Using Options Prices to Infer PDF's for Asset Prices: An Application to Oil Prices during the Gulf Crisis

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Using Options Prices to Infer PDF’s for Asset Prices: An Application to Oil Prices During the Gulf Crisis W R M a Ci P T el lhla ho I I n t r o d u ct O p c i p anr o m p f ti apn oic abroro et rd t ta rcomisaic th p o th u a n T r r thd i s enae t icf r asnsaceo oaypst ly rm p fo th p o rth u a nsuo ari B d m ss ceu reo opot rlpowbst r th p oe tha a p r ca ths i ar t porv mn sut ocre t ette d oi th p osth u t a n Hr ri i dci ioss b e i i mnesr uwnaerl t ti b w a a a s th ed s oi t itpr oubsot u gi t mas nra r t pt t deib s p t b w i eo r an us co v p ohi t hd ore pt atceripir lv sto t ar d i T p s d t a m rte d iaep hi sbuevti a d us fi etlreAm tisiot o an a i p c opi m t dt P p urGuutar ciro W earr li co ot o kt e d s it t r s t b mt o e si r mho c imt bet ov at anu ti W f t th e d s ia c sint whia to tma ic r n a itbommat si i t t i a s p i of rmmgpaha hde o n i t io bm i Wsr aabfi f art i p m o a hao us tahns B latl mkoe t a wolyan hcic ge ko v th m e a r oa ths p e ossrar ma d t oe akeiui bass mna tsr de i o p o s a dm r i sp uiccr uac pt “ Correspondenceshouldbedirectedto C.Thomas,MailStop#42,FederalReserveBoard,WashingtonDC20551;Tel:(202)452-3698; Email:thomasc@frb.gov.TheauthorsarestaffeconomistsintheDivisionofInternationalFinance,BoardofGovernorsoftheFederalReserve System.ThispaperrepresentstheviewsoftheauthorsandshouldnotbeinterpretedasreflectingtheviewsoftheBoardofGovernorsofthe FederalReserveSystemorothermembersofitsstaff.WewouldliketoespeciallythankDavidBates,JonFaust,GregDuffee,ChristianGilles, andEdGreenfortheirextensivecomments.WealsothankAlIanBrunner,NeilEricsson,JeffFuhrer,LudgerHentschel,andGeorgeMoore, aswellasseminarparticipantsatVanderbilt’sOwenSchoolofManagement,OhioStateUniversity,theAmericanFinanceAssociationwinter 1995meetings,theBankforInternationalSettlements,theUniversityofNeuchatel,theDeutscheBundesbankt,heEuropeanMonetaryInstitute, andtheFederalReserveBoardInternationalFinanceDivision.ElizabethVrankovichandDaraAkbarianprovidedvaluableresearchassistance. ‘Otherstudieswhichfocusontheasset’sterminalpdfincludeBreedenandLitzenberger(1978),JarrowandRudd(1982),Shimko(1991), andMalz(1996).

2 R i e frn oc p f i co o pbto tv rifr mFi e ompa acppl ri i p n t r c ri a weeo a ob fabr o e pweSh uo rmoo t alreintcra p o ra t eoi p s ar fu t soet nrd r ha ueo ir e wteic maeilnntvcer t o r i ui th e nl p ha s op- t ais s t prpr t o tim irmaiocmrioc d ica o rs th tr e r(i pi esmnl b p kqra u ra -autr est t tne irvatiio ( p a aS c th re te ea f u o Amx acaorpeea mamerer it di on tiaer d a c f e e fo tlhx p or a op T orsos r i t prit p uer w d ~’ ross b o th p o a Ao o i tm ri o tu r pt e di eronisdtri p oskt-t u a an u t p d l swi th e e rl o t s pse r o ooctta asli imwira (ii i [ r on i bu ()ur m re u t pt eso s h o tq ofot olat imd uriud ol is “ s i s t i sl t th ari o asp-pu m s s a c pr t ihrmiuirl i tuo oc mtpi s (to e tht j x u a um ao Bsap ( )-dsmGi td w umr1i9 pileo ff p o do th uP Gus c utw ae sth mrac prirs ibss eaxomis o rt p t b d f a mr o t I r i di o iro Thhxrats g awce ti dsis nor ib m f i s [V e u or c ll ti , P s o th ltt d reria th t sus r ciptrat t cth brmi c oc ib an b W r e t c th nr e d eeo iiti si ab cfsgta e stt ooivtt avori p p t e Thx m r th rpth t iwi p e eanirc g io cf aroc ui thoinn d h o r y s e e fo thtnpo I a rl dg ta pt i da d ncecatmeditii t at ~Evidencethattherisk-neutralandtrueparametersareverydifferentwouldincludeafindingthattheoilfuturespricedidnotfollowa martingaleprocess.Availableevidence(seeDominguez(1989),Kumar(1992),andDeavesandKrinsky(1992))indicatesthattheoilfutures pricedoesinfactmartingale. Thisevidenceatleastallowsforthepossibilitythatourrecoveredparametersarequiteclosetotheactuarial parameters ‘I_oourknowledge,theonlyattempttoestimatethesedistributionsnon-parametricallyisthatofAlt-SahaliaandLo(1995)whichusesintrada}quotesforEuropeanOptions.

3 t s p o th ru e a ontimb us i pedri e eses o t xv-mo rtrl al Th b o th t e a fre i fn cgl reai eed h Axnierfi ni ib r f e f foal ufo th t s ned oeisuc axt sun htrm tc iebab iori a ca w v co s footh at dm h id ei Mr m s aeqormapftlie ni da ri p ca b qr d Moo ii msucif wi tp t fet do ic sesriiraarm stta g a s aeg p t d np eeiin iv msnar ersr t c pr tooawrmi oma ib g p i c wir oon i on t n d o ei sF vesa shce i rmbt ino bo r st f ib a g t d eoiv th ir s hr tt e p o ormA ve leoipri ariinmdeebu a e b th t nl d eri t o mts e s r i tpr gt g rmiosretarocteiebn d i T m so th i t i ntr huA ifop opsrbm op ut t rmte er teir d ri t ths p pt a a A f r orro ti ha th eait bt oc ie al uitc o ch ti d t s d t th s io th dp at rri th hfa s t e etatpat thor ritct p d o if to a ms e tpi i nToh sf r irtps ltt imaiepaoc ii rtaelc on s s i a th P t Gu C uw e i aiucmr o n nrst f he iasoti pote oca on p o r t uo th ar p apt cr hsasicth ri oocm Th r o th p i o a f r S aeps I op g bo ec f atlrhnle o a A o o amf c cp e u oo ot d r nti otint fu idcsp a ttrurotirtp e S x 11 i p hoelth bi cl b rusc wiu opoau pr s t tiret t tr io d foi f p s S tuI d r rt ep i ti o aiac a bsct trurpt uo mcu ti ptli w S V p th re hor th a ec Apes apilc stsuie ulmre on a icf i S VI e c ti 4However,themethodasawhole,ratherthanaparticularestimate,canbeevaluatedex-postusingEDFtests.SeeFacklerandKing(1990) andSilvaandKahl(1993)forexamples. 5Forexample,Rubinstein(1994)beginswithanestimateoftheriskneutraldistributionatexpiration,usingBreedenandLitzenberger(1978), beforeconstructing,withbinomialtrees,oneprocessconsistentwiththatdistribution.

4 W E s o u th r pber it ty t dl o teitoai !% pr st pea t itor o p i v d Fpo c ( r ith va o tt poer rail si ict vaiuot optetc p o th d ai ( sth s dt b bobar it t trp i e sus aba croe luoitnpp r Fo A s oa m th r pbee tty t dl r teitaai t eo.p st pic i l tiro d o t th e e i p w Ix g rarrt o e eva ewin dept ecrconet mei s p t fo f po r no ju tuhc d r o fih f t prs icaa cte optut urst ri e Txd w th pe e i p wxrd eaarborit a f e t emvat ema rcio m v o a oi g th th f an ppr i ta ivfru i a p tilu d artumuaist o e pT x is fo al sp t p it th iimrr oahagi d aonoc fi t ti fu hcaest a th o e tp x ar b pfo tth o i vaheouwih prc b e aon i tit xpoti t d ai I ths e l tr hsa th r b ota oiw ti bar ounaeeaiutprumta v fo th o a p t lu io Th l b ar we k an oos be w prno utanT upweit bo h ar n So i b w thn i gi i teetweteh oma a pr uivegi i Ap it 1 T fi n le f d o th ( tp oetr u a as naa t t nreio deo tndixopo an le X d th o s p e p W i ti t b tpri no pr t t ioonpdreir ex an d a p l b 8e~e Th t ti et er n i T xT irotng inp ot risod r i ir a c an ths o o d n sf n i pei t i eu‘sr -wachs t edri peta mfe cot th o e ipe‘x* E, d pt e xtiae it p p prr t eoneno eart Txcpt b ar p o throa the sth u u d sas nm i u w da rned camt ptrter sSeeCoxandRoss(1976)foradiscussionoftie risk-neutralvaluationtechnique. 7Therearemartyboundsintheoptionliterature,buttoourknowledgethisisthefirstsetofboundsonthepriceofAmericanoptionsinterms oftherisk-neutralterminaldistribution.PerrakisandRyan(1984)andLevy(1985)deriveboundsforEuropeanoptionsindiscretetimeinterms ofthetruedistribution.Grundy(1991)derivesboundsforthetruedistributionfromthemomentsoftherisk-neutraldistribution.Lo(1987) derivesboundsonEuropeanoptionsintermsofthefirsttwomomentsoftherisk-neutraldistribution.

5 p ru i th oE Th ub an l a b bfse A ppiJ]ca aoouwlpme a g i a ti f o l l ow Ct =m E -X e“ l -x]],[]; . E , ( [m i c, =m 4 E -X e* T -x]]]; . Jf E , ( [m ( P, =m X- e“ ’ E- . E fJ,f , [m i ( P, =m X- e“ ET ‘ . ,[ E f, Jm i Th i b tn b i s t e f aohetus f ocrahi ai rap imunF ti r t w ar c e bo o thno hpa cs oconditionaptl on agivenrridisaltruibnutiotnifuocr f m w h oa th th rf sp m e uawviasr o t t rrip tu vueemsmtiero u i th e ox B pth mp se e a ae c t s fur t r psr Lao,tdieq uamt ti t e o xtho f p p a th ed ue o t xoc riEa t pSti ~p-tX i atiry,[’ur ti r on c r e t b e e tvho cax o otd c et e ca op purctla eoidb nwu ci l t E - X . T i th f i,t i thesmaha li f bo [tfirhiup a lo bo I a a A d o c m db w le tph ie a ao i t r otrEti ndneioop ihceTur s i i th ma l feo th l b i s tec t va o otiws E ouim ovne o utr o p t i on Th u b d fr th l pbpio ooun b t d ff faow u iis t s i i th ma l I o d thnitee is vas o t xsop c a e opeb tet xpf un t t e w xd i b pth l i imo oi p sTr u en wear ncto i a uti de

6 b c th f oo t oe n Txahlua a ps ml our epr ndaidcoow rbii rt w a g t d ei Th o i s ihta sor vta un prtth mp ve i Co btirno t a ut s p w i i e r t th f d e htihoca eu wxchnirncew orne ce i tit l p ( u thi f e p i nno r nd rir t s inr e pee toc un iostox s an tp S th ur a ndo ono imo dd t c la seps th nueicrsrn re er t dly e th ox a th ue op nnr fplec c i r txi e iede twerr sosi -t ta ol p th r oo a c r e s Th te vea otct op urvnt tposeei pr en i h t u th f si i on nb itbehagd in db oi pers dleheifscT e ca xb r b ps ewi thes t p p r a r a eciterao r es rtme em u onn p s c an d e e t i o rb o s peh tleri cWon ea evs ouw hin f a p w g a hr o v fo th ohi o itvhi a t upnr alst ceptI t ghe w h c a om p n a w sal or tr u th t i rn r oc ni epcre u evU ng etr ct p th o v r i pg a i th uot b al iiv c opupo es ns A s i th a wi c p h t o tp up bon raiesiow tti nd u nE vd o thu oi I t se r alp wc us e o o se ttipr u imvpee im nt a o c vp d a we t i en rt r u i t o hinr rhFarei t rireptigyit u b w i wmoip a on da pd f elurna ipet te Eu acpspvmcae I R th eD i c IsI o t v r e i brui E (1 - (4qgi b fo uA o oap m t p bte pteer urni eriiot xp o th i r s np an th pa t ttrd fi fe etre ipcrsri i pe retuOt T ri r t d efi a os p c wt nhei ct cr pt oroaripoi eos tubf tveopns ut v f th b i so a so o t mrodt u ia luclr sexpo nwd tucin ri th o p c p or n t ic ta io gOnJune24, 1993,NYMEXbegantradingonitsAccessSystemwhichallowstradingbetween5pmand8amEST.Theseextendedhours mayjusti~ usingthecontinuous-tradingupperboundfordataafterJune1993.Asapracticalmatter,however,forinterestrateslessthan20 percent.theupperboundsforcontinuoustradingandovernightholdingswilldifferbylessthan0.05percent. .&’

7 Le g ; e, b a p d a foi th pr(f sOf aprt wh etrr rmut u i ib et f e xi op t Tph v 8red e thc d ee m t ai i riw saws es inct t tcrr io A Le 0 r ou e e o e an lesp$ d e rt txa wien reepi t g ;e$,s)e. ma ct 1 P E s 1 o t i 1. in ma T a s e tt s p r e aat po e ce if tn opplhst pmqru dani at c o oth e nd s i Td g stsui a et e tiw wr e ts i nueipma a lobn ti ut b c a Aoon wa t i bm awh u t n a pop ovtprt fandbeut ct ter ur b i i t o ho qo th m ee u u u abanx t i fu vrcmand o t perkecrk u a nt b r dI th me see t su r artx nb or l se r c lvypiqeeus eelr t th o v w bpc t th u hebt a i t leo i il t uopup ra luotnnb xp ce r l t teh o wi b ap s n pth lo ohbe tr T eae tli we ouscteairc ve t e s o r x Ge th pnhip so t u eo aivbnat o lpr ecetdcu ut er ti c w d th wr e s es i ci s c i dc sisp gaopi r ou hgfen htistefsf u an l t o th d p i I po sa w cah t t e per rt wwe itislt T f b icei ut w w, c th es oar i thi el p ta pae si us gtf a co opow turhtalu i th m an fo al pu o th aor ou o thp m i. ( > X tnTe seon we Wio c th s oar i theup ta pan si us ft t coa tphp a uoeelo t m urte an th p t ar i th m i. (f < X) uot ha ne Th a o p ca b wc pi t r o e r tpu tis oict earit ti ra a d anis a e tes a f st r o rr u i l b me ut lo C =i C ” +( - C t[ + [Xii wit’ :c ( P =i P +( - P t,[ +Eii i t ”[ (’i[i A w i = 1 i E [ > X an i = 2ho t fOh er e rw

8 Th e ~ , wi b th r or an er i e es t rowse o t pati o t d pi an n s i th s t tw e r ooyi lwuh i ax i b st omb amu F mMeti th s w ar a a e al o p f ama gi ic c p pl oth wgh nb roatipr tre i b w thn tw eb S dia oo preg aucro cutn thtptt cone p a c e i thr e r qe ls u a ro a ti ti 1 F Fo uo D 1 n i sc t t 1. r io ib I c a f h fo ufo th e o n d s wicotr t btais ti f si tr le p an ae o i r n Fo rt s e e iiasrS e x Np wm sp r pltecas toen f ta p a th o e ipdrx fr a m pt o th l iraiciix d org i(h on astt M M f th Ld o fi fr fs oNr up tmg i giut r nbc r)ia.l ib g = [f + 7c[f ( ‘~3g3[fol w h er k ( U e (5 an q(6 a p s e u c b rw qa f a ionp tiiicuriat o -rbt e p ( aPi ~il wl wz ri = 1 a a(tnhe o m,2( be Jveee“t ) se Th p o tha m e r bxe aod ( ea q( ma ems ubaet pl m th siu o s ne fo al oqi o a girrm c ptua oiim tn fozi tr c o n s t r a in g ThismixtureresumptionwasalsousedbyRitchey(1990)whoderivesEuropeanoptionspriceswhenthepdfisamixtureofIognormals.

9 3 ( Xi = 1; () < Wi < 1, i = 1, 2. x i Th r o teh su o th sm r t t n o per f iaeulm dtut ctrAa a r d c e b id sb n i th tmunou t tm r ot a pioai E ss=ocn~trt H a d o i S i V b w p sle o t ef ec wec biimri veuf uta loe ti ss c od th np O tt ud rteh f as pr rcidhre nut arye t i ete / io xpda an t p r a w e no b vr s Ihia atb onu i dal r mt r dimip weest ct w a t us th f p a a m u eor t blg erio t f t o e dads ur snt is es D o th d an ee ar fs i St tI at iae ou m iclt at I th p s w rc eth fo r e l dc e f v t aiaibstio aac iqo so c s a w a lo teh i a gov a i o p t t eo pg va r Cseh ptOrhatog a xi f fo ho c t th b oar u eelth e go d s fi saiety os dutsin T t thtr p p th d b a th ib bo e i a losn te oua tab atpe oelt acsowaen o p p r t i io ce Fo th da p th o hla 3 da t ep o a tx r tittT- p r wel ir a 7 p T th de bf i thi rf fa t hubsocoou d bct acoernm0 if p O t da eth f p war a u$2 F thci d rii t boennca ( ee s-utr th b $1 an $ teh l b wa d t 20 b teowva oouwe t to xa ertee er d b i th b wfaele th 0.f p o tF te ca e wi str uwnea o arc en$ th d b i th b wfaea 0. pf o o t t ope prebo F rt uwnpe n rwecn s iuth m ffo th l fb t b id on bc teowva ouoi e t et xea er nt th b d f e o o b a i 0. p F ut trvhoe efflacbopr p ( rncd st b $ th e u bse wa2b tt a opp opiurlo t 0)tel cwti maopt t b f t l t 0. op o th aa o e up es ha ct rcpltl nrdi

1 1 S M a Bt 1 e a od n n 1. ch da I o t g th r fr ou rmo e(au t d sufi dMfeuL p st w a r u ae“ o sc p lsmsi t oT mptor coa v aod ndici tsesr o p l i t irth u tc n i pr e of d ahga rc me Brol eaot pr rlmino w i t th f mp a e u wi pbx dr ha frri atpsi lil i d urog irsaT i th a b s B ( s m efol pu E 1 mo achriioodufru p 9S7e pt ti a p h bp d r t p oeA x o vavuinmeet m“ri ept aaer st ltoiwss th q a u p o B p a aanr W dr( o ( o x r B hahnbi 19 t atmme-aer e c an ma ca lu oW cc s a o“muos se mons bill amso st tta at p w b d f ars l d ra oilii ai us rtogsB a ctneg nop t ri pr g o p e e Wp qrrn t 2upe e i ti( aaa cG orat csitiria olvo ( S hd bi em s th rsi o tsLN end r o paqieiu f baufmtvriei o p ( A p r1 fo d p o t B i ta epSe p a Sce pene tiao r st ox T t ar tw d b ih the S f a Me mf Fius te tmwo oda re fo th e e p i xd awr Sie us t B ef qayr rl a fe uamipcpi t p a A o w m MrL us pth we up hia t leoric bo f a igiiAcom o t p a A po Sm rSL ap t ethe t f ssptir riica iecoo ut w bxp d f a s I d r o i iw Mrosga tht antwg nfu r hssip a eixor bu w b d f a m o I rd iil o i S ro i salgx netaw wi norM tue ib t SL u a d t i t a e hfa tf ease ecc fp h Txe nceor re niwon b i w thmetw m ar ps ctc ood hea o rt om imp ta st 10OverdahlandMatthews(1988),whenstudyinga moretranquilperiodintheoilmarket,alsorecoveredtheparametersofasingle Iognormal,thatistheparametersfromthestandardBlack-Scholesoptionpricingmodel.

1 1 D L v i th 1D mi i ait s s t 1. t-a a ri ti B p t thr e e i ois u t notchfo da l i ese a i tm aesd miat f f ufo th d n ii Thc spa no tor btt f t r ti s iloeorru oont bu h m o th d i o air th t so t mfog Et O f -r Xeerna h,[ei bu E O X - f ]] T , tw t ca b w[ a (hEe If >eXr rm- X “Pr [ > X aitax ( - E If, X] oP [f < X] I i ,c th ev rio th we n er[fle i t pr r th f e t s arl a d i ta an i moacntiha r smt it t d np iskoncerrt or e s o f p nup a ui l rp t ml o twhi t opppo picrmir auccr r or it a th d iTh r s b o pt e on cr i ptci ouri abnobnt c or fuo on ti e anx p mpar i th f e o s cobo t s et a l1 t sea ubp gmblo tti il l s 2 th s o bt eae s a er3wt gs t ab ttiwk mhie eesg st ri I p i Xa an X ar rth l ant h s i thow a it c it ruen grhbi t lafo o w b i t po th f ot er il l i l oown E “Pt r[ of obl, E X < . P r, f Xo ~bX O<[~f<, [ ( Xi P r o bt ( An n o d ic u g s t sa t r e fmrbotu c inees e on b xp ut an p ri ( o Fo e1 b fo a 0agi x d b ) wi cai c - s la s motnpr(i1t d oiu o a s so n t o u enr d - iwh nowe bbriov buens if serlti e t thqg d u ri t thi sdaivde tv b ( e rl a 1 i sc leat 0b)u T i i c th an e dh s leri c ts i e usnt iemar t riqiu t espr r b th l e s an a the ohg ts F cr iiboo wsettlro ria al goahn

1 $ a (i a fe i $ na p a fi Sds l oet td ar.l0o b wii ant arr st o s I th t bt th s hr ewaitha ioi orn yoo tkweec f ik on or e xan th p p r The tho s oc bt d ta ii t btaa haws die t trli io i omth f pfo ua o fo thnd r s i Cch tT s i stua tt llp iwonrit t o b e s d heq i rTh sous vli i atma i re ortth Iv ii ix wobougal na th d l r th u a ta wei u i d s thpppyni e t s nree lahnefs if( IV A tr th Oi M) r da i c rk at IV 1 D S o at u rc D o s p fo al c oieo r oatf f ap rtutt idcut o t tipra J 2 1 t M 30 19 h we p ul f9r9arrNu W u r t ouse YMp acht v o th o i e (5 aan q(p6 Th s pru i d tilu aa ette e o e ti dte b a s c e m uo ot r 2 omt m o pl m apd aT e cugiar r frmtie toimtt r o th a o bi ean as vp d t lal me rio t ur a stirnaie po ra f t s p e H t t or aer pt f r ipwi l p a pari eu citreiatudtp vi lme v o a th s s p wh th f m tt h s uamI t ievar erio a litu m oonttt f m th su a c e rr t oo to ko t mut u el pten sl moi nc re tl em on c an s to U r s np p aa t tp destire ari vorariwo sas ngsy q i i t nu d r ah o n ate s te rea ct D J 1 t Mu o 1 ht w uclri 99 arroraoi s9e9 c n F ocne e c oal o np th wte rac p r wir n oape i c nti vo iccnott a n id 11Therewerenopricelimitsintheoptionsmarket. Overtheentiresampletherewerelimitsoncrudeoilfuturespricechangesforall contractsexceptfortheoneclosesttoexpiration. In Decemberof 1990the limitsoncrudeoil futurespricemovementswerewidened substantially. 12WearegratetidtoNYMEXBoardofDirectorsmemberJimZamoraofZAHRTradingandformerNYMEXemployeeBradHomefor theirdescriptionsofthesettlementprices.

I 1 s w e e f t th dax set I a l cter dda ewiro lf crawmo d ioiudetn c e o w xa en frp tht dax s i Tar 1erli c rlssu a eti a lu ctf nf ti e o th c a th oe x anc ftc t l r u acsi I T 1 ab N o Ra o um R o C E o sT Tn t O p ti Sot pptomt Frau Prtr atp R Da O anD p C ( Cootni ( M I M M I M M M Oc 9 7 - 8 4 /125/4 17 42 2124/ 37 1 9/3 No 9 7 - 1 6 /23305 15 57 1/25/ 45 1 4/4 De 9 7 - 1 8 /31510 13 55 1/25/ 44 1 1/3 Ja 9 7 - 1 10 /13889 13 48/ 126/ 42 291 3 Fe 9 8 - 1 10 /3866 / 11 43 52/ 5 3/ 2 3 Ma 9 9 1 / 9 3588/ 112 51 103 0 50 1 /193/ Ap 9 8 - 2 14 /482/7 11 44 2110/ 45 1 8/3 D p fo th s T rba th mr aevcl aic pe atiill a t opoass c e wo u t c x n th d a ft p F ieerla srec a ir ctoshc aca Nct (nto ul o e l p (5 qan (6 th fot ua c noia ik m n o tonipr s A nnliiotr th s c t ar oa t o 6e4 t n da heea t t ort r ve d ar w tr add aci ng s t e 64 mh p i weea p n rEra ie d y a me t r s oti pafoiefo zael e th ses o te p t fr aML ian t sr o tm p a fa aS m te ra et ‘3Oneday’sworthofdatafortheDecembercontractwasalsoexcludedduetoanobviouserrorindataentryonthepartofNYMEX

I 1 Iv.z E~ti~~tio~ T th hP Gu c r m e c o r foo ursaor t dmimisgh o i a ru t p tc r e (e oIcr ewo p n o t-uwi e dif mCerKu ac 2 a s d t P i eGu oi ss e(e d r vue t Sa rsAur fppam ptd aacw an 3 a c oo u cn n ovt ot r is h n (enelaepr d uorst ttitait w o 1o 2 m eu h o v Gi t th igeth c p ic ncoo w c cusa mtu soi t I a tho fo o thhdg ti bne s I oi sref tma prmtir far tib p w l t b d r fr a t i d erri raith coke bsi ea catce -bm t ri m M tih m o c alx e r i a t eca siou xatsI c uorv d ogom iist w b fi th d ( n = n o= O) E wese e. xthat auln xph t -ama t r w oeth te l l mii o c a ahwgrge a t pha nhtti o ear ootr t l Foo e nge o a xI n r a oao a Sa Arrocra motpt f m m in a! th w o th Ie d uoi wi t h igg mo a w tnhiaiginst t roetir r e bn th l ca d o o i S Vsmgp nge t ecp nodrer stasisf ib s e d e th P Gvf cl u e e r en rirs ct is E o MsL an SL wta p wii t e N m A rf um Gat lg ( . F a EO l o a IB R Rg 0 B f Sot pTR 4Uworui-6ars s t (j < ~, < @ , 0.0001 < a A i < *.14 A d wnee p f r bal e ro i sTt va d w e c ur aM i aal th vweert nsci a h veu u wtie tmeEla imva a p t le ( s g o th erit D o e i s a rre qm ttiap eiltatutaIa Th e p s ar ir t i Clho Tih Tl ct mp up etd east st d f u beo uth ML an SL nmn si Gi t dce siod f t si ti Io th BA f gi p o v for t or p e alT tm pdit rit l ri upl ic p 1’Inaddition,each(p,o)pairwasrestrictedsuchthattheprobabilityofthefuturespricereaching$150perbarrelwaslessthan5percent, undereachoftheIognormaldistributions.Theseboundspreventedthealgorithmfromtakingunreasonablefirststeps.

1 th d b i t SL pf e anrtf a hept e e d rctwrie i en ct G th ML e d si iw c c st o upt a i ovlmeo r bo fibma th o p Th d (p r li i th ldoa pi t pl t od shcebean iio ttt u ff ( I f o o p an wth oa ope p Trri p crtt )pottpi prire bi ou w a o e b vw if(a n oa e t g w t hae udn broohatt ieihteet m ir Th nb opl thi d mbu i toxiop pfrt etzp ffea tinre ti ML d ani th a sp W t th tchrr sa M iedn icbwtuia t dustt th p i C O l nh oet. ar V R e s ul Vo S m - M e ma a su Th m o th e d s iefar M sat S wt v i sir a wibma e c t xth a f t p (Tl a cru f pr ri c e b vticututtuosa amie nd e o th m so th d it Th p s meit eeaa r d r mai (bPs i bbcee uftf th m f ML an th me fr SL wa 0ea pro T PM ebe t m frc ML an th a f p a alccu o a cri ntt 0tu. p mt w rtrour Perou ac b th m f eSL an th a f t pr a ea rcotut w0. p m E thtu eeertou ar s t l m sm eiaoh tg d ava nb tti t ieeessu flflfmet arsst ic th t m b t i dae fo ew runt f cpr whe sucl uatnhti prau li T ca b s i C F w p th d habee i thi Molou e f et m afset t fu p a th d m g i rth f o p aaiNo th vut c ic riin o femwtu nt w tr 15TheboundsanderrorsplottedinChartsOneandTwodifferslightlyfromthoseusedbytheminimizationroutinefor the following technical reason: Theminimizationroutinebehavessignificantlybetteriftheobjectivefunctionisdifferentiableandifanalyticderivatives aresupplied.Thederivedformulaefortheboundsarenotdifferentiablesincetheyincludethemaxoperator.Inestimation,weusedalogit weightingschemetoconstructadifferentiableapproximationtothemaxoperatorwheretheweightsonthetwoitemsinthemaxmoveto zeroandoneastheitemsmorefartherapart. Thedataplottedinthechartswereconstructedusingtheactualformulaeforthebounds,rather thanthedifferentiableapproximationt,ogetherwiththeestimateddistributionandweights.

1 a s d u bi b M smes ae t cf t pr r(LNtaw o utt e h n pol pa otir t z w th hc o fo w e rna f t eprr moro cehtxiurto a $1 $ ucg tl $ $ o $ t i2s c 3 o 4fo wh nth. w .0atli ham.0 ro t 00fu a co ct A d a ti we n l s bi th o hcme imhe otvptum afr M sso t d s no b e t eh thx f ayp p ou u qu ri ec tu Th s b i th ML anm Se d i ido tnw casl o t th ar o ri m C F od r me peh p d r erfiv p oeta efnnuatrst b t icme ncab nt th to p u e fr ths O a c t o Jusec IO o( iwe nbee ttonmtcar a th b p u e ofr ths J a c tot O sae oIO i( t nem cttonuontmt a cr P t th o o th c ut r i l q t r d u hieiitotb ita e is flfwi re std th c th e f SLsr c a e t a i ancti sro as cp mi a si momrogan $S pe b w o v a tih d e bri t $rr- as $ wpet bta ho e ri ig C Si a t sh so t l h o t d t b i t igermair f h ett ofet tw e u sth A c t a a e soii T tpr pxam pnl 1. aa am wtr te E > 1 ] f , ML an SL . [Th b pa pfl rPo2r5 [ o>t Ol] f t t1. m A ca b s ofr th c th c d e o ha eefrxe nM i gep d a ls etcit o S w th p th r th f hpo LNwi ri bbu 2 p ia g ilri tul ber ienM il t i S Th r fo th r i v hea a LN i eist bo p pasa o sCu Fpa T l ~ e vi SL s f ra tmo eo t op i l mragr t tm rai rcob s attti ba d mi r us l t tnh ebu ro t ruis iea ma minabem t u ugh odnc ti m T r h a al ethe c h F c5o os o t ol6ann c es roulotrt co nt e f x ML i a p th o SLe Fo 4 oc roobot 6 ct o at p n tio ro tr b a 1 f ML i be b.th o SL T ma thro2me5ilno cov I a p on m o a w u th SnL e wh thsaMsi were cl t t tly tr s imw t t o vth m ea oar ths p e osrra mas d oe twkeh iu bssi n sr tambade

1 i o p o s a dm r i sp uiccr uac pt T d i tih r fthae io t df g ia ea apsh rew tr te th p x e g a rb SL ea rMmL T i rn i t rooi ter cdi ni ghwis b m i fo p m o u pr cator a i- o o incpu fArs - - i T tthTta he fo o u ct a - al c o da af S cho - a me ter (a hnllt -rpo e o -mre $ c t $ 0 ofo ML Fo 0i . m npu S .ph0-8 a m et o a$r000 he c t - o fo ML $Fo t m o S0L o pa p u h.e anvretit pr d00 a er i t SnL di no a d e p il mhaar t ctno rl o ata iA bmoc bti ghab e SL t x t o pth vp e f i ee nca rri(cmnd -er po -$ tte are he o u pt ( -e o o- $ f a o -0me vrturto p he. merr t t 0-3le alob mo t o th d i s tai r i b ut Ta 2 Me P E ri rr ( - P $ A r e ct di I th M O o t oMno C Pu Caal P ( (x x (x ( f<f SL - 0 0 0. ..00 -( 8 ML - - 0 0. 0. .0 0. V. S M t C a o t odm i ~ s ar ti A th d l b i th e tf fre Msf ah th fe twti S oau raep i ma b th c t t d ar ino s fhieaasi s ha f segt T ei niait re c so th SL mm c bpn in w lMLaond Sii est miot c a n n at ‘Thenestingissueisasfollows:Wehavetwocompetingnon-linearmodelsthatexplainav , Denotethemby MLN:y=g[nl,Xz,n~,v,,Pz,y,,al, o;,al, w1,w2IZ]and SLN:y=h[~,o 12],‘ / a data matrix containing strikes and interest rates, g[] involves the weighted bounds and h[]involvestheBAWapproximationII’g[]andhl]didnotrepresent differentfunctionalforms,SLNcouldbenestedwithinMLNwiththerestrictions(n,=nz=O,orp,= p, = L1,and al = 02=~~)

1 th a c s a hy c s mbi us whs fp- ana l u t s raor it omp( qui F ktei a Th s n t fo f t ac osua neosl n or n m omgo on a mt adpaaJ a -nl-P (o J an PA t W us th P te w c t s moo es w hi t fots mp two r e g r e s si 6 j z ~ jp + .(~ - + r ~l es ( w A&l i th p e fo oh r“j fr mo 1 r~rptia t erd o emci 1w ri r t th p e o ma 1fo o rs ‘ e aa otd evppt mf t Mstal e1etc A p pal an ~z rar th f av f o m ‘ frit t aelt mo pt t~ i a v eor c wo l ee t th nf oe p f qiitumaioum 1 angac ir a si ic am eone Th P t i s th t fo a-. iI stesn d ttimtp i fi uvaa fff tittit m s no h e o th he o mx 1 Hd morrouel 1pi r odieentl e elajo a s t i fo a-. gO sSLn ban M t ci bvo sea fa eii mto i 1o omu is c2a T T p th t r afo- a,ha e s e (1 ht beq bplos areftac oputa ean is t d fo e cr Th f otw ca aoy nt taac trod iSr t a mo 1( lruiand o f d b DFr w teh la tw ce tr n M hi a moeod 1 ot lu E t i th- f c svi s ta ti o5lpir a lge erwh tn nter iis i-fs th f c i s oa tih o5 p lg C u nte lud leivi t i fi rrct vfi eafff ML an SL h t e th p e x o SL wrh elt c prr i n ict onT la a e v( d anv MLa ca r iSLi b M tcha dlatejb r aben ej 17SeeMacKinnon(1992)foradiscussionofthesetests.

1 Ta 3 P Te t - s t at SL a M 1 M oda Mo 1 C ot D n P - tt- -v Dst ra P- O 1c 59 t 30. -0 ob 5 3. 0 N o 1 13 v 70. e 0. 1 2m. b 0 D e 1 19 c 30. e 0. 1 0m. b 0 J 1a 25 n 20. -0 ua 2 2. 0 F e 1 25 b 60. r -1 2 4.ua 0 M 2 2a2 10. -1 rc 2 9. 0 A 2 3p0 60. -0 ri 3 0. 0 M ( an th th P tce c1a ha po Kfot sa p99i e rin nnw spop M 2 ha a l n o p o Foa euar i rt l x desa mb wim maamicto et o p a M 2 er m a thveo mx odo a ep b t artc tc pietrs(ieeg ri c w a oi t flonc-t G th lsM itf h 1 pt a iianv ata tps av ti ra t da ha r r3 o p o t am b rp r tu q d t here icetaiougeht P tin T e t t i no th c n w ra th f Ghaas hNioas R l (G eurdeew loi gr M ( aw ~z ar thcd 1 o meKohe 2 wi rr 99i t t pi nenso vmaar 2 ( fo o ‘~ e a a p vp c i a n o t t hdoaodi lus t ee uma ipo 1 Tst en d w eno b c r wi ~o ii s s r v ial i Moraa 1 i coma tet mwp n iv la

2 h e th e f xM 1 T r 4elp p t F od frroyro=laaOb -t es Ta 4 GN F -T SL a M 1 M aodMo 1 ; Fo D n P - F- t -v D te Pr-a ( n,(n ) 9 c 11 10 0. t 4.0. 2 1ob 0 l 1o 11 20 ~ 0.9 0. e 2 2 7. m0b ) 2 e 11 28 c 0.7 0. e 2 2 5. m0b a 2 n11 36 0.6 0u.a 2 3 4. 0 ‘ 2 e 11 36 b 0.9 0. r 2 3 7. u0a d 4 11 33a 0.2 2. 2 3rc 8. 0 ~ 4 11 45p 0.6 0. 2 4ri 2. 0 Th c i tho s a th on th P S c i r w-hl amM ei n ui te jesi V. S E e v l e en ct T th hP Gu c r th eo m o ofr e urs laxar pisg mo pe aho “ hi th m an pn ra a th er e e cxt rk liip o owusvi ce t tnc ipct c C th er o PD frs mthi t m t pi s bmei a aod a mtemiarsri o “ a u t i hon th ml i t nenfne alaor f t h west diuigr rp b th ML an SeL m t o w de ee ‘sV’henSLNistreated as Model1inequation(14),thederivativesofMLNwereevaluatedato, = C2= a, =.2,WI = W2= .5, .8. Giventhatthefuturespricef ag tr d w p2 = In(f) - G: ~z . p, = .95 pl , pl = 1.05 . V2 , Xl = ~3 = j, ~d X2 = notusedintheestimationoftheparametersforeitherMLNorSLN.thesederivatives,givenSLNiscorrect,willnotbecorrelatedwiththe errorsfromSLN When MLNistreatedasModel1inequation(14),thederivativesofSLNweree~aluatedattheestimatedvaluesofpand a fromSLN “l’hisisaconservativeapproachthatincreasesthelikelihoodofasignificantF-statistic.

2 O T O 25h 19 th L c Fu Ti rti orn t Ir nfoaep osdbh a e tt 30x o K 1tp0 oi w u ql a a se wKou eluocehn siwai h l K on w e Thu r a p e o p wreue vs uws t elifurihaaico la n t e ( e x r D $ p pea b e Cih is Se c prel rar3t. P e f at M amb SL fo O 2 ( p acn O 2 ( a tp c us tot boJa obcnoetoan O O 22 m e fxo f a p p we c eu qu rktcri ear tut$ p biganat T n ti o th m w ea m idi si o i dwsnii M g atde nf enarsi ill ib fi p m r b $6o an $7 pee bb a as t a b i werr li Th l o c ai oi pn i NY h rh eo ri o Thgeaisn cc-Jda 1 1 w 1 s g aoe a99c v nhe rvo e o no f orth eleermou rdnm i ann 2 i b vc th th ec e f n ho tole a ts caaor Oo utp d lti ri s p fo th M c e fe $ r wo t sart pr fn t A9.pic choiettl trf $ Th si p o C E t tha e o eha v igPDnraeo t x do s Jplaueur 17 P t th f ai s (a ca br se i t irtfi t p t mioa riwan s ex a f s c i o aam goi m hd n i( ii ajIrar daan fpse o Srl Airc ru o f t ac p p c t th $ i p oub r hlar 4 OusJi 1 aritct 0an-P an ie t d i r an o e ag da thm P hgn a fr Mtte mosu i cl ennae cc t al t f S B J 23 t wa la hda ro LNb i nthe itt PDf eat t mua fe r t a aep d r lt (i epa s 6uwic t m-tto pa ornrComc 2 ib ri V C o n c l u si T p d a m foe u oap phi ev t in t m psti el pri th arro d foi c s p o Tht m i qmrru g ia emt ticbenst ll loodu ti d ti b r sb an fr etw a wir cl po di it i bTla p s u as tr ar ti o a m o t l i d o ius hherx wsg d bt c ntu r i t onori imor bu

I 2 d th P G c u A eth f i orn o tr r p ul di orc sioiint o co sist ba p m s i pr i t o th s r pn lit g u mo roimo iaccetct cohcvo ve p o t T l o s r i a i t svie wp g hri acich lpi emeaupriori in eane ct a th s p t th mb po r haa ca a roOouhm st sheasrkcie b usu thci t r whe w t i s a m e o s m aai a ist1 e r otnpiorucmxaa d he u t on2 i n sa i p vd e eith amensstssa ri tt de dtle irb t gaib I d o (i l gs e d t pin Fe. sre t oit tio fo rxa ebxkrm i m uurtbu i ca b u t s l o th “ p ” O seiig tr hmeo Pema i ocb b u t i th m a oa ps c sri mo peh .k so s anetolsniesm I th a t pth oi m pw fi th tl o ar pri w c pct w otnat m c a oth t a i th mth r amsr iem p ie of ra mgken di obleniita oi p W f t thr e p osoi ic iontoahmaa crdei ni w ois dii tma $ pe b r 5 w i al ca 0a wi om hic r-rn ngoW a arfs tm$m6t is s I at o di sa p a gjo o sc n nh outo da at oddamor m aa rmptt ct tw m h d i o i m I p pf c aavdl t o f mroio t I cterel mmpat iocg i a o mvo th m e a p r oa ts p e olsrra msa d oet akeieais ibmass u n o th i d o p e o su ardm er i s spa t ic ru i ma F e ioxp daa can tm laro sah ii mat ne n ll icxtp fait o w s c c i r nce g r i t hneosum ei i srr afr -ac icre ‘q Deutsche Bundesbank (1995) provides examples usingrelatedtechniques.

2 R e f er A Yi. an A WlLo “ - N E S o o Ssan Dpth t Ima aatein imra li F A P i NB W nr Pa ssS a53oi ( enr cerk 19 ci B a G. an Rr W “o An hE A e npf o- Aal p AfOpal rVamdee ic ox J o F 4 o( 3i 1u n0 9rna81n- B D S “ C o ’8a Wa I E T raEtxTh Fr Op pv Ma es ”Joec ido F 4 ( 1 i 1 n0 9Ju 0 a 9 90 nc -1 B F “ P o C l Cr o J o Taho Fmi n E ou imt 3c(ck1ico 16rnaaod B F. an M S “Tl P co O a Cra h Lp o Jiooicl cko Patoirp bi E 8 ( c6 1 o3 n97 om-676 B D. an R Lr i “ e ot S tz Pe Cla e I t indOereip Pmbr-p ” Coer J o B 5 o( u 6 su1 2 i r9n71- ne C J C. an S A R “T V o Oox a f oAs lSp l P tuta ”Joti ro eorc o F E i 3 ( c 1 n o 1 4a n n975o- cmii D R. an I K e “ P r an aE r ii tRifMa veef Cnrsf O Fu mii”cT E J 1 ( 9on 1 u 3 er 99r -1 na D B e “u I n uCn o Dd f t fo eeMToh Possrc r Dnoetn biva ma B M u R 4 n( o 1e d N1 n e opo 99 s7- thve ba D K “o V amn Eo oiCrf l O F Tnh fCa ”qI Oiuott a M ue cinilt C W P Ri T o F M hri i H itpi I a n ronEnar Strtkan 5 er ( 4 1 8 9 -8 89 F P L. an RaP K “ c oC O a piPkn l At r lei ii sAob bron sger C M o A a J m omA r m oEg e k 7ro(cF 19ur ierti7 odni cu G 1Sr. an I Ra T od I y S s n aab P.Mh zhtAer rto P eg(1 caey G B D “ Pr an th UO u Ar n R p D dnd ss ”i Joic eteitosFi 4trrl ( 1 1 0 9J 4 5 91 ul - 10 J R. an A R a “ AO rV p ud f A pa ptrS r r Plu too ”oJwo brio xi o F E i 1 (c 3 n o 1 a4 n n9o87- cmii K M “ F uA o o C r Oc F mTehP ” Ic cStutruPa riar3a u(r st 1 9 92

2 Lo A W “ S Ue B m fo Oi P - a opupEp pPat ri Jrxop aoay me F E i 1 (c 3n o 1 a7 n n 9o83- cmii L H “ an L B e oU Pu an Ca O oow Va ppS vy D pttun Ap omoc J o F 4 o( i1 u11 n 9 r9na8n 7- M J Ga “ S c pT aMK Ae Rci r eiJooedsn togfEc noreicif L 3 i( 1 t 1 eM 09 r a2a-r 92 tu M A M “ thR P a De r oicFo E s olzb Ra tf vxucotap riPr er bi F R Ba oe Ne Ye M ( d s or19er im er M W R an C P eT “W an P l Rh t Me icea P omc aDr ro ovois C Oi F P D thur Gu C r ” Bo ur ot G ri udo tic Foevur R S I nF yDt Pe ii N 4r st( s apn1n9a cOac ti us O J A. anv H L M e “T Usa o NYr Ot t dF tC pt Oah Porr he Th E J 9 ( 1on 1 3u er 985r-n P S. an Pe J R “ r P B Ori Dya r Ti a Jooupits o Fici ki 3 ( 5 1 9 - 52 R R J “ Oi V fot D a pCNa Mc il Jo itioohrueFasic xRte 1 ( 1 2 W 89 i 5 90 nt -2 R Mu“ B b TI Ji i o Fmn r 4 no( s i7p ee tom u1r9 n7a1li ei S D C “ hI V B P imo D r el iamkpolHea sRa yboImtit b abri O P M D rp o Fe i ia B pti Ein ce amuUse co o rt ni S C o( a 1 Nl u i o t99 f ve ohe rn S E M S. an K H Ki “ oRS la e Caoh O oly pP via ti roab . A J so F s M o e1 ( u s7a ur s 19 tu6mr5k en

2 A I D o eb r ouiv T O I th ah ~ , u a Aen s one di ammo sete a iptdeir rertl rt a u b fo th ca o v p i ot p o t t al dt pe erfuienrt unio st 4 a i g a f Cs~ =m oE i–X e+ E sl f -x Jwfh lovef ~imett a p a e o thxo X i thp o ps ip pr i t (r t tr r t rfica r oioin otinio i i th m t i b i t n r eanp a titma dE d twiemeer er s tx en /t peex p p t e eI t i t x pr rc oaraaptki t ito n t laioup btoir in C, =E f - w ~i th u nEm hviad X] S u ia~a u imsb ro f co th A pu o mi a f P~ =emp oX - , e*5r E l tiEXt -f ic. ~lom 4 P D t Le rJi = {T z, ...z }bsaicm o s o pcoo i ~t. inof unre[ T r eth r pli o eth or wi eTm O a p( s>j -a ( 2e~ E ti einln o J ti r a p e a w th po ca b t roi r hi p ae De d&- ~s ~e a ntid ~en~ O/ ex ‘ >1 v’ Th p i b i Le nC, - mro E d - e ‘a E u f,[-x ctS ~mO 4 s C =Cl , i C S Chl mSt Tw sh th p ( s C,ow~ ( l~y C, T b ~ i C < C~n Vt > 1 d u c ti W i t b ‘ b ep ndo t o e eTifm opxp ex rpt i pe O tio p p t e i p e 1 et Le rxy - f - Xe rB a pi~si iaomri io sut ar y,=Et[y,+]-- Et[‘o] =E -X D m( - m, y eSfi m i co[fax b J i w hn E >em , q( [ avuE m a, (y l[iy S O Th v o a A ca o tema aenlxe i erpt = ma O f - =m( . ( It v on p p t e i a e rix ri lu pi

- 1 2 4 =m fl -X, e ‘b’oE,[ ~] =ma 4 y, , e~~’”E = , ( [m ma E,[fO]-X, e ‘&I“El[max[O, fO-x]]] =c 1“ + w C S O o h m p te niec l et er Tw ; W w t sh th (C s C, ~ ( C s C~ anT va o a Am o ca b w r p a ef r c ot i u l r itot s lo iv c = ( t+ m i y~ e7 E V > a, C =m( 61 + ,[ B a Cs < C th s u m ,” p t io < x ( Ct m y e‘a “% t[+c S fuo C b s t ti” t u ti xl [[ ( C,+,< ma ytd, e’51.1Et+m,axyt, e *5.1 W , . Y, S m z y r y wi emi ( ) ca On rap(yt RH th nlc ac 4 [[ ( C,+ls ma y,+, e 761*1. Et+l rnax m(yt), e‘6’.E, . [ m( S y m wa ca r ri y wi E t e i t ncn pl g [y al ( C,q s m yt~,,e‘ib(’.E,qma m(E,[yO]),e ‘6’.E, . [ m( 4 [ S E > m , r i ( m[ e wi EE (m pc onnct, raE ( t i R [t[ym,[yO ac ( S e‘a < 1 e‘5 oE s Ei , a e7 oE [ c bncr, m f t [min(yem m w c it v i h t aa nh lu gi ou

( B i e txw ha p e e cr t at at C, < ma [ y, e*6 “E = . t + (,[ 4 m E -X e~a “E ~ O f -x ~=+c A[l[ T c S Two an th th pm hi p te ro l et C T o No th nr t a i di nu E - s E O f -X,[ . ,[ ( F t th ca i w th p ir t d afki fhhi ers s m b trX s t t i i ( n i s eThereAis a it ~.qtt. Vbt < St ,1eu‘3’.E~ ri , fa- .1> E -li Ja th u b ca b w a p ou r it pe Ct =e‘6’”E O f -X f a, 8 < S [m ( T th l a th i a b t n im r egkoi at zte yid twe er s /e li ( & C =E O f -x t,[# w i th u nE hv d o t u o i s r icaplt c opou I ( h wi e A th q ol 1 u .1al E - = E O f -X, > e’, -E O f - [f [m t[ w a l th u b he egta t u pp nE ou avicqvuad ur is Th p fo pu i i t t abdro wi y - X e- r y -ntf e-p N A th p il d oit o tt bai roi b bheo tntreresc M an T ( p a ah 1 lerl g ptr owmh oevd9e9 mi r t niienr na g i S H o th t e i c ex ve ti

2 Appendix II E D I. M o de E (5 an q(6 i th te giu th p e a f pur qa t ca i t uaoioict b d i A oI eTh arp r he rfu pt er c ive peeanod nndv C = C ( W +( - C + , Wi Ec’[ ( P =wi P U +( - P t[ + [X tWi ep ’[ S fuo th b bu e s ( o aqt n tshii l u utn ot atu)- ti E O z = E z Izz O][ Pr >O y p em w ier qi te O tax teriua ic d i s t r i b u t io C =I m E , e‘5 ( , & P , ] + [[f r[ f O\ “ i ( ( - s ma E , e” ( , @if . i [] +~~C Fr f fO O lf 4 1 p =wi m X- e ~s ( - l E f O+ E, ,[< X ]) ( ( - m X e‘ ( w-- E , E[ +~fPi, O ,[ l f OS 4 1 I i a t th d s foi f p s s(f i a tmhua o Irruim iitxui o bu gn g = + n [f l g,( ‘~3g3[fol w h er

2 U th p o th rI s d o o it te p isg tin ete nvno r xprt ib o i e p( aqn ( mea busA frAu2aaf b2. att ol st io 3 CT; ( E =~ Z -e p +— [f 2 i=l [1 I 11 ln - (4 P =~ ‘ “ 1ra ‘ [ fO i=l [ I ( L 1 rII 1 r 0 +2 1 ex — ( 3 2 2 E l = n “ z 1 P r [f i=l E[ l =E[ - E[ l fO fO (A . . w @r th ce nh up d fmir or Tsu epuerr e t n lsaea nri qcut w i t o th pr o tha m e i A fi r d i paotd i brmt mnreeff te z(~he~em ~f~e Iognomaldistributionis~xp[u+&/2}.Calculationof(A2.9)~d (A2.10)usedi 3. fi G nt ~d Ryzhikra (1980).

. 3 d i ofth maf o e i e r p( e aq (n eCt u A2i noA2ara at nbs on il m ii g e n wh iar n d m an eephi Trz alr anwat raeihve tov ti ma o w a l p a o p ae f p g it rr o i o atlxl si im 1 l o - g i t m( ax 1 ( ‘y + ex (A m s I a o .x + - g x I i oy o t [x g ma it SL M I od Th SL M us th a p o Bod p aa rWh r(1o ( o xt neim e th o p xi t o th pp p o raias erI ti rr ( a oc inIamo neos gn th BA a p fo th pp o a Ar cao o mi rwxi a i erpt mri at E t[ (A C =c [ +A E ,~[ ,2 wh[ E[ f O ,[f* ], C =E - X wh E ,[ ~[ ( l w h er c [ s E E-e- .q 1 ) [t[-[ f d Ol [E], X e‘ @ [ d , [ E ,[ A * s & - ‘ d *

3 J 8 1 1 cr ( - e xp % = [~ 2 E +C ,[ In — x d l - [[ E , [ f O] ‘ @ d 2 =d [ l E- % [, [E f ,[ O] II 0 E =e p + ,[ Th t f i th “ c c p oa iersr f mri i a mmool t ic ti cc pl ( - ‘ .@ * .f* ,a [d f - x =C + [ f ’ ( ,a q* Th f fo th p ooa pu i s t th rf tri cai s B fmude miN al C wa u t s fo0 ~ T i t e 5ol smeo p sa ~ , d t ApJrw er pr im t th NA m ia E n l Gi i t o 0g mp fo i o( 4Uptriza(A erqi an ( 1 i no d i th l f o fA2 4d)o e abp r we ugie a foernsc ti C =l o g.(R io ,[( t + m A2a ~ f* ( ( 4“ 1) 1- “ f , .l(R o ( E, o A2 gl

3 Ch 1 W o B o O P U Bo -L o pp we C * Paunt ofOptionprice - - AbsoluteDiffcmce + 0.8 ).07 0.7 I1 I MM 0.6 I ‘, I \ I \ 0.s 1 \ \ 2.05 \ \ \ \ MM \ \ \ \ 0.3 / \ 0.03 \ \ \ 0.02 0.2 0.01 0.1 0.00 0.( 15 20 25 30 35 S tr P + PmcntofOptionPrice - - AbsoluteDifference * 0.8 ~€ÿ0ÿ.0ÿ7à 0.7 .0.06 0.6 ,0.05 0.5 / / / / ~c0.04 0.4 / / / / .0.03 0.3 / / / / / .0.02 0.2 / / 0 / / / .0.01 0.1 0 / / 0 -“ &#45;&#45;&#45; 0.00 0.0 15 20 25 30 35 40 S tr

3 - I I I ..-.-..-- .-...---------------------, ,=------ -.--.--..--- ------------------- 0 . 0 0 0 1 0 - v 0 i N0 8 0 0

C ha R e R p f arSi Dee s esu nt I D F m enu pl nc — Mixamofbgnomafa -- SinglLeognonnal 0.09, 0.08 0.07 O.c% 0.05 0.04 \ 0.03 \ f \ 0.02 / / 0.01 / / \ . 0.00 0 10 23 30 40 50 60 70 80 9 F P ( d u o l ri tu l a nj Ca P E ri rr n htixtureofLognormals -- UpperBound A SingleLognomIal .... Lowr Ihml 0.15 0.10 A A A 005 - A- - . 7 “m .A 8 t . n ‘.- -m=-~..~ . .6 : - *- ~- ~-= * ~,~ 0.00 .. ..-. ---------.....n ------------------- -- .a~- -- --’ --- . ~- -0.05 A -0.10 A -0,15 A -0.20 AA AA -025 20 25 30 35 S tr Pu P E r rr ic n MixtureofLognormals -- UpperBound A SingleLognonnal ---- ~>u,~~,lunfj 0.15, AA A A 0.10 A A A 0.05 A A A 0.00 * –h- .+-–: ...,. - ~ - E- g l * . :. :....-....-... - .......... n . -0.05 A -0.10 -0.15 A -0.20 I 15 20 25 30 35 S tr

I 3 “ & * * + + 5! I

3 c ha I D F en u nc J 1 O C ct on — M b Srngkbgncmmi 0.18 h\ 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 30 40 50 60 70 F Pr ( d u o l tu l ar O 1 J C an on — M S L 0.20 0.18 0.16 0.14 0.12 0.10 0.08 0.06 ~ / 0.04 0.02 0.00 o ‘ 3 4 s 6 7 F Pr ( d u o l tu l ar

R ‘1i C hg ha h Ir aF ~pl c)c utne-H Ap C o nt o E t Fxu by Pce — ~ ~ - F R ” - – S 70 65 60 55 so \ 1’; \ 45 % / \. u 40 \~ ~ I \ . . ..-, 3 3 5 0 . . - . ,“ , ; $-.- . ,’. . ,. . .- ...’ , , , ; “ * . ., , . . ., . “. , . , .... .. , , ‘, ,, .... * b .. .’. - # ‘, ,. , .. , * ‘u , ,. 2s , , , , , ,-- .,*.-...,. -,\ . l 1 1 I 1. A N D Jan Feb 1 1991 Tr Pr t P W E t F P — M ofLognormals - - SiogleLognonnal 0.35 , 0.30 k ,1 I 1= ‘ \l I 0.20 I I II 0.15 0.10 0.05 \ ‘k Au 1 I I N 1 D I Jan 1. Feb 1990 1991

c ha M o W I D F O 2 J C — M - S I I 0.07, 0.06 t 0.05 0.04 \ 0.03 0.02 I I I I / / / ~ 4 0 30 40 50 60 70 F Pr ( d u o l tu l ar O 2 J C — M ofbgnonnak -- SingleLognonmal 0.07 0.04 0.05 / \ O.w / \ / / I 0.03 0.02 0.01 0.00~ 20 30 40 50 60 70 ID F Pr ( d u o l tu l ar

3 Ch 8 S o A W I D F M o L ( i )So L ( g- xt Aop Cino no gn J 1 a nu Ja 1 0.10 0. [ 0. , 0. , 0.07 0. , / \ I ‘ 0.06 O. I ‘ I 0.05 0. , I I 0.04 0. , I I 0.03 0. , 1 1 0.02 0. I 0.01 0. 0.00 0. 0 2 4 d 80 Futuresprice(doUam/barrel) - (dollarqbaml) J 1 a nu Ja 2 0 ~ “1 0 ~ 0.09 , 0.08 0.08 0.07 0.07 . \ 0.06 0.06 \ \ 0.05 0.05 , I \ / 0.04 O.M I J 0.03 0.03 , I I 0.02 0.02 1 I / I 0 m 4 6 80 FuturesPrice(dollarskmrel) FuturesRiu (dollm+mei) J 1 a nu 0.09 0.08 0.08 0.07 r\ 0.07 , I 0.06 0.06 . 0.05 I 0.05 ~d I 0.04 I 0.04 . I 1 I 0.03 / 0.03 . I I I 0.02 I I 0.02 L I 0.01 I 0.01 I I 0.00 m 40 60 80 0.00 0 -- 20 40 60 80 FuturEsPrim (d F (