Bounds and Prices of Currency Cross-Rate Options

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1 Bouns an Pics of Cuncy Coss-Ra Opions an-lin Chung an Yaw-Hui Wang Dpamn of inanc Naional Taiwan Univsiy Taiwan Ocob 005 vis Januay 007 Absac This pap ivs h picing bouns of a cuncy coss-a opion using h opion pics of wo la olla as via a copula hoy an psns h analyical popis of h bouns un h Gaussian famwok. Ou opion picing bouns a usful bcaus hy a gnal in h sns ha hy o no ly on h isibuion assumpions of h sa vaiabls o on h slcion of h copula funcion; hy a pofolios of h olla-a opions an hnc a ponial hging insumns fo coss-a opions; an 3 hy can b appli o gna bouns on las. Th mpiical ss suggs ha h a psisn an sabl laionships bwn h mak pics an h sima bouns of h coss-a opions an ha ou opion picing bouns obain fom h mak pics of opions on wo olla as an h hisoical colaion of wo olla as a highly infomaiv fo xplaining h pics of h coss-a opions. Moov h mpiical suls a consisn wih h picions of h analyical popis un h Gaussian famwok an a obus in vaious aspcs. ywos: Opion picing opion bouns xchang as coss-a colaion copulas. JEL Classificaion: 3 4 G Cosponing auho. Dpamn of inanc Naional Taiwan Univsiy No. c. 4 Roosvl Roa Taipi 067 Taiwan. Tl.: yhwang@managmn.nu.u.w. W a inb o anonymous fs Joao Amao Maos Anonio Camaa an Bing-Hui Lin fo hlpful commns an suggsions; an also o h smina paicipans a Naional Chngchi Univsiy Naional Chiao Tung Univsiy Naional Taiwan Univsiy Naional Tsing Hua Univsiy Yuan Z Univsiy h Euopan inancial Managmn Associaion Annual Ming 006 h 4 h Bachli inanc ociy Wol Congss h inanc Managmn Associaion Annual Ming 006 h Taiwan inanc Associaion Annual Ming 006 an h Naional Taiwan Univsiy Innaional Confnc on inanc 006. W hank h Naional cinc Council of Taiwan fo financial suppo. This pap was viw an accp uing h nu of h pas Eioial Boa in which pofsso Giogio zgo ac as Managing Eio of h Jounal of Banking an inanc.

2 Bouns an Pics of Cuncy Coss-Ra Opions Absac This pap ivs h picing bouns of a cuncy coss-a opion using h opion pics of wo la olla as via a copula hoy an psns h analyical popis of h bouns un h Gaussian famwok. Ou opion picing bouns a usful bcaus hy a gnal in h sns ha hy o no ly on h isibuion assumpions of h sa vaiabls o on h slcion of h copula funcion; hy a pofolios of h olla-a opions an hnc povi ponial hging insumns fo coss-a opions; an 3 hy can b appli o gna bouns on las. Th mpiical ss suggs ha h a psisn an sabl laionships bwn h mak pics an h sima bouns of h coss-a opions an ha ou opion picing bouns obain fom h mak pics of opions on wo olla as an h hisoical colaion of wo olla as a highly infomaiv fo xplaining h pics of h coss-a opions. Moov h mpiical suls a consisn wih h picions of h analyical popis un h Gaussian famwok an a obus in vaious aspcs.

3 . Inoucion In h opion picing liau sachs a no only ins in picing bu also in bouning h opion valus. Th a many usful chniqus ha can b mploy o iv opion picing bouns. o xampl Mon 973 Gaman 976 Lvy 985 an Guny 99 us h abiag-f appoach o iv opion picing bouns. Richkn 985 Richkn an uo 989 Basso an Pianco 997 Mahu an Richkn 000 an Ryan 003 us lina pogamming mhos o iv opion picing bouns. In aiion o h abov wo yps of chniqus som oh appoachs such as opimizaion mhos an sicions on h volailiy of h picing knl hav also bn us in h liau. Mos if no all of h pvious suis iv opion picing bouns by icly using h pic infomaion such as h pic isibuion o pic pocss of h unlying ass. In conas o h pvious liau his suy uss h opion pics of h la olla as o iv h picing bouns fo h coss-a opion. In oh wos w boun cossa opion valus using h mak pics of h olla-a opions. In his sns h ia of his pap is clos o ha in h saic hg liau Ca Ellis an Gupa; 998 whby h xoic opions a pic an hg in ms of h pics of sana opions. inc h is a iangula laionship bwn h foign xchang as among h cuncis Taylo an Wang 005 show ha i is plausibl o sima isk-nual nsiis RNDs an opion pics of a coss-a un h U olla masu using h mak Th moivaion fo oing his is as follows. I is gnally obsv ha opions on olla-nomina xchang as a a un saisfacoy liquiiy whil coss-a opion maks a much lss liqui. Thus h picing bouns obain fom h liqui mak pics of olla-a opions a usful fo picing hging an abiaging. Boh Taylo an Wang 005 an his pap commnc h analyss un h foign olla isk nual masu o pic h coss-a opions. As suggs by Bakshi Ca an Wu 007 i may b b o spcify h gnic picing knls in ach couny o iv h pics of ivaivs. Nonhlss Taylo an Wang 005 analyically show ha h pics of ivaivs un iffn masus a quivaln whn h law of

4 pics of wo la olla-a opions. Insa of icly xploing h opion picing fomula his suy also using h olla isk-nual masu ivs h picing bouns fo coss-a opions by uilizing h xchang opion picing bouns impli in h copula hoy. 3 om analyical popis of h bouns un h Gaussian famwok a also psn in his pap. Compa wih h pvious suis ou objciv is no o iv igh bouns bu ah o gna infomaiv bouns fom usful mak pic infomaion olla-a opion pics. Ou opion picing bouns conibu o h liau in a las h aspcs. is of all alhough ou picing bouns a no igh hy a gnal in h sns ha hy o no ly on h isibuion assumpions of h sa vaiabls o on h slcion of a copula funcion. conly ou picing bouns hav conomic manings bcaus hy a pofolios compos of h olla-a opions an somims also compos of spo olla as an hnc povi ponial hging insumns fo coss-a opions. inally ou picing bouns a also usful fo gnaing bouns on las. Th mpiical ss of ou picing bouns a conuc using h pics of opions on foign xchang as among h U olla uo an poun sling. W fis show ha h a song an sabl laionships bwn h mak pics of coss-a opions an h picing bouns obain fom h mak pics of opions on h wo olla as. Boh of h abov fining an h analyical analysis in cion moiva us o un h gssion mols o masu h xn wh h coss-a opion pics can b xplain by ou picing bouns an h colaion bwn h wo olla-as. on pic hols. In o o uiliz h concp of h xchang opion his aicl fomulas h picing poblm un h olla masu. 3 Th ails of h copula hoy can b foun in Jo 997 an Nlsn 999. Chubini Luciano an Vchiao 004 fis apply h copula hoy o iv h picing bouns fo h xchang opions.

5 Ou mpiical suls inica ha h picing bouns sima fom opion pics of wo olla as an h colaion of wo olla as can povi highly significan infomaion abou 85% fo xplaining h coss-a opion pics acoss las. Ou suls a immun o h assumpion of h RND isibuion fo h olla as h mak volailiy lvl an h chang in h cuvau of h impli volailiy funcion. inally w monsa how o calcula bouns on las using ou picing bouns. Th main of his pap is oganiz as follows. cion ivs opion picing bouns fo h coss-a opion psns hi analyical popis un h Gaussian famwok an shows how o boun h la of h coss-a opion using h iv picing bouns. Daa an h mpiical mhoologis fo gnaing h isk-nual nsiis an opion picing bouns a psn in cion 3. cion 4 iscusss h mpiical suls whil cion 5 conclus h pap.. Bouns of h Pic an Dla of h Coss-Ra Opion.. Bouns of h Pic of h Coss-a Opion By applying h éch bouns in h copula hoy Chubini al. 004 show ha h sup-plicaion bouns of h opion o xchang on ass fo h oh ass a compos of h pics of h univaia opions on h wo iniviual xchang asss. 4 W fis show ha h payoff of a coss-a opion un h olla masu is quivaln o ha of an xchang opion wh h wo unlying isky asss a h cosponing olla as. ollowing h sam logic as in Chubini al. 004 w us h isk-nual picing appoach o iv h picing bouns fo h coss-a opion. 4 Howv Chubini al. 004 only iv h low boun fo on paicula pobabiliy coniion. Th shoul b an alnaiv fomula appli o h oh pobabiliy coniion. ou quaion 4 fo hs wo pobabiliy coniions. 3

6 Consi opions whos payoffs pn on h xchang as among h following h cuncis: U ollas UD Biish pouns GBP an uos EUR. W no h olla pic of on poun a im by / an likwis h olla pic of on uo a h sam im is no by /. Th coss-a pic of on poun in uos is hn givn by = un h no-abiag agumn. / / / Now consi a Euopan call opion wh h hol has h igh o buy fo a im T. Un h olla masu o fom h viwpoin of U.. sins h abov opion is inical o an opion o xchang / T ollas fo / T ollas a im T. Hnc a coss-a call opion un h olla masu is quivaln o an opion o xchang on ass fo h oh ass an is olla payoff quals / / max T T 0. This payoff can b -aang as follows: / T / T / T max[min 0]. Hnc h cun olla pic of an xchang opion is min by h isk-nual picing appoach as follows: Call / = / 0 T / / Callmin T wh Call min 0 T psns h pic a im of a call opion on h minimum of an wih sik pic 0 an mauiy im T. Applying h éch bouns in h copula hoy w a abl o iv h upp low boun of h minimum call opion pic an hus h low upp boun of h coss-a opion pic as follows. Poposiion. Th upp boun of h coss-a opion pic in ollas is: + / / Call / = Call T + Pu " T 3 4

7 wh is a consan saisfying ha / + / = x x i = i x i is h cumulaiv isibuion funcion an " = /. L b a consan which solvs / = /. Th low boun of h coss-a opion pic in ollas is hus: Call / / / ' Call T Call T if = / T / T / ' / + Call T Call T / u < / u fo u < ohwis. 4 ' wh = /. Poof: Plas s Appnix A. om quaions 3 an 4 w obsv ha ou picing bouns fo coss-a opions a pofolios of h cosponing olla-a opions an may also b of h spo asss. Thfo iffn fom mos opion picing bouns in h liau h iv picing bouns hav conomic manings. Moov h ivaion of ou coss-a opion picing bouns os no ly on h isibuion assumpion of wo olla as an h slcion of an appopia copula funcion. Thfo on can apply h chniqu uiliz h o iv h pic bouns fo any Euopan-syl ivaivs whos payoffs can b aang as h sam yp as ha of an xchang opion. inc h coss-a is complly min by h oh wo olla as un h iangula abiag laion a naual qusion o ask is how h payoff of h coss-a opion is la o h payoffs of h oh wo olla-a opions. If his laionship can b spcifi on is abl o apply h spanning appoach of Bakshi an Maan 000 o pic o o povi picing bouns fo h coss-a opions using h pics of wo olla-a opions. In Coollay w show ha h colaion opions consi in Bakshi an Maan 000 povi a low boun fo h coss-a opion pic. Th poof of Coollay is availabl fom h auhos upon qus. 5

8 Coollay : Th pic of a coss-a call opion wih sik pic is boun blow by h pic of a colaion opion wih h following payoff: wh = x y. max / T 0 max x / T y 0 No ha Coollay las h low picing boun of a coss-a call opion o h pic of a olla-a call wih a payoff of max / T x 0 an h pic of a olla-a pu wih a payoff of max / y T 0. Thfo Coollay suggss ha h olla-a opion pics an hnc ou picing bouns may b infomaiv fo xplaining h coss-a opion pics. La in cion 3 w will un a gssion mol o invsiga h xplanaoy pow of ou picing bouns... Analyical Popis of h Picing Bouns un h Gaussian amwok Whn h wo olla as follow a bivaia lognomal isibuion hn un h iangula abiag laion h coss-a also follows a lognomal isibuion. Thus h xis closfom soluions fo h opion pics of wo olla as an ou picing bouns. To hav som insighs on ou picing bouns w invsiga h analyical popis of hs bouns whn h wo olla as follow a bivaia lognomal isibuion. Dno h clos-fom soluions of wo olla-a opion pics a im nomina in U ollas as / C an C / spcivly. Poposiion shows ha h upp an low bouns hav clos-fom soluions un h bivaia lognomal isibuion assumpion. B / B / Poposiion. Assum ha wo olla as / an / follow a bivaia lognomal isibuion wih a colaion cofficin of ρ an volailiis p ya of / an / spcivly. Th upp an low bouns nomina in uos of h coss-a call opion 6

9 wih a sik pic of hus hav clos-fom soluions of C B / / + / an C / B / / spcivly. Poof: Plas s Appnix B. Un h bivaia lognomal isibuion assumpion h iangula abiag laion implis ha h coss-a opion pic nomina in uos is C B / / wh + / = / / - ρ / /. Thfo Poposiion is inuiivly u bcaus + / / / / /. Whn h colaion bwn wo olla as is high low h coss-a opion pic is clos o h low upp boun. Moov Poposiion implis ha whn h impli volailiy cuvs of wo olla-a opion pics a fla hn h impli volailiy cuvs of ou upp an low bouns fo coss-a opions a also fla..3. Bouns on h Dla of h Coss-a Opion Givn h sima picing bouns in ms of impli volailiis i is plausibl o iv bouns on h coss-a opion s la using Poposiion 5 of Bgman Guny an Win 996. Assum ha h volailiy funcion s is a funcion of h unlying ass pic s an im only. L an spcivly no h low an upp bouns on volailiy c s an c s spcivly psn h mak o accua call pic an is la an bs c an c spcivly san fo h Black-chols call pic an is la. Bgman bs al. 996 iv bouns on h opion s la as follows. Poposiion 5 of Bgman al If fo all s an s hn s c s c bs bs bs s wh s solvs c s = c s c bs + c bs bs bs bs s s s an s solvs c s = c s c s s s. 7

10 Th la bouns of Bgman al. 996 a u fo gnal Makovian iffusion pocsss. Whn h coss-a opion s valu oay is known Bgman al. 996 show ha h bouns on is la can b snghn as follows. Poposiion 6 of Bgman al If fo all s an s hn fo any s an such ha on knows cs c bs s c s c bs s wh s solvs c s = c bs bs s + c s s s an s solvs bs c s = c s s s s. c bs Ou picing bouns a icly applicabl o Poposiion 5 of Bgman al. 996 an hus h bouns on h las can b obain saighfowa. o insanc h impli volailiy of ou low upp boun povis an sima of fo applying Poposiion 5 of Bgman al Whn h mak pic of h coss-a opion is known ou upp boun can b us in conjuncion wih Poposiion 6 of Bgman al. 996 o obain igh bouns fo las. La w will calcula bouns on h las whn ou picing bouns a appli o Poposiions 5 an 6 of Bgman al Daa an Empiical Mhoologis 3.. Daa Th pimay aa us in his aicl a aily opion pics ha a quo as Black-chols impli volailiis fo h cuncy opions / / an /. W mak us of a confinial fil of OTC opion pic mi-quos suppli by h aing sk of an invsmn bank in Lonon. 5 Ou cuncy opion aa cov h pio fom 5 Mach 999 o Januay 00. Th OTC quos a fo all h foign xchang opions co a h n of h ay in Lonon. Th aa inclu opion pics fo svn xcis pics bas upon las 5 om slmn pics a availabl fo coss-a opions a in h Chicago Mcanil Exchang bu hy cospon o almos no aing volum. Consqunly w ly on ov-h-coun OTC opion pics wih which w hav h sam im-o-mauiy opion aa vy ay. To h bs of ou knowlg such pics a no availabl in h public omain. 8

11 qual o an 0.9. Th im o mauiy of h opions is on monh wih which opions in h OTC mak a mos fqunly a. W also us h spo xchang as of / / an / an h uo-cuncy ins as poxis of iskf as of an co by Daaam as h inpus fo all lvan calculaions. Th summay saisics of h quo impli volailiis show ha all impli volailiy funcions xhibi a smil shap wih h lvl fo h / opions bing h highs whil h lvl fo h / opions a h lows. Th low sana viaions of h quos imply ha h lvls of impli volailiis fo hs h xchang a opions o no chang much as im gos. Th skwnss is posiiv an h kuosis is clos o 3 which os no pn on h monynss of h opions. 3. Empiical Mhoologis fo Gnaing h Bouns Bcaus an a min by h isk-nual nsiis of wo olla as w us h obsv mak pics of Euopan call opions on / an / an a paamic isibuion spcificaion o sima hi isk-nual nsiis. Onc h isk-nual nsiis a obain an can b calcula wih a numical mho such as h Nwon- / / / / = Raphson mho o solv = an + spcivly. W a hn abl o pic olla-a opions wih all siks an g picing bouns of h coss-a opions using quaions 3 an 4. In his pap w us h gnaliz ba nsiy of h scon kin GB o sima h RNDs of wo olla as. 6 Th GB nsiy has fw paams bu i psvs many 6 Many yps of univaia RNDs hav bn popos incluing lognomal mixus Richy 990 an Mlick an Thomas 997 gnaliz ba nsiis Booksab an MacDonal 987 muli-paam isc isibuions Jackwh an Rubinsin 996 an nsiis iv fom fiing splin funcions o impli volailiis Bliss an Panigizoglou 00. Poviing ha opions a a fo a ang of xcis pics ha ncompass mos aas of h isk-nual isibuion i is ocumn ha sval flxibl nsiy familis povi simila mpiical simas. Th bouns sima wih h lognomal mixus RNDs a compa wih fo h obusnss chck. 9

12 siabl popis: gnal lvls of skwnss an kuosis a allow h shaps of h ails a fa laiv o h lognomal nsiy an h a analyic fomula fo h nsiy is momns an h pics of opions. uhmo h paam simaion of h GB nsiy is asy an h sima nsiis a nv ngaiv. Th ails abou h simaion of h GB nsiy can b foun in Booksab an MacDonal Empiical Rsuls Th mpiical suis in his aicl conain fou pas. W fis analyz h popis of ou picing bouns an hi laionships wih h mak pics of h coss-a opions. con w invsiga h xplanaoy pows of h picing bouns an h colaion bwn wo olla as fo h mak pics of h coss-a opions. Thi som obus analyss fo h accuacy of ou suls a povi. inally givn h sima pic bouns of h cossa opions w monsa how o boun hi las using h appoach popos by Bgman al Empiical Picing Bouns of h Coss-Ra Opions In o o hav a sanaiz compaison all h mak pics an picing bouns a conv ino h Black-chols impli volailiis. Th picing bouns of h -monh cossa opions wih svn iffn sik pics las a sima vy ay. All low bouns a min by h scon alnaiv of quaion 4 bcaus h impli volailiis of / a always lag han hos of / uing ou sampl pio. 7 Th scipiv saisics of h sima picing bouns an h mak impli volailiis acoss las a shown in Tabl. W also show h voluion of h sima picing bouns an h mak 7 Th analyical popis of ou low boun un h Gaussian famwok suggs ha if h volailiy of / is ga han ha of / hn u > fo u < an vic vsa. / / u 0

13 impli volailiis in igu. As h pans acoss las a vy simila igu psns h sul wih a la of 0.5 only. As shown in igu h mak impli volailiy always lis wihin h sima bouns an h voluion of h mak impli volailiy of h coss-a / opion xhibis a simila pan o hos of h sima bouns. As h foign xchang mak bcam mo volail fom 999 o 000 h boun ang fin as h iffnc bwn h upp boun an h low boun un wi as im wn by uing h pio. Tabl suggss ha h lvl h man an h volailiy of h upp bouns a almos h sam acoss las wih an xmly shallow smil. In conas h low boun an h mak impli volailiis xhibi cla smil shaps acoss las wih h low boun smil bing p han h mak impli smil. To xplo h laionships bwn h opion mak pics an h sima bouns w fuh look a h bhavio of h iffnc bwn h upp boun an h mak impli upp ang an h iffnc bwn h low boun an h mak impli low ang. Thi scipiv saisics acoss las a illusa in Tabl. Boh h lvl an vaiaion of h upp angs a lag han hos of h low angs acoss las. W also invsiga h laionships bwn h angs an h colaion of wo ollaas as Poposiion suggss ha h high h colaion is h clos h mak impli volailiy will b o h low boun. W gss h upp ang an h low ang spcivly on h colaion an po h slop cofficin simas in Tabl. 8 Th suls claly inica ha h upp low ang is significanly posiivly ngaivly associa 8 Th colaion cofficins a sima using h ynamic coniional colaion DCC mulivaia GARCH mol popos by Engl 00 wih h hisoical im sis aa of wo olla spo as. Th fac ha colaions bwn financial asss a usually im-vaying has impoan implicaions in many ways such as pofolio hging an mulivaia ass picing. This mol ovcoms h complxiy of convnional mulivaia GARCH mols in compuaion by icly moling h im-vaying colaion as a coniional pocss. Th pocu of using h DCC GARCH mol o gna h im-vaying colaion sis is ail in Engl 00.

14 wih h colaion of wo olla as acoss las; i.. h high h colaion is h clos h mak impli will b o h low boun. This fining is consisn wih h analyical popis of ou picing bouns. In summay h low bouns xhibi a smil shap whil h upp bouns an h mak impli volailiis a laivly fla acoss las. Boh h upp an low bouns xhibi acabl an psisn laionships wih h mak pics of coss-a opions. 4.. Picing Bouns Colaion an h Coss-Ra Opion Pics inc i has bn foun boh analyically an mpiically ha h a psisn laionships bwn h mak pics of h coss-a opions an ou picing bouns w fuh us a gssion mol o masu h xn wh h coss-a opion pics can b xplain by ou picing bouns. W gss h mak impli volailiis on h upp an low bouns. 9 Th gssion mol is spcifi as: Mol : MIV = c + β + UB + β LB ε 5 wh MIV UB an LB spcivly no h mak impli volailiy of h -monh coss-a opion on / h upp boun an h low boun a ay an ε is h siual m. 0 Th simas fo his mol a shown in Panl of Tabl 3. om Panl of Tabl 3 w fin highly significan gssion cofficins of β an β. Th ajus R s a vy high an ang fom 0.7 o 0.77 acoss las. I is noicabl ha 9 Th infomaion conn of ou picing bouns fo coss-a opions is simila o ha of h pics of opions on h cosponing wo olla-as. Thfo h impli volailiis of h wo olla-as hav h ponial o povi simila xplanaoy pow fo h coss-a impli volailiy as ou bouns o. Howv accoing o ou analysis h impli volailiis of / an / a highly cola abou 0.8. Thus a sious mulicollinaiy poblm occus whn icly gssing h impli volailiy of / on hos of / an / alhough is ajus R is jus slighly low han Mol. As ou picing bouns fo coss-a opions a lina combinaions of h pics of wo olla-a opions wih paicula sik pics ou bouns povi a soluion o h mulicollinaiy poblm by ansfoming wo highly-cola impli volailiis o wo lss associa bouns. As a sul using h bouns insa of h olla-a impli volailiis nabls ou analysis o b mo vali an liabl. 0 As h bouns a sima vy ay fom h -monh opions h aa us fo h gssion mol a aily aa.

15 β ahs o a small ang bwn 0.33 an 0.39 whil β angs fom 0.6 o In oh wos h upp boun conains almos h sam lvl of infomaion conn fo h coss-a opions acoss la whil h low boun conains iffn lvls of infomaion conn acoss las. In sho w confim ha h a song an sabl laionships bwn h mak pics of coss-a opions an h picing bouns sima fom h mak pics of h opions on wo olla as. om h analyical popis in Poposiion w fin ha no colaion infomaion is us in h calculaion of h picing bouns of h coss-a opions fo which w only uiliz h pic infomaion of h opions on wo olla as iniviually. Howv Dissn Manhou an Vilkov 007 analyz h laionship bwn h pics of sock inx opions an h pics of iniviual sock opions inclu in h inx an hy fin h lvanc of colaion isk an h associa pmium fo sock inx opions picing. Inspi by hi suls his pap inclus an xa xplanaoy vaiabl h hisoical colaion of wo olla as ino Mol o s whh h colaion is abl o povi aiional xplanaoy pow. Thus h gssion mol is moifi as: Mol : MIV = c + β UB + β LB + β3co + ε 6 wh Co is h DCC colaion cofficins of wo olla as a ay. Whn h colaion of wo olla as incass h vaianc of h coss a cass an hus h cossa opion pic also cass. Thfo h gssion cofficin of h hisoical colaion β is xpc o b ngaiv. 3 Th gssion suls fo Mol a shown in Panl of Tabl 3. I is claly sn ha h colaions of wo olla as povi incmnal infomaion in xplaining cossa opion pics as all ajus R s incas by abou 0% in compaison o Mol. Th gssion cofficins fo h colaion acoss las a significanly ngaiv an consis- 3

16 n wih ou xpcaion. uhmo ou suls a in lin wih h analyss an finings of Dissn al As volailiy is usually highly psisn i is xpc ha incluing h on-pio lagg volailiy as an inpnn vaiabl will incas h goonss-of-fi. Howv in his suy wha w inn o invsiga is how wll h coss-a opion pic can b xplain by h olla-a pic infomaion only i.. wihou pvious coss-a opion pic infomaion ah han how wll h mol can b spcifi. Thfo w only us h sima upp an low bouns as h xplanaoy vaiabls in his suy. o compaison w also inclu h on-pio lagg coss-a impli volailiy as an aiional xplanaoy vaiabl in Mol. Th unpo suls show ha vn wih h aiional xplanaoy vaiabl which is highly cola wih h pnn vaiabl h cofficins of h upp an low bouns an h colaion a sill significan a h % lvl an hi signs a sill consisn wih h hoical xpcaions. Du o h significan in-sampl xplanaoy pow of h sima bouns an colaion o h mak pics of coss-a opions w a ins in h pfomanc of ou mpiical mols in h ou-of-sampl picion. Givn h sima paams of h pvious mol Mol w pic h cun impli volailiy fo h coss-a / opions fom h cun mak pics of h olla-a opions an h hisoical colaion. Th acual an pic impli volailiis of h coss-a opions fo h la of 0.5 a shown in igu. Th suls fo oh monynss a vy simila an hus omi. Th picion os fin as h absolu valus of h acual valus minus h pic valus of impli volailiis fom Mol a gnally small. Th avag os acoss las a abou 0.3% which is small han h bi-ask spa in h OTC mak. In As xpc h sul has an almos pfc goonss-of-fi R 0.96 owing o cocing h consiabl fis-o sial colaion in h siuals of Mol. 4

17 aiion h volailiis of h picion os a vy small as wll abou 0.3% implying ha h mol pfoms consisnly wll acoss im. In summay h picing bouns sima fom opion pics of wo olla as an h colaion of wo olla as can povi highly significan infomaion fo xplaining h coss-a opion pics acoss las. Ou suls a valuabl sinc ou picing bouns which a pofolios of olla a opions a applicabl o pacical usag no only fo pic xplanaion bu also fo hging paiculaly whn h al-im coss-a opion pics a unobsvabl Robusnss Analysis To invsiga whh ou suls a obus w fis chck whh h sima bouns ly on h assumpion of h RND fo h olla as an hn w analyz whh h ouof-sampl picion os fom Mol a snsiiv o sampl slcion h impli volailiy lvl an changs in h cuvau of h impli volailiy funcion. To chck whh h sima bouns pn on h isibuion assumpion of h olla-a w assum ha h RNDs of wo olla as follow h lognomal mixus isibuion an hn compa h bouns sima un his assumpion wih hos un h GB isibuion assumpion. As shown in Tabl 4 h iffncs bwn h bouns sima using hs wo iffn RND assumpions a saisically insignifican acoss las a h 0% significanc lvl alhough h iffncs of h low bouns a lag han hos of h upp bouns. To chck whh sampl slcion affcs ou finings w -o h ou-of-sampl picion of Mol fo wo vnly ivi sub-sampls. Alhough h picion os a slighly high in h scon sub-pios 0.3% vs. 0.35% on avag h pans acoss 5

18 las a basically h sam. In oh wos ou fining os no pn on h sampl slcion. As h volailiy of xchang as incass ov ou sampl pio i is naual o chck whh h incasing volailiy affcs h accuacy of infomaion povi by ou picing bouns. Moov alhough h avag impli volailiis of all xchang as xhibi a smil shap h slops of h impli volailiy cuvs vay fom bing ngaivly slop o posiivly slop uing ou sampl pio. This implis ha isk nual skwnss an kuosis chang subsanially vy ay. Thfo o fuh chck whh h ou-ofsampl picion o of Mol pns on h volailiy lvl o h chang in h cuvau of h impli volailiy funcion w fis calcula h impli volailiy skwnss an kuosis using h Thom of Bakshi apaia an Maan 003 an hn un h following gssion mol: E c α E β X = wh E nos h pcnag picion o an X psns h impli volailiy skwnss o kuosis lvl sima using h appoach of Bakshi al.003 a im. Th AR spcificaion is moiva by h high fis-o auocolaion of picion os. Th simas a po in Tabl 5. All β cofficins a insignifican un h 0% significanc lvl. igu 3 inicas ha h RNDs of h coss-as a fa-ail avag kuosis quals 3.3 an slighly ngaivly skw avag skwnss quals igu 3 also shows ha h impli skwnss changs noicably ov im. Nvhlss Panl of Tabl 5 suggss ha h is no cla vinc suppoing ha h picion os acoss las a affc vn hough impli skwnss changs much. imilaly Panl 3 of Tabl 5 shows ha h impli kuosis has lil impac on h picion os acoss las. 6

19 In summay h accuacy of infomaion povi by ou picing bouns is immun o h mak volailiy lvl an h chang in h cuvau of h impli volailiy funcion. Oh poxis of h mak volailiy lvl such as h impli volailiis fo iffn monynss lvls a also us an h suls no po h a almos unchang Bouns on Dla of Coss-a Opions Givn h sima picing bouns in ms of impli volailiy i is plausibl o boun h coss-a opion s la using ou picing bouns wih Poposiions 5 o Poposiion 6 of Bgman al. 996 whn h call pic of h coss-a is unknown o known. W ak h a-h-mony ATM coss-a call opion a on Jun as an xampl an pic is la bouns in igu 4. Th soli lins in h scning o a h Black-chols pics compu as a funcion of h unlying ass pic using volailiis of h upp boun h mak impli volailiy an h low boun spcivly. Whn h coss-a call pic is unknown is la is boun bwn 0.06 an ash lins. Whn h call pic is known h la bouns bcom igh an angs fom 0.48 o o lins. 5. Concluing Rmaks Insa of picing coss-a opions icly his suy las h opion picing bouns o h pics of h cosponing olla-a opions. Ou picing bouns a iv fom a gnal sul of h copula hoy an hus o no ly on h isibuion assumpions of sa vaiabls. Diffn fom mos opion picing bouns in h liau ou coss-a opion bouns a funcions of h opion pics an somims also h spo pics of wo olla as. Th suls fo oh monynss lvls fin as h sik pics ivi by h fowa pic a simila. o xampl h la of h call opion wih h monynss lvl of.035 OTM angs fom o fom 0.07 o whn h opion pic is unknown known. 7

20 Using h pics of opions on foign xchang as among U olla uo an poun sling fo h mpiical ss w show h psisn laionships bwn h mak pics of h coss-a / opions an ou picing bouns. Ou picing bouns an h colaion bwn wo olla as povi 85% of h infomaion in xplaining h pics of h coss-a opions. Thfo ou suls a usful fo isk managmn an ivaiv picing paiculaly fo hos having coss-a isk xposus an whn only h cun opion pics of wo olla as a availabl. Th chniqu uiliz o iv ou coss-a opion picing bouns can b appli o any Euopan ivaiv scuiy whos payoff can b aang as h sam yp as ha of an xchang opion. o xampl on can iv h picing bouns fo quano opions using h copula appoach appli in his pap. 3 uh analyss la o h picing bouns of h oh yp of xchang opions a lf o ins as fo fuu sach. 3 Th fomula of quano opion picing bouns a also iv by h auhos an a availabl upon qus. 8

21 Rfncs Bakshi G. Ca P. an Wu L ochasic isk pmiums sochasic skwnss in cuncy opions an sochasic iscoun facos in innaional conomis. Jounal of inancial Economics fohcoming. Bakshi G. apaia N. an Maan D ock un chaacisics skw laws an h iffnial picing of iniviual quiy opions. Rviw of inancial uis Bakshi G. an Maan D panning an ivaiv-scuiy valuaion. Jounal inancial Economics Basso A. an Pianco P Dcasing absolu isk avsion an opion picing bouns. Managmn cinc Bgman T. Guny B. an Win Z Gnal popis of opion pics. Jounal of inanc Bliss R. R. an Panigizoglou N. 00. Tsing h sabiliy of impli pobabiliy nsiy funcions. Jounal of Banking an inanc Booksab R. an MacDonal J A gnal isibuion fo scibing scuiy pic uns. Jounal of Businss Bn D. an Liznbg R Pics of sa-coningn claims implici in opions pics. Jounal of Businss Ca P. Ellis. an Gupa V aic hging of xoic opions. Jounal of inanc Chubini U. Luciano E. an Vcchiao W Copula Mhos in inanc. John Wily an ons. Dissn J. Manhou P. an Vilkov G Opion-impli colaions an h pic of colaion isk. Jounal of Banking an inanc fohcoming. Engl R Dynamic coniional colaion: A simpl class of mulivaia gnaliz auogssiv coniional hoskasiciy mols. Jounal of Businss an Economic aisics Gaman M An algba fo valuaing hging pofolios. Jounal of inancial Economics Guny B. D. 99. Opion pics an h unlying ass s un isibuion. Jounal of inanc

22 Jackwh J. C. an Rubinsin M Rcoving pobabiliy isibuions fom opion pics. Jounal of inanc Jo H Mulivaia Mols an Dpnnc Concps. Lonon Chapman & Hall. Lvy H Upp an low bouns of call an pu opion valu: ochasic ominanc appoach. Jounal of inanc Mahu. an Richkn P Minimum opion pics un casing absolu isk avsion. Rviw of Divaiv Rsach Mlick W. an Thomas C Rcoving an ass s impli PD fom opion pics: An applicaion o cu oil uing h Gulf cisis. Jounal of inancial an Quaniaiv Analysis Mon R. C Thoy of aional opion picing. Bll Jounal of Economics an Managmn cinc Nlsn R.B An Inoucion o Copulas. Nw Yok ping. Richy R Call opion valuaion fo isc nomal mixus. Jounal of inancial Rsach Richkn P On opion picing bouns. Jounal of inanc Richkn P. an uo On sochasic ominanc an casing absolu isk avs opion picing bouns. Managmn cinc Ryan P Pogssiv opion bouns fom h squnc of cunly xpiing opions. Euopan Jounal of Opaion Rsach Taylo. J. an Wang Y.-H Opion pics an isk nual nsiis fo cuncy coss-as. Woking pap Lancas Univsiy & Naional Cnal Univsiy. 0

23 Appnix A. Divaion of h Pic Bouns fo h Coss-Ra Opion L P x i an b h pobabiliy h cumulaiv isibuion funcion an h olla isk-f ins a spcivly. Du o Bn an Liznbg 978 h pic of an opion on h minimum of wo isky asss can b xpss as: Call min / / 0 T = T = = 0 T T Pmin 0 0 P C / / / > x x / / / > x x > x x x x A. wh C is a suvival copula 4 an x = x. Accoing o h éch bouns in h i copula hoy i is u ha max u + v 0 C u v min u v sinc C u v is a copula. Consqunly h upp an low bouns of h minimum opion a givn as h following spcivly: i Call Call + min min / / / / 0 T = 0 T = T T 0 0 min max / / x / x + / x x x 0 x. A. inc i u is a casing funcion of u an is a consan which solvs / + / = i is u ha / u + / u fo u. Thfo h low boun of h minimum opion is: Call = = = min T T 0 0 T / / T / / / u u + u u / Call 0 T = u u / T T T 0 T 0 / 0 T + max / / u u u u + u / T u + T / T 0 Call u 0 u 0 u / u u / T T. A.3 4 If wo unifom vaiabls U an V a join wih a copula funcion C hn h join pobabiliy ha U an V a ga han u an v spcivly is givn by a suvival funcion: P U > u V > v = u v + C u v = C u v.

24 ubsiuing quaion A.3 ino quaion an applying h pu-call paiy w obain h upp boun of h coss-a opion pic as: Call / + = / = Call = Call T / / Call min / T + Pu T + Pu / / / 0 T T " T A.4 wh " = /. Assum ha h xiss a consan such ha / = /. If / u < / u fo u < hn i is saighfowa o show ha h upp boun of h minimum opion is: Call = = = + min / T T / 0 0 T / / / u u + u u - Call 0 T = / T T T 0 / / min / u u u u + T + Call u / T / ' T u u / u u. A.5 ' wh = /. ubsiuing quaion A.5 ino quaion yils h low pic boun of h coss-a opion as: Call / = / = Call T / Call + min / T Call / / 0 T ' T. A.6 imilaly if / u > / u fo u < hn on can iv ha: Call Call + min / / = / / T Call 0 T = / + Call / T T. / T / Call T + Call / / ' T ' T A.7 Q.E.D.

25 B. Poof of Poposiion Whn h wo olla as follow a bivaia lognomal isibuion is acually h / isk-nual pobabiliy ha h Euopan call opion on / wih a sik pic of will b xcis bcaus: / = = N / = P / / T / < wh = T an is a consan saisfying ha: ln. = / + / /. Thfo in Poposiion N / / / + N / =. inc N x + N x = is u on can show ha: Thus h soluion of fo / / / = /. A.8 is: / / / / + / / / + / = xp / + / / / + / / / /. A.9 om quaions 3 an A.8 h upp boun is givn by: / + = / N N N / / / / / / / N / / N N / / / / /. No ha h abov upp boun is nomina in U ollas an is valu in uos is: / N N /. A.0 / / / ubsiuing quaion A.9 ino quaion A.0 yils h upp boun i..: N / / /. / / / / = C B + + N / / + / 3

26 o bviy w iv h low boun only fo h cas wh. / / > inc saisfis ha / / / / N N = on can show ha is soluion is:. xp / / / / / / / / / / / / / / / / + + = A. inc / / > i is saighfowa o show ha whn u <. ln ln / / / / / / / / / / u N u N u N u N = + < + = Thfo h low boun is min by h fis alnaiv of quaion 4 5 i..:. / / / / / / / / / / / / / / / / N N N N N N = + Th abov low boun is nomina in U ollas an is valu in uos is:. / / / / / N N A. ubsiuing quaion A. ino quaion A. yils h low boun i..:. / / / / / / / / / / = C N N B Q.E.D. 5 On h oh han if h volailiy of / is small han ha of / hn h low boun is min by h scon alnaiv of quaion 4. 4

27 igu : Th Impli Volailiis fom Mak Pics an h Esima Bouns fo Coss-Ra Opions This figu shows h voluion of h Black-chols impli volailiis fom h mak pics an h sima bouns of h coss-a / opions wih h la of 0.5. Th coss-a opion picing bouns a sima by calibaing quaions 3 an 4 using h opion pics of wo olla as / an /. Dla: 0.5 Upp Mak Low /5/99 05/5/99 07/5/99 09/5/99 /5/99 0/5/00 03/5/00 05/5/00 07/5/00 09/5/00 /5/00 igu : Acual an Pic Impli Volailiis fo Coss-Ra Opions This figu consiss of h voluion of h acual an pic Black-chols impli volailiis of h cossa / opions wih h la of 0.5. Th acual impli volailiis a back ou fom h mak pics of opions. Th pic impli volailiis a obain fom Mol in cion 4 using h mak pics of opions on wo olla as / an / an h hisoical DCC colaion of wo olla as. Dla: 0.5 Acual Pic /4/99 06/4/99 08/4/99 0/4/99 /4/99 0/4/00 04/4/00 06/4/00 08/4/00 0/4/00 /4/00

28 igu 3: Impli kwnss an uosis fo Coss-Ra Opions This figu consiss of h voluions of h impli skwnss an kuosis of h coss-a / opions. Th impli skwnss an kuosis a calcula using Thom of Bakshi al Th suls inica ha h isk-nual isibuions of h coss-as a fa-ail avag kuosis quals 3.3 an slighly ngaivly skw avag skwnss quals kwnss /4/99 06/4/99 08/4/99 0/4/99 /4/99 0/4/00 04/4/00 06/4/00 08/4/00 0/4/00 /4/00 uosis /4/99 06/4/99 08/4/99 0/4/99 /4/99 0/4/00 04/4/00 06/4/00 08/4/00 0/4/00 /4/00 6

29 igu 4: Dla Bouns of a Coss-Ra Opion This figu consiss of h upp an low bouns on h la of h ATM coss-a / call opion a on Jun whn h coss-a opion pic is unknown ash lins o known o lins. Th la bouns a calcula using h Poposiions 5 an 6 of Bgman al Upp nown: Low nown: 0.48 Upp Unknown: Low Unknown:

30 Tabl : ummay aisics of h Impli Volailiis fom h Mak Pics an h Esima Bouns Panl : Upp Bouns Dla Man Dv Panl : Mak Implis Dla Man Dv Panl 3: Low Bouns Dla Man Dv This abl consiss of h summay saisics of h impli volailiis fom h mak pics an sima upp an low bouns of h coss-a / opions acoss las. Th opion bouns a sima by calibaing quaions 3 an 4 wih h opion pics of wo olla-as / an /. Tabl : ummay aisics of h Esima Upp Rangs an Low Rangs Panl : Upp Rangs Dla Man Dv β Panl : Low Rangs Dla Man Dv β This abl consiss of h summay saisics of h sima upp angs an low angs of h coss-a / opions acoss las. Th upp angs an low angs a h isancs bwn h upp bouns an mak implis an bwn h low bouns an mak implis spcivly. In aiion h paam simas of h following gssion mol a povi. R = α + βco + ε wh R is h upp o low ang Co nos h colaion bwn h wo olla-as an ε is h siual m a im. Th colaions a gna by h DCC mol of Engl 00. Th numbs in h panhss a -saisics. 8

31 Tabl 3: Explanaoy Pow of Esima Bouns an Colaion o Mak Impli Volailiy Panl : Mol Dla β β Ajus R Panl : Mol Dla β β β Ajus R This abl consiss of h gssion suls of h following wo mols: Mol : Mol : MIV = c + β UB + β LB + ε MIV = c + β UB + β LB + β Co + ε. 3 H MIV an no spcivly h mak impli volailiy of an opion on / UB LB Co h upp boun h low boun an h hisoical DCC colaion bwn / an / a ay an ε is h siual m. Th numbs in h panhss a -saisics. Tabl 4: Robusnss Analysis fo Opion Bouns Panl : Upp Bouns Dla GB Mixus p-valu Panl : Low Bouns Dla GB Mixus p-valu This abl consiss of h mans of h upp an low bouns of h coss-a / opions acoss las which a sima using wo iffn RND assumpions GB an log-nomal mixus fo h olla-as. In aiion h p-valus of h man qualiy ss fo h wo assum isibuions a povi. 9

32 Tabl 5: Robusnss Analysis fo Ou-of-sampl Picion Eos Panl : Rgssion of Eos on Impli Volailiy Dla β Panl : Rgssion of Eos on Impli kwnss Dla β Panl 3: Rgssion of Eos on Impli uosis Dla β This abl consiss of h paam simas of h following gssion mol us o analyz whh h ou-of-sampl picion os fo Mol pn on volailiy skwnss o kuosis. E = c+ α E + β X + ε wh E nos h picion o in pcnag an X is h impli volailiy impli skwnss o impli kuosis a im sima using h Thom of Bakshi al Th numbs in h panhss a -saisics. 30