro he arke prces of opons.. Inroducon he a of hs paper s o develop an arbrage-free ul-facor jup-dffuson odel for coodes. he coody could be for exaple crude ol anoher peroleu produc gold a base eal naural gas or elecrcy. Before urnng our aenon o coodes s worh reflecng on he developen of neres-rae odels. he paper by Vascek (977) nroduced an equlbru ean reverng neres-rae odel no he leraure. By nroducng a e-dependen ean reverson level hs becae he exended Vascek odel (Babbs (99) Hull and Whe (993)) whch could auoacally f any nal er srucure of neres-raes. Black e al. (99) used a slar dea n a non-gaussan seng. hese odels focused prncpally on nsananeous shor raes. Furher research (Babbs (99) Heah e al. (99)) developed no-arbrage odels (ncludng ul-facor versons) evolvng he enre yeld curve conssen wh s nal values. here are any parallels beween he above neres-rae odels and odellng fuures (or forward) coody prces. Soe of he coodes leraure (Gbson and Schwarz (99)) has focussed on equlbru odels wh he frs facor beng he spo coody prce and he second facor beng he nsananeous convenence yeld whls Schwarz (997) nroduced a hrd facor wh sochasc neres-raes. However hese odels leave he arke prce of convenence yeld rsk o be deerned n equlbru and are no necessarly conssen wh any nal er srucure of fuures (or forward) prces. Subsequen odels (by analogous echnques o neres-rae odellng) have been conssen wh any nal er srucure. See for exaple Corazar and Schwarz (994) Carr and Jarrow (995) Beaglehole and Chebaner () lersen and Schwarz (998) lersen (3) Clewlow and Srckland ()(999) wh he laer parcularly focussng on evolvng he forward prce curve. In hs respec our paper s closes n spr o Clewlow and Srckland (999) hough we also ncorporae sochasc neres-raes (and jups). As a general rule aenon has osly focused on pure dffuson odels. Jups were ncorporaed no neres-rae odels n Babbs and Webber (994)(997) Bjork e al. (997) and Jarrow and adan (995). See also eron (976)(99) Hoogland e al. () Duffe e al. () and Runggalder (). arallelng hese odels jups have also been nroduced no odels for coody prces n Hllard and Res (998) Deng (998) Clewlow and Srckland () Benh e al. (3) and Casassus (4) alhough usually n odels whch are no necessarly conssen wh any nal er srucure.
Our odel wll aep o nroduce a ul-facor jup-dffuson odel wh sochasc neres-raes whch s conssen wh any nal er srucure. We now urn our aenon o hs odel by frsly oulnng feaures of he coodes arkes. I s an eprcal fac (Bessebnder e al. (995)) ha os coody prces see o exhb ean reverson. Furherore s also eprcally observed n he case of elecrcy ha he prces of shor-daed (close o delvery) conracs exhb sharp spkes. he pac of hese prce spkes s uch lower for conracs wh a greaer e o delvery. Oher coodes such as ol and naural gas can also exhb prce spkes alhough hese end o be of a saller agnude. However we also observe ha he arke prces of opons on any coodes ply Black and Scholes (973)/Black (976) volales whch vary wh he srke of he opon. ha s arke prces ply a volaly sle or (ore usually) a volaly skew. One way o accoun for volaly sles and skews s hrough a jupdffuson odel. Unlke fnancals asses whch are held for nvesen purposes coodes are held n order o be consued or used n an ndusral process (alhough gold and o soe exen oher precous eals can be and are held for nvesen purposes hey are also used for soe specals ndusral purposes). he noon of convenence yeld s nroduced for coodes. Loosely speakng s a easure of he value of physcally holdng a coody raher han beng long he coody hrough he forward or fuures arkes. For exaple an end-user of a coody ay well choose o sore soe of (as a ype of self-nsurance polcy) n order o nse dsrupon f here s a proble wh supply. he convenence yeld also plcly accouns for he cos of sorage of he coody and he cos of nsurng he coody. I s observed eprcally ha convenence yelds are usually hghly volale. Furherore convenence yelds are usually posvely correlaed wh he value of he coody (Lence and Hayes ()). here s a acro-econoc nerpreaon o ean reverson and convenence yelds hrough lnkage o supply and deand and nvenory levels: When prces are low soe producers ay sop producng whch wll end o cause prces o rse. If prces are hgh soe consuers ay sop consung whch wll end o cause prces o fall. When nvenores are low shorages are ore lkely whch ends o ncrease boh he value of he coody and he perceved value of physcally holdng he coody (as opposed o beng long he coody hrough he forward or fuures arkes). hs laer can be nerpreed as ncreasng he convenence yeld. he reverse arguen holds f nvenores are hgh. We would lke our odel o ncorporae all of he above sylsed observaons of he coodes arkes. Exanaon of our ul-facor jup-dffuson odel wll show ha capures all of he above effecs. We wll also assue ha neres-raes are sochasc. When neres-raes are sochasc fuures coody prces and forward coody prces are no longer he sae. In hs paper we wll work wh boh fuures and forward prces bu osly wh fuures coody prces. We wll assue ha arkes are frconless. ha s connuous radng s possble and we assue ha here are no bd-offer spreads n he coodes arkes or n he bond arkes. Of course we do no assue ha he coody can be sored or nsured whou cos snce s precsely hese coss whch gve rse o he noon of convenence yeld. We wll assue ha arkes are free of arbrage. I s well known (Harrson and lska (98) Duffe (996)) ha under hese assupons here exss an equvalen arngale easure under whch fuures prces are arngales. In he case of a dffuson odel f here are suffcen fuures conracs (and rsk-free bonds and possbly forward conracs) raded hen any dervave (such as an opon) can be nsananeously hedged or replcaed by a dynac self-fnancng porfolo of fuures conracs (and rsk-free bonds). he arke n our odel s hus coplee. In hs case he equvalen arngale easure s unque. However n he case of a jup-dffuson odel he arke ay be eher coplee or ncoplee. If he arke s ncoplee hen he equvalen arngale easure would no be unque. In he case of ncopleeness we wll assue ha an equvalen arngale easure s fxed by he arke hrough he arke prces of opons and we wll call hs (by an abuse of language bu for he sake of brevy) he equvalen 3
arngale easure (raher han an equvalen arngale easure). I s also possble for he jupdffuson odel o lead o a arke whch s coplee. he crcusances n whch he jup-dffuson odel gves rse o a coplee arke are specfed n secon 6. he reander of hs paper s srucured as follows. In secon we wll provde noaon and nroduce he odel. In secon 3 we wll relae o sochasc convenence yelds and o ean reverng coody prces. In secon 4 we wll dscuss how he odel can be used n connecon wh one Carlo sulaon. In secon 5 we wll derve he prces of sandard opons n seanalycal for n our odel. In secon 6 (whch a leas parally logcally precedes secon ) we derve no-arbrage condons for our odel explan he crcusances under whch our odel leads o coplee and ncoplee arkes derve paral negro-dfferenal equaons sasfed by he prce of coody dervaves and relae fuures coody prces o forward coody prces. Secon 7 s a shor concluson.. he odel of fuures coody prces Noaon: Le us explan soe noaon. All jup-dffuson processes are assued rgh connuous. ore explcly H ( ) l H ( u ) = ncludes he effec of any jup a e. he value of u H ( ) jus before a jup a e s H ( ) l H ( u ) a SDE we ean ( ) ( ) dh H =. When we wre u. For he sake of brevy however we shall always wre dh ( ) H ( ) ( ) ( ) dh H. n We defne oday o be e and we denoe calendar e by ( ). In hs secon and n secons 3 o 5 we wll work exclusvely n he equvalen arngale easure (whch as already ndcaed ay n fac no be unque). As already ndcaed f he equvalen arngale easure s no unque we wll assue ha one has been fxed by he arke and we wll call hs he (raher han a) equvalen arngale easure. We denoe expecaons a e wh respec o he equvalen arngale easure by Exp [ ]. Sochasc evoluon of neres-raes: We assue ha neres-raes n our odel are sochasc. Le us nroduce soe noaon. We denoe he (connuously copounded) rsk-free shor rae a e by r (connuously copounded) nsananeous forward rae a e o e by ( ) denoe he prce a e of a (cred rsk free) zero coupon bond aurng a e by ( ) = exp f s ds. By defnon we denoe he f and we. All references o bond prces n hs and subsequen secons are of course references o rsk-free zero coupon bond prces. 4
We assue ha (under he equvalen arngale easure) he shor rae follows he exended Vascek process (Babbs (99) Hull and Whe (99)(993)) naely dr α ( γ r ) d dz = r σ r or equvalenly (Babbs (99) Heah e al. (99)) he dynacs of bond prces are d ( ) ( ) = r σ α r d + ( exp( α ( ) )) dz r d + σ ( ) dz r r Noe ha dz ( ) denoes sandard Brownan ncreens. We assue ha σ r and. (equaon.) α r are posve consans and γ s defned so as o be conssen wh he nal er srucure (e he er srucure of neres raes oday e ) whch we ake as gven. Defne he sae varable X = σ exp( α ( s) ) dz ( s) (noe ( ) = I can be shown (Babbs (99)) ha: r σ r ( s ) r f ( ) = f ( ) + σ ( s ) ds X X ). (equaon.) (equaon.3) Coodes: We denoe he value of he coody a e by C. he value of he coody oday s C. he value of he coody s usually ered he spo prce. However n hs paper we shall generally use he expresson value of he coody because n soe coody arkes he spo prce s no always exacly easy o defne. Now we urn our aenon o fuures coody prces. We denoe he forward coody prce a e o (e for delvery a) e by ( ) F. We denoe he fuures coody prce a e o (e he fuures conrac aures a) e by H. I can be shown (Cox e al. (98) Duffe (996)) ha n he absence of arbrage ha F ( ) = Expexp r ( ) ( s) ds C (equaon.4) and ( ) [ ] H = Exp C (equaon.5) 5
where o repea C s he value of he coody a e. A key o odellng coody prces when neres-raes are sochasc s o recognse ha n hs case fuures coody prces and forward coody prces are no he sae. Indeed equaons.4 and.5 show ha fuures prces are arngales wh respec o he equvalen arngale easure whereas when neres-raes are sochasc forward prces are no. Noe ha equaons.4 and.5 are conssen wh ( ) = C = H ( ) and F( ) C = H ( ) F = (equaon.6) We ake as gven our nal er srucure (e he er srucure oday e ) of fuures coody prces. ha s we know H for all of neres ( nerpolaon of he fuures prces of a fne nuber of fuures conracs). ) (perhaps n pracce by In soe odels he dynacs of he value of he coody are posed and hen equaons.4 and.5 would be used o derve he dynacs of forward coody prces and fuures coody prces. By conras our odel wll pos he dynacs of fuures coody prces. In oher words fuures conracs are no dervaves bu nsead are he prve asses of our odel. We wll shorly pos he dynacs of fuures coody prces ( ) H n he equvalen arngale easure (conssen wh he arngale propery of equaon.5). We wll hen oban he dynacs of C = H. he value of he coody va he relaon Le us consder why equaon.6 has o be vald: Consder a fuures conrac aurng a e. C < H hen would be possble o creae a rsk-less arbrage by If were he case ha Wh hs assupon f were he case ha buyng he coody and sellng he fuures conrac. Now whls s easly possble o ake shor sales of fnancal asses shor sales of coodes are eher very dffcul or (ore lkely) possble. herefore we assue ha here are non-saaed agens n he coodes arkes (for exaple ol copanes nng copanes ol refneres ndusral end-users of coodes ec) who hold he coody n srcly posve quanes and who would sell her holdngs f was profable o do so. C > H hen hese agens would creae a rskless arbrage by sellng her holdngs of he coody and buyng he fuures conrac. Snce our C H C = H. odel assues no arbrage us be he case ha =. Lkewse Now we nroduce he nsananeous fuures convenence yeld forward rae ε ( ) va he relaon a e o e C = H exp ε s ds. (equaon.7) s= = exp ε s ds Defne s= ε. (equaon.8) We noe ha we gnore any pac of qualy ng and locaon opons whch are soees ebedded n coodes fuures conracs. 6
hs defnes wha we call a fcous fuures convenence yeld bond prce. We call fcous because no such bond acually exss nor do we assue exss. I s solely a aheacal consrucon defned by analogy o neres-raes and real rsk-free bonds va equaons.7 and.8. We can also wre H ( ) ε ( ) ( ) C =. (equaon.9) We nroduce sandard Brownan ncreens denoed by We denoe he correlaon beween dz ( ) and dz ( ) by ρ beween dz and k k = j. dz Hj by Hj dz for each k k =.... for each k and he correlaon ρ for each j and k j =... and = ρ f We also nroduce osson processes denoed by N for each =... wh N whose nensy raes are λ. We assue ha λ are deernsc funcons of a os and hey us be posve for each =... for all. We also assue ha each of he N are ndependen of each oher and each s ndependen of each of he Brownan oons. We nroduce b for each... call hese jup decay coeffcen funcons. We nroduce γ for each... jup condonal on a jup n Hj = whch are non-negave deernsc funcons. We = whch are paraeers whch deerne he sze of he N. We wll call he γ he spo jup apludes. A rsk of coplcaon bu for he sake of brevy we wll consder wo possble specfcaons for he spo jup apludes and n urn hese are lnked o wo possble specfcaons of he jup decay coeffcen funcons. For each... = we assue ha eher: Assupon. : he spo jup apludes are assued o be (known) consans. In hs case he jup decay coeffcen funcons Or: b are assued o be any non-negave deernsc funcons. Assupon. : he spo jup apludes are assued o be ndependen and dencally dsrbued rando varables each of whch s ndependen of each of he Brownan oons and of each of he osson processes. In b are assued o be dencally equal o zero e hs case he jup decay coeffcen funcons b for all. Reark.3 : Noe ha for each we assue eher assupon. or assupon. s sasfed. For dfferen could be a dfferen assupon (e f we have ore han one osson process we can x he assupons). 7
Reark.4 : he ovaon for hese assupons wll be descrbed n deph n secon 6 bu we can provde a bref suary here. We wll show ha s no possble n general n he absence of arbrage o have boh jups whose apludes are rando varables and sulaneously have jup b ) whch are no dencally zero. Hence we assue ha all he decay coeffcen funcons ( osson processes sasfy eher assupon. or assupon.. We wll develop he odel wh assupons. and. n parallel snce he choce of hese assupons scarcely alers he developen. For each N E N denoes he expecaon operaor a e condonal on a jup occurrng n. If for a gven he spo jup aplude s consan (assupon.) he expecaon operaor s se equal o s arguen. We are ovaed by he presence of ( ) C ε ( ) applyng Io s lea o ( ) arngales n he equvalen arngale easure. n he denonaor of equaon.9 he effec of and by he knowledge ha fuures coody prces are Assupon.5 : We assue ha he dynacs of fuures prces n he equvalen arngale easure are: dh H ( ) ( ) k = ( ) dz ( ) dz =σ σ + exp γ exp b ( u) du dn = λ EN exp γ exp b ( u) du d = (equaon.) where ( ) ndependen of H ( ). σ for each k k =... are deernsc funcons of and and are Reark.6 : Fuures coody prces are arngales n he equvalen arngale easure. Reark.7 : In he absence of jups he dynacs of fuures coody prces n he equvalen arngale easure are very slar o hose of forward prces n Clewlow and Srckland (999) (alhough we also ncorporae sochasc neres-raes). When = (and n he absence of jups) equaon. gves dynacs for fuures coody prces whch are essenally dencal o hose n lersen and Schwarz (998) alhough hey ake he sarng pon of her odel he dynacs of spo coody prces and convenence yelds. Alhough we ndex he spo jup apludes γ wh he assupons. and. boh ply ha her oucoes do no depend on e he ndex sply refers o he e a whch a jup ay occur. Noe ha λ s he rsk-neural (e under he equvalen arngale easure) nensy rae of for each and furherore (n he case of assupon.) he dsrbuons of he spo jup apludes γ are also defned wh respec o he rsk-neural equvalen arngale easure. We assue ha a any gven nsan no ore han one of he osson processes jups. N 8
Defne for each e ( ) λ EN exp γ exp b ( u) du (equaon.) Noe hs expresson s deernsc rregardless of wheher he spo jup apludes are as n assupon. or n assupon.. By he for of Io s lea for jup-dffusons appled o equaon. and usng equaon. d = k= k = k ρ σ σ Hj d + σ k= j= k = + γ exp b ( u) dudn e ( ) d = = ( ln H ( )) σ ( ) + σ ( ) ρ σ ( ) σ ( ) d dz σ ( ) dz Hj (equaon.) and where we have used he usual convenon ha f he upper ndex s less han he lower ndex n a suaon hen he su s se o zero. Reark.8 : Equaon. enables us o beer descrbe he sze of he jup when one happens. When here s a jup n N ln H ( ) changes by γ exp b u du. Le us brefly consder he plcaons of hs. When here s a jup he log of he fuures coody prces nfnesally close o aury jup by delvery ( ) years ahead jup by γ exp b ( ) (and provded b γ exp u du b prces do no jup a all. he effec of he funcon. However he log of he fuures coody prces for u du. Consderng he l as ) hen very long-daed fuures coody (whch s assued always non-negave) s o exponenally dapen he effec of he jup hrough fuures coody prce enor. hs sees o be n lne wh eprcal observaons n he coodes arkes (hs s parcularly a feaure n he case of elecrcy). In he case of assupon. of he fuures coody prces across dfferen enors. Reark.9 : Noe ha for each prce H ( ) wll rean posve. Le us reurn o he odel: b and jups cause parallel shfs n he log γ can ake any value n ( ) Now rewre equaon. our SDE for ln H ( ) for ln ( ) negral for fro o hen: ln H and he fuures coody H s nsead and hen rewre n k = k = ( ) = ln H ( ) σ ( s ) + σ ( s ) ρ σ ( s ) σ ( s ) ds 9
k ρ σ σ + σ σ ( s ) ( s ) ds ( s ) dz ( s) ( s ) dz ( s) Hj Hj k = j= k = + γ exp b ( u) dudn e ( s ) ds (equaon.3) s s = s = By dfferenang wh respec o we ge he dynacs of he value of he coody ln : C H ( ) ( s ) ( s ) ln H d( H ( ) ) d σ σ ln = + ( s ) ( s ) dsd + σ σ k = k ( s ) ( s ) σ Hj σ + Hj ( s ) Hj ( s ) dsd ρ σ + σ k j = = + ( s ) ( s ) σ s dsd + s dsd k σ ρ σ k ρ σ = = ( s ) ( s ) σ σ + dz s dz ( s) d k = k σ ( ) + σ ( ) ρ σ ( ) σ ( ) + ρ Hjσ ( ) σ Hj ( ) d k= k= k= j= k = ( ) dz ( ) dz +σ σ γ b exp b ( u) du dn d + γ dn e ( s ) e ( ) d ds d = = s s = s = (equaon.4) dz Noe ha he fnal dffuson er vanshes e σ ( ) = snce ( ) = σ. Noe ha n general (exanng he fourh lne of equaon.4) ln H ( ) would be non- arkovan bu we would lke k = σ ( s ) dz ( s) σ o be such ha ( s ) dz ( s) ln H lnc s a arkov process n a fne nuber of sae varables. We consder he funconal for ( s ) = η + s χ s exp a u du s σ (equaon.5)
for each k k =... where ( s) η χ ( s) and a u are deernsc funcons. Recall (equaon.) he sae varable X = exp( α ( s) ) dz ( s) And defne he sae varables: Y = dz ( s) r σ. r σ (equaon.6) X = s a u du dz s Y χ exp (equaon.7) s = ( s) dz ( s) η. (equaon.8) Noe Y ( ) = and X ( ) = ( ) = Y for all k. r Defne for each... = X = γ exp b ( u) du dn (noe N N s s s X = ). (equaon.9) dx = b b u du dn d + γ dn hen γ exp. N s s s (equaon.) = hen we can show (snce k = σ ( s ) dz = dx k= k = σ and usng equaons. and.3) ha ( s) σ ( s ) + dy X d dz ( s) d + σ ( ) dz σ ( ) dz k = σ k = k = ( s ) σ = dx + dy + r f s dsd (equaon.) hen usng equaon.3 and equaons..6.7.8 and.9 we have he followng expresson for he value of he coody : C a e C H ( ) We noe ha wll becoe clear laer ha n order o avod a poenal degeneracy we ay pu η for all k excep one (or cobne ers of he for η dz ou equaons below n full o ease noaon. ) bu we wll wre
H = k = k = k exp ρ Hjσ s σ Hj s ds exp X k= j= k= exp X N e ( s ) ds = = H exp σ ( s ) + σ ( s ) ρ σ ( s ) σ ( s ) ds ( + Y ) + ( X Y ) (equaon.) α r In a slar anner we can oban he followng expresson for he evoluon fro e o e of he fuures coody prce o e n ers of he sae varables: H = k= k = k exp ρ Hjσ s σ Hj s ds k= j= exp Y + exp a u du X k= k= H exp σ ( s ) + σ ( s ) ρ σ ( s ) σ ( s ) ( ( ) ) exp α r exp X Y α r α r exp exp b ( u) du X N e ( s ) ds = = ds (equaon.3) hs shows ha H ( ) C and ( ) H are arkov n a fne nuber of sae varables 3. Reark. : Wh he help of resuls n secon 4 (specfcally equaon 4.6) s sraghforward o verfy by drec calculaon usng equaons. and.3 ha [ ] ( ) = ( ) Exp C Exp H H whch confrs conssency wh equaon.5. 3 We noe ha s sraghforward o cobne he Y ( ) and Y no a sngle sae varable. We could do hs bu prefer no o n order o axse he nuon behnd he odel. However shows ha H ( ) C and ( ) H are n fac arkovan n + + sae varables.
3. Sochasc convenence yelds and ean reverng coody prces Our a n hs secon s o gve resuls abou sochasc convenence yelds and ean reverson n our odel whch show ha our odel s able o capure he sylsed observaons of he coodes arkes ha were ade n secon. Frsly we provde a aheacal lea. Lea 3. : ln H ( ) = f ( ) ( ) ε roof : We noe fro equaon.9 ha. (equaon 3.) C H ( ) = exp ε ( s) ds = C exp ( f ( ) ) s ε s ds ( ) s= s= Now ake logs and hen he paral dervave wh respec o. roposon 3. : he dynacs of he value of he coody are as follows. If we defne ( s ) ( s ) σ σ εr ε ( ) σ ( s ) + 4 σ ( s ) ds k= k σ Hj s σ s ρ + s Hj σ s σ Hj ds k = j= σ s σ s ρ σ s ds ρ σ s ds k = k= σ ( s ) dz ( s) k = e ( s ) + γ sb exp b ( u) du dns + ds = s = hen (equaon 3.) ( ln ( )) = εr d H r d k σ ( ) + ρ Hjσ ( ) σ Hj ( ) d k= k = j= + σ ( ) dz + γ dn e ( ) d k = = = and. (equaon 3.3) dc = + C ( r εr ) d σ ( ) dz k = 3
+ ( exp( γ ) ) dn e ( ) d = =. (equaon 3.4) roof : u equaon 3. no equaon.4 hen wh soe algebra and equaons. and 3. we oban equaon 3.3. Usng Io s lea for jup-dffusons gves equaon 3.4 Reark 3.3 : Noe ha he SDE n equaon 3.4 has a drf er whch (by consrucon) s of an enrely falar for. In order o ge a greaer nuon o he odel we are also neresed n he dynacs of he fcous fuures convenence yeld bond prce ε ( ) whch we dsplay n proposon 3.4. roposon 3.4 : he dynacs of he fcous fuures convenence yeld bond prce are: ( ) ( ) d ε = µ ε d χ exp a ( u) du dz ε k = + exp γ exp b ( u) du dn ( e ( ) e ( ) ) d = = where (equaon 3.5) k ( )( ) µ ε + η + χ + ρ η + χ η + χ ε k= ρ r Hj Hj Hj k = k = j= η + χ exp a ( u) duσ ( ) σ ( ) + ( ) η + + χ exp a u du η χ k= k ρhj η + χ exp a u du ηhj + χ Hj k = j= roof : Fro equaon.9 we have ( ) ε =. Now use Io s lea for jupdffusons. ( ) ( ) H C Reark 3.5 : Whls hs expresson appears que long s concepually sraghforward as he volaly er for he Brownan oons n he SDE has a slar for o ha n he SDE for rsk-free bond prces n a facor Gaussan neres-rae odel (Babbs (99) Heah e al. (99)). Of course he drf of a rsk-free bond (or any non-dvdend payng raded asse) n he equvalen arngale easure s equal o he rsk-free shor rae. hs does no apply o he drf of he fcous fuures convenence yeld bond prce however snce s a aheacal consrucon no he prce of a real raded asse. In addon we see ha fcous fuures convenence yeld bond prces exhb jups b are dencally excep n he specal case ha for all he jup decay coeffcen funcons equally o zero for all. 4
roposon 3.6 : he dynacs of he nsananeous fuures convenence yeld forward rae ε ( ) a e o e are: ( ) σ σ dε ( ) = ρ σ ( ) ρ σ ( ) k= k = σ ( ) + σ ( ) d k σ ( ) ( ) σ Hj + ( ) Hj ( ) σ + ρ k σ + σ = k= j= χ a ( ) exp a ( u) dudz e ( ) + γ b ( ) exp b ( u) du dn + d = = + k= ( ) d Hj ( ) σ ( ) d (equaon 3.6) roof : Apply Io s lea o equaon 3.5 wh (fro equaon.8) ( ) ( ) ln ε ε =. roposon 3.7 : he dynacs of he fuures convenence yeld shor rae ( ) analogous o neres-raes) are: ( s ) ε (usng ernology ( s ) σ σ ε ( ) ε ( ) σ ( s ) + 4 σ ( s ) ds k= k σ Hj s σ s ρ + s Hj σ s σ Hj ds k = j= σ s σ s ρ σ s ds ρ σ s ds k = k= σ ( s ) dz ( s) k = e ( s ) + γ sb exp b ( u) du dns + ds. = s = roof : We noe ha χ a ( ) exp a ( u) SDE for ε ( ) for ( s) for fro o. du σ = ( ) and hen rewre our ε nsead and hen re-arrange ers and hen rewre hs SDE n negral 5
Reark 3.8 : We noe hs expresson for ε ( ) s he sae as he expresson gven earler for ε r n equaon 3. (whch ndeed should be) e εr ε ( ). hs jusfes our noaon for ε r and ε ( ) (e jusfes our choce of ε r n equaon 3. and shows s conssency wh equaon.7). Noe ha he fuures convenence yeld shor rae ε ( ) a e follows a ean reverng jup-dffuson process drven by Brownan oons and osson processes. In secon we noed ha eprcal evdence suppors he vew ha he value of a coody s posvely correlaed wh convenence yelds. he followng proposon derves hs correlaon. roposon 3.9 : he correlaon a e beween he log of he value of he coody and he ε s gven by nsananeous fuures convenence yeld forward rae ( ( )) ( )( ( ( ))) C ov dε d lnc V ar dε V ar d ln C where ( ε ) C ov d d ln C = ρhjχ a ( ) exp a ( u) du ( ηhj + χ Hj ) d k = j= + Var γ b ( ) exp b ( u) du dnγ dn = and V ar ( dε ( )) = ρhjχ exp a ( u) du χ Hj exp ahj ( u) dud k = j= + Var γ b ( ) exp b ( u) du dnγ b ( ) exp b ( u) du dn = and V ar ( d ( ln C )) = ρhj ( η + χ )( ηhj + χ Hj ) d k = j= + Var ( γ dnγ dn ) (equaon 3.7) = roof : Iedae fro equaons 3.4 and 3.6 6
a for each k k... Reark 3. : Now s clear ha reverson rae. We would herefore expec each a = s playng he role of a ean o be non-negave for all k and for all and a leas one of he o be srcly posve. We also requre he jup decay coeffcen funcons b ( ε ( ) ( )) u for each o be non-negave. When hs s he case s easy o see ha he correlaon correl d d lnc beween he log of he value of he coody and he nsananeous fuures convenence forward rae wll always le beween zero and uny provded ha he correlaon arx ρ s posve defne. hs posve correlaon s n lne wh he eprcal evdence noed Hj n secon. Reark 3. : Furherore perfec posve correlaon would only occur n he case ha our odel reduces o a one-facor odel (eher dffuson or jup). Reark 3. : A correlaon of zero only happens when b for all... a for all k k... = and =. However nspecon of he SDEs for he fcous fuures convenence yeld bond prce he nsananeous fuures convenence yeld forward rae and he nsananeous fuures convenence yeld shor rae show ha f hese laer are boh rue hen all hese varables e he fcous fuures convenence yeld bond prce he nsananeous fuures convenence yeld forward rae and he nsananeous fuures convenence yeld shor rae would be deernsc. ha s n hs specal case hey would have no dffuson volaly and no jups. Reark 3.3 : In a sense s he presence of non-zero ean reverson rae funcons (e a ( ) ) whch akes fuures convenence yelds have a dffuson volaly and s he exsence of non-zero jup decay coeffcen funcons (e b ( ) ) whch akes fuures convenence yelds have jups. In he specal case ha for all he jup decay coeffcen funcons are all dencally equal o zero (for exaple when all sasfy assupon.) fuures convenence yelds have no jups. Of course hs s nuve n vew of equaons.7.8 and.9 snce n hs case when here are jups here s a parallel shf n he log of he fuures coody prces across dfferen enors. Reark 3.4 : Noe ha (usng equaon 3.7) he volaly of he value of he coody a e a for does no depend on he volaly of bond prces or neres-raes nor does depend on any k k =... nor on b ( ) for any... lne wh he nuon behnd he consrucon of our odel. =. hs s enrely expeced and n Reark 3.5 : Noe ha he volaly of he value of he coody a e depends on η bu neher he volaly of he fcous fuures convenence yeld bond prce ε ( ) nor he volaly of he nsananeous fuures convenence yeld forward rae ε ( ) a e depend on η for any k (alhough her drf ers do). he followng proposon provdes furher nsgh no our odel because shows ha he log of he value of he coody exhbs ean reverson. roposon 3.6 : he log of he value of he coody s a ean-reverng sochasc process whose SDE s of he for: d H k = ( ln C ) = a ( Λ( ln H ( ) ) ( ln C )) d + σ ( ) dz 7
+ = = γ dn e ( ) d (equaon 3.8) where ( ln H ( ) ) Λ s defned by ( ) ln H ah Λ( ln H ( ) ) + ah ln H ( ) + Ψ( ) a X ah ah + ah Y + ah X Y + X k = k = α r α r + ah X N ah e ( s ) ds = = e ( s ) b X N ds = = (equaon 3.9) where ( ) Ψ s a deernsc funcon whch depends only on easly obaned a he expense of soe edous algebra). k = and (whose exac for s roof: We use our expresson for C H ( ) (equaon.) and ake logarhs and our SDE for d( ln C ) (equaon.4) ogeher wh equaon. o elnae one of he sae varables X. he choce s arbrary bu o be defne we elnae X H. We oban equaon 3.8 Reark 3.7 : hs shows ha ln C follows a ean reverng jup-dffuson process wh a long run ( ) ean reverson level of Λ ln H ( ). Bu we can see ha ( ln H ( ) ) Λ s sochasc and s self also a ean reverng jup-dffuson process. In oher words he log of he value of he coody follows a ean reverng jup-dffuson process whose long run ean reverson level s also a ean reverng jup-dffuson process. Reark 3.8 : Noe also how X and Y appear n equaon 3.9. hs shows ha sochasc neres-raes can also conrbue o hs ean reverson. hs s especally nuve f rsk-free neresraes are negavely correlaed wh he Brownan oons drvng he value of he coody. If for exaple (noe he for of equaon 3.4) he Brownan oons drvng he value of he coody ncreases hen he rsk-free shor rae wll end o decrease whch wll end o reduce he drf er on he SDE for he value of he coody whch wll hen ceerus parbus end o cause he value of he coody o drf down. he reverse arguen also holds. If he Brownan oons drvng he value of he coody decrease hen he rsk-free shor rae wll end o ncrease whch wll end o ncrease he drf er on he SDE for he value of he coody whch wll hen ceerus parbus end o cause he value of he coody o drf up. Reark 3.9 : We sress agan ha he fcous fuures convenence yeld bond prce s a aheacal consrucon. We do no assue ha such a bond really exss. Noe also ha n our odel we have no had o ake any addonal assupons abou he naure of he sochasc evoluon of nsananeous fuures convenence yeld forward raes or he fuures convenence yeld shor rae. her dynacs arse naurally fro he assupon of he dynacs of fuures coody prces n equaon.. 8
Reark 3. : Noe also ha we have no had o ake any assupons abou he naure of he arke prce of rsk assocaed wh fcous fuures convenence yeld bond prces nsananeous fuures convenence yeld forward raes or he fuures convenence yeld shor rae. Reark 3. : How can we suarse hs odel? We have a ul-facor jup-dffuson odel. he value of he coody s drven by he ). he fcous fuures Brownan oons ( dz ) plus he osson processes ( dn convenence yeld bond prce and he nsananeous fuures convenence yeld forward rae are also drven by he sae Brownan oons and (excep n he specal case ha b for all ) he sae osson processes. he log of he value of he coody follows a ean reverng jupdffuson process whch ean revers o a ean reverson level whch s self a ean reverng jupdffuson process. I s noeworhy ha sochasc neres-raes can also conrbue o he ean reverson process. Fuures coody prces are also drven by he sae Brownan oons plus he Brownan oon drvng neres-raes and bond prces plus he osson processes. he correlaon beween he log of he value of he coody and he nsananeous fuures convenence yeld forward rae s posve. 4. one Carlo sulaon In hs secon we show how we can sulae fuures coody prces. he key o hs wll be o sulae he sae varables snce hen we can use equaon.3. one Carlo sulaon of he dffuson sae varables s sraghforward (see Babbs (99) Depser and Huon (997) or Glasseran (4)). So now we exane how we can sulae he jup sae X. varables N Recall ha we ake no assupons abou he spo jup apludes hey sasfy eher assupon. or hey sasfy assupon.. Alhough we ndex assupons ean ha he oucoes of Frsly for fuure reference we defne for each φ b u du ( ) exp γ do no depend on. γ oher han ha for each γ wh hese. (equaon 4.) Recall ha for each... he process sars a zero e = he process has ndependen ncreens. he expeced nuber of jups n N has a osson dsrbuon wh nensy rae λ. N and every e a jup occurs he process ncreens by one. N over he e perod u du. o s λ Now by he defnon of a non-hoogenous osson process he probably ha here are n he osson process N n he e perod o s: n jups 9
r ( = ) = exp λ u du (equaon 4.) n! N n u du We now sae a very useful aheacal proposon. roposon 4. : Suppose ha we know ha here have been λ n n jups beween e and e. S S... S. he condonal jon densy funcon of he Wre he arrval es of he jups as n arrval es when he arrval es are vewed as unordered rando varables condonal on N = n s: ( S s S s Sn ) sn N n ( s ) ( s )... ( sn ) r = & = &...& = = = λ λ λ λ u du n (equaon 4.3) roof : he above resul s proved n for exaple arln and aylor (975) n he case ha he nensy rae s consan and he exenson o a e-dependen deernsc nensy rae s sraghforward (and herefore he proof s oed). hs s an poran resul because now s sraghforward o sulae X he nuber of jups N. Frsly we sulae n up o e. here are several ways gven a rando nuber generaor whch produces rando nubers unfor on ( ) o sulae he nuber of jups n a gven e nerval of a non-hoogenous osson process (for exaple see Glasseran (4)). Usng equaon 4.3 we can sulae he arrval es. (hs s parcularly sraghforward f n are unfor on ( ) ). S S... S n of he Now noe ha equaon.9 he defnon of X n jups beween e and e λ s consan snce hen he arrval es condonal on N ples ha n n X N S b u du S S = S = = γ exp = γ φ ( ) n = hen. (equaon 4.4) X =. We nclude hs case n equaon 4.4 by usng he usual convenon ha If N a suaon s zero f he upper ndex s srcly less han he lower ndex. I only reans o sulae γ (n he case of assupon. he jup szes are known consans and n he case of assupon. hey are ndependen and dencally dsrbued whch eans hey do no depend on he arrval es) and hen we oban N X fro equaon 4.4. In order o sulae fuures coody prces we also need he fnal deernsc er n equaon.3. For each usng equaon.:
exp e ( s ) ds = exp λ ( s) ENs exp γ s exp b ( u) du ds s (equaon 4.5) Noe ha he negral n equaon 4.5 would n general have o be done nuercally bu s a sple one densonal deernsc negral whch can be pre-copued before enerng he one Carlo sulaon. We wll use he followng proposon n secon 5. roposon 4. : Exp exp exp b ( ) u du X N e s ds = = = (equaon 4.6) roof : Use equaons 4. and 4.3 and sandard resuls abou condonal expecaons. Reark 4.3 : Noe (leavng asde he ssue of any errors n he evaluaon of he deernsc negral n equaon 4.5) ha here are no dscresaon error bases n he sulaon of fuures coody prces n our odel as here gh be n soe odels nvolvng he sulaon of non-gaussan sochasc processes (for dscussons on hs opc see Babbs () or Glasseran (4)). 5. Opon prcng Our a n hs secon s o derve he prces of sandard opons and o do so n a for suable for rapd copuaon. he key o hs wll be he observaon ha condonal on he nuber of jups and her arrval es (and wh a suable assupon abou he spo jup apludes) fuures coody prces are log-norally dsrbued a whch pon falar resuls coe no play (see also eron (976) and Jarrow and adan (995)). We wll derve he prces of sandard European opons on fuures fuures-syle opons on fuures sandard European opons on he spo and sandard European opons on forward coody prces. Laer n hs secon we wll provde soe nuercal exaples whch llusrae our odel. We wll also show ha we can rapdly (ypcally of he order of /5 h of a second per opon dependng upon he requred accuracy) copue he prces of sandard opons. o acheve our goals we wll have o ake an assupon abou he dsrbuon of he spo jup apludes γ. here are hree cases of neres ha offer he prospec of racably. he frs s o assue he spo jup apludes are consans as n assupon.. Assupon. can be spl no wo possble cases whch gve our second and hrd cases of neres. he second case s o assue ha he γ are dscree rando varables wh a fne (n pracce sall) nuber of possble values. We wll exane hs case n secon 6. where we wll see can be consdered as a parcular case of he frs. he hrd case s o assue he spo jup apludes γ are norally dsrbued. We wll exane he frs and he hrd cases n hs secon. For each : In he case of assupon. he spo jup apludes are assued o be equal o β a consan.
In he case of assupon. he spo jup apludes are assued o be norally dsrbued wh ean β and sandard devaon υ (and n hs case b ). Clearly n he case of assupon. expγ exp b s u du s log-norally dsrbued s and usng sandard resuls for he expecaon of he exponenal of a norally dsrbued rando b ) varable we have (pung ENs exp γ s exp b ( u) du = exp β + υ s (equaon 5.) A generc opon prcng forula: Our a s o value a e a European (non-pah-dependen) opon aurng a e wren. on he fuures coody prce where he fuures conrac aures a e and Condonal on he nuber of jups s s... s... es n n =... n he e perod o = of hese jups hen (usng equaons 4. and 4.4): n exp exp b ( u) du X N ( ) = exp βφ ( s ) = and he arrval (equaon 5.) n he case ha assupon. s sasfed for hs ; or: n he case ha assupon. s sasfed for hs... exp exp = hen b ( u) du X N ( ) s log-norally dsrbued wh ean exp n β + υ = exp n β + υ = (equaon 5.3) Defne he ndcaor funcons for each... (.) = = f assupon. s sasfed for hs and and (.) (.) = oherwse = f assupon. s sasfed for hs and (.) = oherwse.
roposon 5. : he fuures coody prce ( ) fuures prce ( ) H a e (where =... n he e perod o hese jups s log-norally dsrbued wh ean H a e o e condonal on he ) and condonal on he nuber of jups s s... s... and he arrval es n n = of n H ( ) exp (.) βφ ( s ) + (.) n β + υ e ( s ) ds = = = ( ) ( ; ; ) = H V n (equaon 5.4) where ( ; ; ) V n n exp (.) βφ ( s ) + (.) n β + υ e ( s ) ds = = = And where: In he case of assupon. beng sasfed for a gven exp = exp exp (equaon 5.5) e ( s ) ds λ ( s) ( ( φ ( s ) β ) ) ds (equaon 5.6) And n he case of assupon. beng sasfed for a gven exp e ( s ) ds = exp exp β + υ λ ( u) du (equaon 5.7) roof: Equaon 5.4 follows edaely fro equaon.3 aken ogeher wh equaons 5. and 5.3. Equaons 5.6 and 5.7 use he defnons n equaons 4. and.. Reark 5. : Noe ha n general would be necessary o copue he negral n equaon 5.6 nuercally. he negral of he nsananeous varance of log of ( ) jups n =... n he e perod o =... of hese jups s Σ ( ) where H s a e s condonal on he nuber of and he arrval es k = k = s s... s n Σ ( ) σ ( s ) + σ ( s ) ρ σ ( s ) σ ( s ) ds k + ρ σ σ + υ ( s ) ( s ) ds n (.) (equaon 5.8) Hj Hj k = j= = 3
Consder a non-pah-dependen European opon wren on he fuures coody prce. he opon aures a e and he fuures conrac aures a e. Le he payoff of he opon a e ( ) be D H ( ) for soe funcon Condonal on he nuber of jups D. n... = and he arrval es =... of hese jups he value of he opon a e s (where ): Exp exp r ( u) du D ( H ( )) n s s... sn ;... = ( ) Reark 5.3 : In vew of proposon 5. gven he payoff ( ) s s... s n (equaon 5.9) D H of he opon a e we wll be able o use sandard resuls (for log-norally dsrbued prces) ogeher wh equaons 5.4 5.5 and 5.8 o calculae he expecaon n equaon 5.9. We defne for each Q n u du u du ( ; ) exp λ n! Noe ha ( ; ) λ n (equaon 5.) Q n s jus he probably ha here are he e perod o for each... =. n jups n he osson process N n roposon 5.4 : he prce of he opon a e s: n = n = n = n = n = n =... Q ; n Q ; n... Q ; n... Exp exp r ( u) du D( H ( )) n s s... s ; =... λ ( s ) λ ( s )... λ ( s ) n dsds... ds dsds... ds... ds ds... ds n λ ( u) du n = n n n (equaon 5.) roof: I follows edaely fro he resuls of secon 4 (n parcular equaon 4.3) equaons 5.9 and 5. and sandard resuls abou condonal expecaons. Reark 5.5 : Noe ha n he specal case ha for a gven =... he spo jup apludes sasfy assupon. he negral over he arrval es of he osson jups wll be splfed as he negrand becoes ndependen of he arrval es of ha gven osson process. 4
Reark 5.6 : Furher n he specal case ha assupon. s sasfed for all... opon prce a e s splfed o: n = n = n = n = n = n =... Q ; n Q ; n... Q ; n Exp exp r ( u) du D( H ( )) n; =... = he (equaon 5.) Reark 5.7 : Furherore n hs las specal case V ( ; n ; ) equaons 5.3 and 5.7): also splfes o (usng ( ; ; ) V n = = exp exp β + υ λ ( u) du exp n β + υ (equaon 5.3) Usng equaons 5.4 5.5 5.8 and 5. we are now n a poson o wre down he prces of varous sandard opons. Our specfc opon prcng forulae wll coe fro subsung a specfc for for equaon 5.9 no equaon 5.. We sae he resuls whou proof bu for full deals and ehodologes see eron (973)(976) Babbs (99) An and Jarrow (99) Duffe and Sanon (99) Jashdan (993) Jarrow and adan (995) and especally lersen and Schwarz (998). Sandard European Opons on Fuures: Suppose ha we wsh o value a e a sandard European (call or pu) opon on he fuures coody prce. he opon aures a e and he fuures conrac aures a e where. he payoff of he opon a e s ax( η ( H ( ) )) where s he srke of he opon and η = f he opon s a call and η = f he opon s a pu. he prce of he opon a e s: n = n = n = n = n = n = η... Q ; n Q ; n... Q ; n... H ( ) V ( ; n ; ) exp A( s ) ds N ( ηd ) N ( ηd ) λ ( s ) λ ( s )... λ ( s ) n dsds... ds dsds... ds... ds ds... ds n λ ( u) du = where n n n (equaon 5.4) 5
A d ( s ) ρ σ ( s ) σ ( s ) σ ( s ) σ ( s ) k = ln ; ; Σ ( H ( ) V ( n ) ) + A( s ) ds + Σ ( ) ( ) d d Σ ( ) and V ( ; n ; ) s as n equaon 5.5 and ( ) We recall n he above forula ha ( ) Σ s as n equaon 5.8 s he prce of a zero coupon bond a e aurng a e e a opon aury. In he specal case ha assupon. s sasfed for all =... he opon prce forula splfes n vew of equaons 5. and 5.3. Fuures-syle Opons on Fuures: Suppose ha we wsh o value a e a fuures-syle opon (call or pu) on he fuures coody prce. Fuures-syle opons are raded on soe exchanges. he key pon abou fuures-syle opons s ha hey are slar o fuures conracs n ha hey go undergo connuous reseleen (n pracce daly reseleen) wh a ark-o-arke procedure and as wh fuures conracs here s no nal cos n buyng a fuures-syle opon. We assue ha he fuures-syle opon aures a e and he fuures conrac aures a e where. he fuures-syle opon prce (e s delvery value) a e s ax( η ( H ( ) )) where s he srke of he opon and η = f he opon s a call and η = f he opon s a pu. However he gans and losses of he fuures-syle opon are reseled connuously (n pracce daly) durng he lfe of he fuures-syle opon conrac. I can be shown (see eron (99) Duffe (996) or Duffe and Sanon (99)) ha he fuures-syle opon prce a e s he prce of a sandard (e non-fuures-syle) opon a e whch has a payoff of exp r ( u) duax ( η ( H ( ) )) a e. Hence he fuures-syle opon prce a e s: Exp exp r ( u) du exp r ( u) du ax η H ( ) = Exp ax ( η ( H ( ) )) ( ( ) ) 6
Hence we can show ha he fuures-syle opon prce a e s: n = n = n = n = n = n =... Q ; n Q ; n... Q ; n... η H V ; n ; N ηd N ηd = where d ( s ) ( s )... ( s ) λ λ λ n λ u du n ln ; ; ( H ( ) V ( n ) ) + Σ ( ) Σ( ) ( ) d d Σ and V ( ; n ; ) s as n equaon 5.5 and ( ) ds ds... ds ds ds... ds... ds ds... ds n n n (equaon 5.5) Σ s as n equaon 5.8. Reark 5.8 : Noe ha he prce of a zero coupon bond does no appear n equaon 5.5. Reark 5.9 : I can be shown (usng he ehods of eron (973) Duffe (996) Duffe and Sanon (99) and Jashdan (993)) ha s never opal o exercse Aercan fuures-syle opons on fuures prces before aury whch eans ha European fuures-syle opons on fuures prces and Aercan fuures-syle opons on fuures prces always have he sae prce (hs apples respecvely o boh calls and pus). Hence equaon 5.5 s equally vald for boh European and Aercan fuuressyle opons on fuures prces. Sandard European Opons on he spo: Suppose ha we wsh o value a e a sandard European (call or pu) opon on he spo coody prce. he opon aures a e. he payoff of he opon a e s ( ( C ) ) ax η where s he srke of he opon and = η = f he opon s a pu. Noe C F ( ) =. η f he opon s a call and 7
he prce of he opon a e s: n = n = n = n = n = n = η... Q ; n Q ; n... Q ; n ( η ) ( η )... F V ; n ; N d N d = where d ( s ) ( s )... ( s ) λ λ λ n λ u du n ln ; ; ( F ( ) V ( n ) ) + Σ ( ) Σ( ) ( ) d d Σ ds ds... ds ds ds... ds... ds ds... ds n n n and V ( ; n ; ) s obaned fro equaon 5.5 and ( ) 5.8 (wh ). (equaon 5.6) Σ s obaned fro equaon Sandard European Opons on Forwards: Suppose ha we wsh o value a e a sandard European (call or pu) opon on he forward coody prce. he opon aures a e and he forward prce s o e where. We denoe he srke of he opon by and we wre η = f he opon s a call and η = f he opon s a pu. here are wo possble payoffs: (Noe hese are opons on he forward coody prce and are no o be confused wh he way ha soe opons n he coodes arkes acually work where he delverable s a srp of forward prces over a perod of e). We consder frs he case where he payoff of he opon a e s ( ( F( ) )) ax η. 8
he prce of he opon a e s: n = n = n = n = n = n = η... Q ; n Q ; n... Q ; n... F ( ) V ( ; n ; ) exp B( s ) ds N ( ηd ) N ( ηd ) λ ( s ) λ ( s )... λ ( s ) n dsds... ds dsds... ds... ds ds... ds n λ ( u) du = where B s s s s k = σ s σ s σ s d ( ) ρ ( σ ( ) σ ( )) σ ( ) ( ( )) ( ) n n n ln ; ; Σ ( F ( ) V ( n ) ) + B( s ) ds + Σ ( ) ( ) d d Σ ( ) and V ( ; n ; ) s as n equaon 5.5 and ( ) Σ s as n equaon 5.8. (equaon 5.7) We consder secondly he case where he payoff of he opon s also ( ( F( ) )) ax η bu now he payoff occurs a e. hs eans he payoff s he sae as a payoff of ax( η ( )( F( ) )) a e. he prce of he opon a e s: n = n = n = n = n = n = η... Q ; n Q ; n... Q ; n ( η ) ( η )... F V ; n ; N d N d = where ( s ) ( s )... ( s ) λ λ λ n λ u du n ds ds... ds ds ds... ds... ds ds... ds n n n (equaon 5.8) 9
d ln ; ; ( F ( ) V ( n ) ) + Σ ( ) Σ( ) ( ) d d Σ and V ( ; n ; ) s as n equaon 5.5 and ( ) Σ s as n equaon 5.8. Noe ha as wh equaon 5.4 equaons 5.5 5.6 5.7 and 5.8 can all be splfed n he specal case ha assupon. s sasfed for all =... as ndcaed pror o equaons 5. and 5.3. Nuercal exaples and copuaonal ssues: he above resuls are very useful as hey also allow he possbly o calbrae he odel hrough dervng pled paraeers fro he arke prces of opons. Clearly for calbraon purposes rapd copuaon s poran. We wll now llusrae our odel wh a oal of 8 nuercal exaples and also dscuss copuaonal ssues surroundng he rapd copuaon of opon prces usng equaons 5.4 o 5.8. he probables n he osson ass funcons wll rapdly end o zero once he nuber of jups s greaer han he ean nuber of jups. herefore copuaon es n he case when all he osson processes sasfy assupon. wll ypcally be very sall (a leas when he nuber of osson processes s no oo large). When all or soe of he osson processes sasfy assupon. s necessary o copue he negrals over he arrval es. he os approprae ehod would see o be o use one Carlo sulaon of he arrval es (we sress only of he arrval es no of he osson jups nor he dffuson processes whch can be done analycally). hs s he ehod we use n he nuercal exaples below. Alhough hs gh sound copuaonally nensve he sulaon s jus of he arrval es of he jups. In any cases he varaon of he negral wh dfferen arrval es wll be que sall leadng o sall sandard errors. hs gh ypcally be he case for opons whch are deep n or ou of he oney or when he jup decay coeffcen paraeers b ) are close o zero. In addon o nse sandard errors we used he ehod of anhec ( varaes and we also used equaon 4.6 as a conrol varae usng he opal-weghng/lnearregresson ehodology descrbed for exaple n chaper 4 of Glasseran (4). he opon prces n ables 5 and 8 (see our nuercal exaples below) were all copued usng 5 one Carlo sulaons. he deernsc negral n equaon 5.6 was copued usng he rapezu rule wh 5 pons. Usng a uch larger nuber of pons confred ha he poenal errors n he opon prces n ables 5 and 8 due o he approxaon nheren n copung hs negral were n all cases less han. whch s neglgble copared o he sandard errors repored. In he exaples below he suaon over he osson probably ass funcons was runcaed when boh he proporonal and absolue convergence of he opon prce were less han.. Copuaons were perfored on a desk-op p.c. runnng a.8 GHz wh crosof Wndows rofessonal wh Gb of RA wh a progra wren n crosof C++. We now llusrae our odel wh a oal of egh exaples labelled exaples o 8 he resuls of whch are n ables o 8 respecvely. We wll spl he no wo caegores exaples o 3 and hen exaples 4 o 8. In all egh exaples we assue ha he fuures coody prces o all aures are 95 and he neres-rae yeld curve s fla wh a connuously copounded rsk-free rae of.5 (as n lersen and Schwarz (998)). Alhough any of he paraeers n our odel can be e-dependen (and ndeed ay be useful o allow for hs o capure for exaple seasonaly (see lersen (3))) we wll llusrae he odel 3
wh consan paraeers. In order o ach he paraeers of lersen and Schwarz (998) whose se-up s slghly dfferen o ours bu enrely equvalen n he wo facor pure-dffuson case we choose o have wo Brownan oons (n addon o he Brownan oon drvng neres-raes) e = and η H =.66 η H =.49 /.45.3877596 χ H =. χ H =.49 /.45 a H =.45 σ =.96 α =. r r ρ HH =.85 ρ H =.964 ρ H =.43 Noe he negave value of χ H s arfcal n order o ach he lersen and Schwarz (998) daa and could be ade posve by cobnng η H and η H no one er and akng conssen adjusens o he correlaons n he obvous anner. Also as rearked afer equaon.5 when calbrang our odel would n general be necessary o pu order o avod a degeneracy. η for all k excep one n We consder frsly exaples o 3. Exaple s pure-dffuson and exaples and 3 are wh jups. he pure-dffuson exaple s effecvely dencal o ha used n lersen and Schwarz (998). We value sandard European call opons (usng equaon 5.4) on fuures conracs whose aures are.5 years afer he aury of he opon. We prce opons wh srkes 75 8 95 5 and aures equal o.5.5.75 3 years (here are 3 opons n oal). Exaple : In exaple we prce opons n he pure-dffuson case (usng equaon 5.4 reduced o he no-jup case). he resuls are n able. Clearly he resuls are exacly as n able of lersen and Schwarz (998) (we have exra opon aures and exra srkes) snce we have (albe n a slghly dfferen for) he sae dffuson paraeers. Now we nroduce jup processes for exaples and 3 bu keep he dffuson paraeers as n exaple. he paraeers of our processes are purely for llusraon Exaple : In exaple we assue ha here s one osson process has consan paraeers: λ =.75 β =. b =. = and sasfes assupon. and he paraeers are only for llusraon. he value of b s roughly equvalen o he effec of a jup beng dapened o approxaely 37.8 % of he jup sze over half a year whch sees plausble. 3