Modeling Contract Form: An Examination of Cash Settled Futures. Dwight R. Sanders. and. Mark R. Manfredo *

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1 Modlng Conra orm: An xamnaon of Cash ld uurs Dwgh R andrs and Mark R Manfrdo * Papr prsnd a h NCR- Confrn on Appld Commody Pr Analyss orasng and Mark Rsk Managmn Lous Mssour Aprl Copyrgh 00 by Dwgh R andrs and Mark R Manfrdo All Rghs Rsrvd Radrs may mak vrbam ops of hs doumn for non-ommral purposs by any mans provdd ha hs opyrgh no appars on all suh ops * Dwgh R andrs s an Asssan Profssor of Agrbusnss onoms a ouhrn Illnos Unvrsy Carbondal Illnos Mark R Manfrdo s an Asssan Profssor a Arzona a Unvrsy

2 Modlng Conra orm: An xamnaon of Cash ld uurs Absra hs rsarh prsns an nuv nrpraon and xprsson for prng ash sld fuurs onras In parular h ho of h avragng prod for h undrlyng ash ndx s valuad or xampl h avragng prod for h Lan Hog fuurs onra s wo days whras s hry days for h d funds onra Dos h ho of h avragng prod mak a dffrn? Undr ran assumpons h bhavor of h fuurs pr pror o nrng h xpraon or avragng nrval s ndpndn of h lngh of h nrval for sorabl ommods bu s no for non-sorabl ommods ywords: ash slmn onra form nw fuurs onras Inroduon h Chago Mranl xhang s lan hog fuurs onra ash sls o a wo-day avrag of h undrlyng ash mark; whras h Chago Board of rad s d funds fuurs onra sls o h monhly avrag of h undrlyng ash mark Clarly forms of slmn an vary wdly aross ash sld fuurs onras h ho of h xpraon nrval for ash sld fuurs onras g wo days for lan hogs vrsus 0 alndar days for d funds mpas how h fuurs pr bhavs n rms of volaly prng and bass onvrgn Whn an xhang dvlops a ash sld fuurs onra hy ar fad wh hs problm: wha should b h form of fnal slmn? h obv of hs rsarh s o provd nsgh no pr bhavor basd on h lngh of h xpraon nrval pfally h rsarh dvlops a smpl analyal xprsson for h prng of ash sld fuurs onras hrby dvlops an nuv framwork n whh o xamn alrnav forms of ash slmn Opmal fuurs onra dsgn nvolvs wo dsn sps ashan rs an undrlyng ass mus b dnfd orn or rud ol ond h slmn mhansm of h fuurs onra mus b spfd physal dlvry or ash slmn Rsarhrs hav xpndd onsdrabl ffor o drmn hararss n h undrlyng ass mark ha ar mporan for sussful fuurs onras Blak Numrous as suds hav dnfd nsans whr hs ondons hav no bn m and onras fal Gara hompson and Wldman Howvr far lss rsarh ffor has xplord h form of h fuurs onra n drmnng s suss or falur and ha ln of rsarh has prmarly fousd on mbddd opons n dlvry-sld fuurs Marnz-Garmnda and Andrson Ohr rsarhrs hav fousd on alrnav ways o onsru undrlyng ash nds hs rsarh assums ha a ash sld onra s opmal and hn xamns alrnav sasal mhods o onsru h undrlyng ash ndx g Ca and Ln hs ln of rsarh s mporan baus addrsss a op ha aadm rsarhrs ofn ak as a gvn; y s a ral problm for praonrs and ndusry rsarhrs prsonal nrvws Cash slmn rls on good undrlyng ash prs whh ar aggrgad aross m o ra a slmn ndx Jons; Garbad and lbr h onra dsgnr mus sl an xpraon nrval: h amoun of m ovr whh o alula h fnal slmn ndx hs rsarh xamns how h ho of h xpraon nrval mpas h prng of ash sld

3 fuurs onras for sorabl and non-sorabl ommods W dvlop an analyal xprsson for h fuurs pr as a funon of h xpraon nrval h rsul s no basd on an agn s opmzaon problm Rahr assums ha ash slmn s opmal and hn drvs a prng xprsson xampls and dsusson buld h nuon bhnd h analys h rsuls of hs rsarh ar mporan for wo rasons rs s a rsarh ara ha has no prvously rvd a gra dal of anon ashan ond provds a gud for xamnng ash sld fuurs prs h Commody uurs radng Commsson rqurs ha h alulaon produr for ash sld fuurs onras provds safguards agans manpulaon m dvrsfaon s on suh safguard hus h ho of xpraon nrvals s mporan from a prforman and rgulaory sandpon Mhodology Cash lmn ovr an xpraon Inrval In hs son h problm and rmnology s prsnd rs a slmn m s dfnd as h da a whh h fuurs onra ass o rad and h pr s s qual o h avrag spo pr ovr h prodng -day xpraon nrval hs slmn produr s larfd n h followng mln whr h m subsrps rprsn days no h -lngh xpraon nrval o h spo pr h day pror o nrng h xpraon nrval s 0 h frs day no h xpraon nrval s and s h fnal day n h xpraon nrval or h slmn day m -day xpraon Inrval Nx dfn a fuurs onra ha ash sls a m o h avrag spo pr ovr h prodng -day xpraon nrval for quaon h fuurs pr a any m for slmn da s dnod as On h slmn day h fuurs pr by dfnon quals h avrag spo pr ovr h pror radng days or xampl say ha quals h 0 h and las busnss day of Dmbr 0 hn h fuurs onra sls o h avrag spo pr ovr h prodng 0 busnss days ha s h avrag spo pr for h monh: for 0

4 h Cas of orabl Commods How wll h fuurs onra rad pror o and durng h xpraon nrval? Analyally hs s answrd by akng xpaons of quaon abov In an ffn and raonal mark h fuurs pr a m for slmn m s qual o h xpd slmn pr o n h as of h ash sld fuurs onra for quaon a xpandng quaon w g h followng rlaonshps: [ ] quaon b [ ] quaon Muh of h nuon n undrsandng h radng hararss oms from quaon hs quaon larly shows ha a fuurs onra ha xprs o a -day avrag s quvaln o a srp or qually wghd porfolo of daly onras ah xprng on hr rspv day durng h xpraon nrval o h undrlyng spo ndx Basd on smpl arbrag argumns h followng rlaonshps hold for sorabl ommods Hull p 5: for known pas and prsn valus quaon a for > unknown fuur valus quaon b In quaon a h xpaon for pas and prsn spo prs s smply hr ralzd valus Whl n quaon b h xpaon for unknown fuur valus mus qual h urrn spo pr adusd for arryng oss; ohrws arbrag opporuns would xs or h purposs of hs papr rprsns h oal daly arry oss sorag nsuran nrs and onvnn yld as a proporon of h pr and s assumd o b onsan h m n sorag s hn - days quaons and an b usd o analyz h radng hararss of a fuurs onra ha sls o a -day avrag as dfnd n quaon I s nrsng o xamn h fuurs pr a a fw ky pons n m s h abov mln Pror o nrng h xpraon nrval < h fuurs prng xprsson an frs b wrn drly from quaons b and :

5 5 hn subsung quaon b for h xpd spo prs w g quaon quaon o pror o nrng h xpraon nrval < h fuurs pr wll qual h urrn spo pr ms h avrag proporonal os of arry o and hrough h xpraon nrval or xampl assum s days pror o nrng a 0 day xpraon nrval - and 0 Usng quaon w g h followng rlaonshp If h urrn spo pr s and 005% hn h fuurs pr s qual o 0 h urrn spo pr ms h avrag proporonal os o arry o and hrough h xpraon nrval h sond m prod of nrs s h fuurs pr bhavor durng h xpraon nrval Agan ulzng quaons and h fuurs pr an b prsnd as follows: quaon 5 Inuvly as w mov hrough h xpraon nrval - h aual prs o b usd n alulang h fnal m slmn pr ar bomng known o a any arbrary m h fuurs pr wll qual a wghd avrag of h known spo prs frs rm n

6 6 quaon 5 plus h xpd spo prs for h rmandr of h xpraon nrval sond rm n quaon 5 As m passs an nrasng numbr of spo prs usd n alulang h ash slmn avrag bom known hus h fuurs pr onvrgs oward h -day avrag slmn pr and volaly rlav o h undrlyng spo pr dlns Imporanly h orrlaon bwn h fuurs and h undrlyng spo ash pr wll also dln durng h xpraon nrval o llusra hs pross s usful o work hrough a oupl of ky days and xampls Consdr h frs day of h xpraon nrval n quaon 5 o on h frs day of h slmn prod h fuurs pr wll qual h spo pr on ha day ms h avrag proporonal os of arry hrough h rmanng - days n h xpraon nrval All ls qual hs s h day whn fuurs pr wll hav h narrows bass o h undrlyng ash or spo pr Pror o hs day h fuurs pr wll rfl addonal arryng oss Afr hs da h fuurs pr norporas known spo prs usd n alulang h ulma slmn pr; hus wll no fully rspond o subsqun hangs n spo mark prs or xampl onsdr day 5 n a 0-day xpraon nrval A h nd of day 5 h fuurs pr wll qual h avrag of h frs fv days of h xpraon nrval plus h xpd spo pr for h rmanng 5 days h valus for h frs fv days ar known and fxd hrfor h fuurs pr s lss rsponsv o hangs n h spo pr han was on say h frs day of h xpraon nrval

7 Now onsdr day 5 n a 0 day xpraon nrval h frs 5 days usd n alulang h ash slmn pr ar known and fxd hrfor h hang n h spo pr ovr h rmanng fv days has a rlavly small mpa on h slmn pr nally on h aual xpraon day 0 h fuurs pr quals h ash slmn ndx as dfnd n quaon h fuurs ulmaly ash sl o h avrag undrlyng ash or spo pr rpord ovr h -day xpraon nrval h Cas of Non-orabl Commods In hs son w xamn fuurs onras on non-sorabl ommods ha ash sl o h avrag of a spo ndx ovr a -day xpraon nrval h sam noaon and namng onvnons ar ulzd as n h pror son Clarly h bggs dffrn bwn sorabl and non-sorabl ommods s ha h onom arbrag ondons n quaon b do no apply o non-sorabl ommods o h fuurs pr wll purly rfl h spo pr xpd o prval ovr h xpraon nrval hs may or may no b losly algnd wh h urrn spo pr Agan dfn a fuurs onra ha sls o h avrag spo pr ovr h prodng - day xpraon nrval quaon : 5 for quaon hn h fuurs pr s agan dfnd by quaon 7

8 8 quaon quaon b quaon a for No howvr ha w anno mpos h arbrag ondon n quaon b hs an lad o vry dffrn onlusons onrnng h lngh of h xpraon nrval and h bhavor of h fuurs pr g varably of Unlk for sorabl ommods quaon anno b smplfd o a funon of h urrn spo pr as s n quaon for sorabl ommods hrfor h fuurs pr wll n fa rfl h avrag xpd pr durng h xpraon nrval As prvously sad a fuurs onra ha xprs ovr a -day nrval s quvaln o an qually wghd porfolo of daly-onras In h as of sorabl ommods arbrag opporuns guaran ha hr s rally only on varabl ass pr n h porfolo h spo ash mark and s arrd hrough m In onras baus sorag arbrag s no possbl h fuurs onra for non-sorabl ommods rprsns dsnly prd asss As you nras h porfolo dvrsfs and volaly of h porfolo h fuurs onra dlns hs s an mporan dsnon n h dsgn of ash-sld onras nally h bhavor of h fuurs pr durng h xpraon nrval wll b smlar o ha xprssd n quaon 5 whou h arbrag rsrons o for h followng quaon appls: quaon 6 As wh sorabl ommods durng h xpraon nrval h fuurs onra wll ssnally b a wghd avrag of known prs for and hos prs xpd o prval durng h rmandr of h xpraon nrval for Agan as m lapss hrough h xpraon nrval h onra wll onvrg o h -day avrag and volaly wll dln

9 ummary and Conlusons hs papr xamns h radng hararss of a fuurs onra ha ash sls o an undrlyng spo or ash ndx In parular w dfn a slmn m on whh h fuurs onra sls o a smpl avrag of h undrlyng spo ndx ovr h prodng -day xpraon nrval In hs framwork a smpl prng quaon s dfnd for boh sorabl and non-sorabl ommods h prng quaons ar nuv n h sns ha h dfnd fuurs onra s smply a srp or qually wghd porfolo of daly onras ah xprng on hr rspv day durng h xpraon nrval o h undrlyng spo pr h prng quaons suggs h followng rsuls for sorabl ommods assumng a onsan os of arry rs on any day pror o nrng h xpraon monh h fuurs pr wll qual h spo pr on ha day plus h os-of-arry o and hrough h xpraon monh ha s for sorabl ommods arbrag opporuns guaran ha hr s rally only on varabl pr n h porfolo h spo ash mark and s arrd hrough m hrfor h asss hld n h srp ar hghly orrlad wh ll dvrsfaon I sands o rason ha fuurs pr wll xhb day-o-day volaly roughly quvaln o h daly volaly of h undrlyng spo ndx ond h narrows bass bwn h fuurs and h undrlyng spo ndx wll our on h frs day of h xpraon nrval Pror o ha da h fuurs wll rfl mor arryng oss Afr ha day h fuurs wll rspond sluggshly o hangs n h undrlyng spo ndx as known prs ar bng norporad no h xpd avrag slmn pr hrd durng h xpraon nrval h fuurs pr wll nrasngly bhav lk an avrag of h undrlyng spo ndx; whr h numbr of days usd n alulang h avrag nrass as w mov hrough h xpraon monh hrfor h daly snsvy of h fuurs pr o daly hangs n h spo ndx wll dln hrough h xpraon monh hus pr volaly wll gnrally dln hrough h xpraon nrval In onras o sorabl ommods non-sorabl asss anno b arrd or arbragd hrough m hrfor a fuurs onra ha ash sls ovr a -day xpraon nrval s rally a srp of daly onras whr ah daly onra ould b a rahr unquly prd ass dpndng on h prshably of h undrlyng ommody or nsan h pr of lv al oday s no nssarly d o h xpd pr hry days hn hrfor hr prs may bhav qu dffrnly hs s spally ru n marks ha dsplay srong sasonaly g hogs o a hry day xpraon nrval 0 may nlud non-orrlad prs; whras a wo day xpraon nrval would probably rfl muh mor hghly orrlad prs hus h shorr xpraon nrval for non-sorabl marks would rsul n a mor volal fuurs pr Cash sld fuurs onras appar o b ganng n populary s rn Nw York Mranl xhang and Mnnapols Gran xhang announmns o s mporan ha rsarhrs dvo addonal ffor o hs ara hs wll hlp o assur ha propr slmn hos ar mad o m rgulaory rqurmns and an also nras h probably of nrodung sussful fuurs onras whh an ulmaly nras soal wlfar lbr 9

10 h prsnd rsarh onrbus o hs ara by provdng prng quaons for ash sld fuurs onras hs quaons whl rlavly smpl and nuv provd a sarng pon for furhr rsarh hs rsarh an nlud addonal analyal work o solv xplly for rlaonshps among suh hngs as h lngh of h xpraon nrval and h volaly of ash sld fuurs Or h prng quaons an b usd n smulaons for proposd fuurs onras In hr as hs ssus ar mporan o onra dvloprs and h das prsnd hr may asss n h nroduon and undrsandng h pr bhavor of nw ash sld fuurs onras Rfrns Blak DG uss and alur of uurs Conras: hory and mpral vdn Monograph rs n nan and onoms 986- alomon Brohrs Cnr for h udy of nanal Insuons Gradua hool of Busnss Admnsraon Nw York Unvrsy 986 Bollman hompson and P Gara An Analyss of h Prforman of h Dammonum Phospha uurs Conra NCR- Confrn Appld Commody Pr Analyss orasng and Mark Rsk Managmn Prodngs 996 pp88-0 Carlon DW uurs Marks: hr Purpos hr Hsory hr Growh hr usss and alurs Journal of uurs Marks 98:7-7 Ca J and D Ln smang Cash lmn Pr: h Boosrap and Ohr smaors Journal of uurs Marks 7997:67-6 Commody uurs radng Commsson Applaon for Cash ld uurs Conras hp://wwwfgov/fls/da/analyss/dabdo pmbr 00 Garbad D and WL lbr Cash lmn of uurs Conras: An onom Analyss Journal of uurs Marks 98:5-7 Gray RW Why Dos uurs radng ud or al: An Analyss of ld Commods Radngs n uurs Marks: Vws from h rad dd by A Pk Chago Board of rad 978 Hull JC Inroduon o uurs and Opons Marks hrd don addl Rvr Nw Jrsy Prn-Hall 998 Jons J h onoms of uurs and Opons Conras Basd on Cash lmn Journal of uurs Marks 98:6-8 Luhold RM JC Junkus and J Cordr h hory and Pra of uurs Marks Lxngon Massahuss Lxngon Books 989 Marnz-Garmnda J and JL Andrson Hdgng Prforman of hrmp uurs Conras wh Mulpl Dlvrabl Grads Journal of uurs Marks 9999:957-9 Mnnapols Gran xhang Mnnapols Gran xhang Ownrs Approv Cash-ld Corn and oyban uurs; prng Wha ral Opons and v-cn rk Prs Bgn radng oday hp://wwwmgxom/nws/nws_rlass/rlas6hm Jun 00 Nw York Mranl xhang Nw York Mranl xhang Board Announs Plans o Inrodu Brn Crud Ol uurs Conra hp://wwwnymxom/ Aprl 00 Prson P Obsrvaons on Cash lmn NCR- Confrn Appld Commody Pr Analyss orasng and Mark Rsk Managmn Prodngs 995 pp- 0

11 andor RL Innovaon by an xhang: A Cas udy of h Dvlopmn of h Plywood uurs Conra Radngs n uurs Marks: Vws from h rad dd by A Pk Chago Board of rad 978 lbr WL Innovaon Compon and Nw Conra Dsgn n uurs Marks Journal of uurs Marks 98:-55 ashan Opmal uurs Conra Dsgn Quarrly Rvw of onoms and nan 5995:5-6 hompson P Gara and L Wldman h Dms of h Hgh ruos Corn yrup uurs Conra: A Cas udy Journal of uurs Marks 6996: 697-7

Term Structure of Interest Rates: The Theories

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