On Single and Multiple Currency Multifactor LIBOR Market Models: Application to Currency Options

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1 Ieaoal oal of ee a Reseah IR IN Ole: Impa ao : 3358 O gle a Mlple ey Mlfao IBOR Mae Moels: Applao o ey Opos Wee Hülma ss Mahemaal oey elsasse 45 H-84 üh zela Asa: I he maseam Gassa seg of log-omal foa aes a ompehesve spefao of sgle a mlple ey mlfao IBOR mae moels s eae I les a -eph peseao of mpoa ol a e esls o oelao maes he evalao of Gees s oe h he RM algohm o llsae a Moe-alo smlao of Eopea ey opo pes a elas fo a oss ey mlfao IBOR mae moel s ese agas o aalyal asympo expaso fomlas olg h a o-gassa opla oloo e sho he exsee of sgle ey o-fao IBOR mae moels of aay meso h aay mags Keyos: IBOR mae moel ; foa aes ; oelao max ; Gees ; ey opo ; vesal opla ; lea la opla Ioo he Bla moel also alle Bla-76 moel fs pesee [4] s a vaa of he Bla-holes opo pg moel I has ee esge o pe pmaly opos o fe oas o opos ees ae aps a floos a sapos hs moel has ee geealze o a lass of log-omal foa moels e he amg of IBOR mae MM moels eg es [7] eo 7 O o s evoe o a ompehesve spefao of some sgle a mlple ey mlfao IBOR mae moels hh exe paly he moels osee es [7] hape 4 a Kaa a äel [37] eo s a -eph svey of mpoa ol a e esls o oelao maes he peseao pe se of he sgle a mlple ey mlfao IBOR mae moels follos he eos 3 a 4 he evalao of he Gees s oe h he RM algohm of Glassema a hao [] as popose eo 5 o llsae e es eo 6 he Moe-alo smlao of Eopea ey opo pes a elas fo he oss ey mlfao MM of eo 4 ase agas he aalyal asympo expaso fomlas y Kaa a äel [37] I he olg eo 7 a ef oloo o he o- Gassa seg of geeal IBOR mae moel s gve Beses he poeal se of omal vaae-mea mxes e espeally emphasze he possly o exe IBOR mae moels sg oplas I hs fameo e sele heoem 7 he exsee of sgle ey o-fao IBOR mae moels of aay meso h aay mags Befoe sag he spefao of mlfao IBOR mae moels s mpoa o as hehe hee s a mae evee fo hem As he omme afe 8 shos oe somemes maes he smplfyg spefao of a oe-fao IBOR mae moel A ovos savaage of sgle fao sohas moels s he fa ha saaeos oelao eee ees aes a oly e oe hs oass h he ssaal evee ha he em se of ees aes s ve y mlple faos hee fo o eve moe fo hh oe shol osl he eve ale y a a gleo [] Befoe fomlag he spefao of mlfao IBOR mae moels s mpoa o sss he avaages a savaages eesz a elsse [5] meo fve ales ha ompae sgle a mlfao moels s favo of mlfao moels ogsaff aa-laa a haz [4] lam ha sho-ae moels ease of mss-spefe yams lea o s-opmal exese saeges a ole ha he oss o Wall ee fms of follog sgle fao exese saeges ol e seveal llo ollas eo favo of a sgle fao moel Aese a Aease [3] lam ha he exese saegy oae fom a popely alae sgle fao moel oly leas o sgfa losses he apple a o fao moel h esse Klaasse a Meleeg [4] ae he fs o vesgae hege pefomae hey sho ha f he me of hege smes s eqal o he me of faos he mlfao moels opefom sgle fao moels If hoeve he lage se of hegg smes s se hh s he ase pae he sgle fao moels pefom as ell as mlfao moels ems of ela hegg of Eopea sapos oh a Gpa a Rhe [6] sho fo he ase of he me of hege smes eqal o he me of faos ha hghe fao moels pefom ee ha loe fao moels ems of ela hegg of Eopea sapos a Eopea sapo sales hese esls ae hs osse h hose of esse e al [4] fh Gpa a ahmayam [] sho ha sgle fao moels pove aae pg he alae o he volaly smle o hegg hoeve a seo sohas Volme 3 Isse Noveme 4 se ese Ue eave ommos Ao BY ape I: O

2 Ieaoal oal of ee a Reseah IR IN Ole: Impa ao : 3358 fao yels ee pefomae ha a sgle fao moel alae o smle A I-eph Reve of oelao Maes ome mpoa esls o oelao maes fo se he oex of s maageme a opo pg ae evee a smmaze ome of hem ae ely ely apple eo 4 peses a ee e algea algohm pespeve hs aea Val oelao Maes a peal eomposo A symme x oelao max h agoal ees s a val oelao max f a oly f s posve semefe e s eema s o-egave hs s he ase pove all he egevales of ae o-egave ha s o ese ha a gve max s ee a val oelao max oe ofe apples he QR algohm hh pala allos eemao of he egevales a he egeveos of he max ppose ha a se efe a o eah efe fhe By oso he QR max eomposo Q R hols hee Q a R s a ppe agla max Oe sgshes eee o ases ase : val oelao max hogh eao of he QR algohm oe ges he speal eomposo B B B hee ag s he agoal max of egevales a s he max of egeveos eg Olve [44] Moe pesely le Q R e he QR max eomposo a se Q Q R R R Q hogh eao se Q R R Q 3 he ovegee popees ag a Q Q as ae ell-o Rema If s sly posve efe ha s hee exss aleavely a qe loe Volme 3 Isse Noveme 4 se ese Ue eave ommos Ao BY agla max sh ha he so-alle holesy eomposo If s oly posve semefe ha hs eomposo s o qe hogh somemes apple s maageme a opo pg he speal eomposo shol e pefee ove he holesy eomposo hs s espeally he ase fo Moe alo smlao IBOR mae moels ase : val oelao max a leas oe egevale I hs sao s possle o os a val oelao max Ĉ ha appoxmaes h espe o a gve om losely he gve val oelao max he sese ha he sae ˆ s small eogh Amog he may vese solos o he so-alle eaes oelao max polem he follog meho s espeally smple a aave peal eomposo meho Reoao a äel [55] mlaly o ase hogh eao of he QR algohm oe ges he agoal max of egevales ag a he max of egeveos s of he gve val oelao max ose o he agoal max of o-egave elemes of efe y ' agmax a he agoal salg max ag h o-zeo elemes s max eg B ' oe sees y oso ha ˆ B B s oh posve sem-efe a has agoal elemes hh eals a val oelao max Example : speal eomposo meho fo a 3x3 val oelao max o llsae h a meal oee example ose he 3x3 val oelao max h egevale max a egeveo max ape I: O

3 Ieaoal oal of ee a Reseah IR IN Ole: Impa ao : 3358 he speal eomposo meho yels he exelle appoxmao Ĉ he fee he egevales hh have o e se o zeo a he smalle asole vale hey ae he ee he speal eomposo meho ll e he lmao of he speal eomposo meho s es llsae as follos Example : lm of speal eomposo meho fo a 3x3 val oelao max ose he 3x3 val oelao max h egevale max a egeveo max he speal eomposo meho yels he aae appoxmao 5 5 Ĉ 5 5 By he ay s o easy o f 5 5 feasle vaae oplas h sh a pese oelao max see he olg eo 7 fo moe eale explaaos Ra eo of oelao maes he polem of a eo fo pg a faal po ha epes o a lage me of asses os he he asses ae moele as oelae logomal poesses I pala hs seg les pg of ees ae evaves a IBOR mae moel a opo pg mlmesoal Bla-holes maes ho o h sohas ees aes eg aho [5] [7] [8] o llsae ose a moel h oelae logomal pe poesses / W W W 3 e a sgle mease Hee eoes he pe of he - h asse s volaly a W eoes he assoae vg Boa moo Boa moos ae oelae h oeffe he oelao oeffe eee he es of asses a he max shol e a val oelao max ha s s posve sem-efe h agoal elemes he em sas fo he f em o mpleme a sheme 3 y Moe alo Volme 3 Isse Noveme 4 se ese Ue eave ommos Ao BY smlao oe apples he speal eomposo B B h B a x max If eoes he -h o veo of B he he eomposo eas Isea 3 s moe ovee o mpleme he sheme 4 / hee he ae o epee Boa moos No fo example f oe moels a 3 yea Bema sapo h aal all a payme aes he he moel osss of 3 aal foa IBOR aes o lage ees ae oelao maes sally almos all vaae say 99% a e ae o oly 3 o 6 Boa moo faos heefoe 4 oas a lage me of almos ea Boa moos ha os expesve ompaoal me o smlae Isea of ag o ao all Boa moos smlao s oe h a smalle me of faos say h ypally eee a 6 he sheme 4 s hs eplae y / 5 he x max B s a eomposo of hs appoah mples ha he a of B s less ha o eqal o o faal oelao maes hs a eso s geeally o sasfe heefoe a appoxmao ll e eqe hee ae o possle ays o poee s oe mgh appoxmae he ovaae max eo oe mgh appoxmae he oelao max hle maag a exa f o he volales I pae he volales ae sally ell-o eg Bla-ype pg fomlas o mae qoes heefoe oe appoxmaes he oelao max ahe ha he ovaae max he eaes lo-a oelao max polem osss o f he x posve sem-efe max eaes o he gve x max hogh osae opmzao m se o a 6 he mpoae of hs polem fae has ee eogze y seveal eseahes le Reoao [53] Hgham [4] a hag a W [65] o example Reoao [53] paameezes he se of a oelao maes a apples ppal ompoe aalyss A o oa a feasle po hs ma smple a popla meho goes a o ly [8] hag a W [65] have poe o shoomgs of he A meho a have exee o oa he gloally eaes po Hoeve he lae meho ofe oes o ovege o mve hese oveees Gs a eesz [] as ell as eesz a Goee [49] have osee ovel ehqes o solve hs polem ape I: O

4 3 ll-a a ee a paameezaos Ieaoal oal of ee a Reseah IR IN Ole: Impa ao : 3358 We follo Bgo [5] Gve s a fll-a x val oelao max I s haaeze y ees a me ha a e oo hgh fo paal pposes heefoe a pasmoos paame fom fo has o e fo hh s ase o a ee me of paamees hee ae o possles Oe mgh ea a fll-a oelao max h a smalle me of paamees o a ee-a oelao max 3 ll-ra oelao aameezaos A fll-a paame famly s osee y hoemaes a offey [6] hey efe oelao hogh a fe seqee of posve eal mes / / / a se 3 / 7 small hages Noe ha s he oelao eee foa aes h he hghes sae I s also he mmal oelao possle he moel he paamee epeses he ae of eoelao A evao of fomla 9 s fo aham [45] Appex B eo fo 3 assme ha le o a sagh le a se Impove sale fll a o-fao paameezao exp l f l f hs sem-paame oelao se sasfes he follog oos hh a e oseve o oelao esmae fom mae aa: oelao eeases fo easg may evals he foa ve es o flae a oelao eases h lage maes omally fo eah ege p he oelao s easg p Ohe paes ha mgh e mpoa applaos o he IBOR mae moel ae oe oelao eeases he movg aay fom he agoal alog a o o olm ha s mlaly he movg off he agoal he oppose eo oe has he me of eqe paamees s vess he ees eqe fo a geeal oelao max hs paame fom a e haaeze y a seqee of o-egave mes sh ha } exp{ 8 Aga oe has Moeove he mage of oavy of he seqee l 7 s eeasg o zeo ahe ha eg osa as 9 Aohe smple possly s gve y Reoao [54] lassal o-fao expoeally eeasg exp{ } e-oelao of foa aes h easg sae es oas ahe ha zeo hee epeses asympoally he oelao eee he aes h hghes sae e oo s o flflle Reoao [54] sggess he follog mofao Mofe hee-fao expoeally eeasg exp{ max } aameezaos ae oae y mposg oos o he s s seg all s eqal exep he las oe oas hogh hage of vaale he follog moel ale fll a o-fao easg alog sagoals exp l 9 l hs paameezao s sale he sese ha elavely small movemes he -paamees ase oly elavely Volme 3 Isse Noveme 4 se ese Ue eave ommos Ao BY As poe o y hoemaes a offey [6] hs foal fom may fal o efe a val oelao max hs polem s esolve y Reoao [54] o ays hee-fao paameezao exp{ e R o-fao sqae oo paameezao exp{ m ape I: O } 3 } 4

5 Ieaoal oal of ee a Reseah IR IN Ole: Impa ao : Ree-a oelao paameezaos A ffee paameezao appoah o oelao alao ha allos fo meso eo as explae eo osss of shg o a max ha has o fll a eg Bgo [5] eo 3 a aham [45] eo 44 Aga ose he speal eomposo h egevales ag a ohogoal max of egeveos e A o oa fhe A A Replag he x max A y a sale - a x max B sh ha B B oe s le o he ompaoally moe avaageos ee a smlao sheme 4 Of ose oe s lef h he polem of hoosg a sale paame fom fo B sh ha B B hee exs o popla mehos fo hs pal ompoe aalyss A e e he max h he smalles egevales se eqal o zeo a efe B B B o oa a val oelao max sh ha esale he ees / 5 he egeveo of h hghes egevale s alle ppal ompoe a has he mos sgfa flee ag he egeveos h hghes egevales esls a -a oelao max h mmm loss of fomao o example Alexae a vov [] se A h 3 a lassfy he oespog egeveos o e l a vae oes o moe fomao ao A he eese eae s efee o he ee oveve y A a Wllams [] see also olffe [34] Hype-sphee eomposo Reoao [53] A -a oelao max s efe y B B h os s s 6 he paamee se s efe hogh agla ooaes A eo of he me of paamees a e oe y hoosg I hs speal ase Bgo [5] oas he follog expesso -a olm-homogeeos agles paameezao 7 Volme 3 Isse Noveme 4 se ese Ue eave ommos Ao BY Hee oe has B B B h 8 he speal ase fo all yels a oelao max h all ees eqal o hs yels a oe-fao IBOR mae moel ha s somemes aope as sag po he max 8 s gaaee o e a val oelao max he o veos of B a e vee as ooaes lyg o a hype-sphee he agles a e hose y mmzg he mea sqae eo 9 o aheve a eease he me of fee paamees oe ms have hee ae some polems lef ope hs fomlao o example he fll a ase oe ees agle paamees heeas s half of hese paamees ae eee o haaeze a oelao max Rapsaa e al [5] solve hs polem sg a geome epeao of he agles paamees 33 oelao Max ompleo olems ose he follog -mesoal sysem of sohas ffeeal eqaos f V W V a V V W W W W V W V h a Boa moos W W V e he fos s s o oe volaly-elyg pa oe oas a ypal sohas volaly moel o ese he o-fla mple volaly sfae Eopea opo s pes eg Heso [3] hesey a o [9] a höel a h [58] he oelao max of he sysem has he follog se?????? h efe oeffes mae? he oelao max ompleo polem s ssse a solve Kahl a Gühe [36] Kahl [35] a äel a Kahl [3] A geeal appoah o ompleo polems s Koa a ooe [38] eo 453 ape I: O459 46

6 Ieaoal oal of ee a Reseah IR IN Ole: Impa ao : Algohm Valao a Geeao of Val oelao Maes I ase oe s o eese he eemao of he egevales a he egeveos of a oelao max s o eessay o apply he QR-algohm fom eo o ese a gve s a posve sem-efe max Aleavely hee s a vey smple es hh a hghly e eommee fo sofae mplemeao I fa hee s a smple po epeseao of he eema of a oelao max ems of s paal oelao oeffes hh ae efe y he esve fomlas eg Yle a Keall [64]: ;3 ;3 ;3 ;3 ;3 ;3 he eema of a oelao max s gve y eg aho [6] fomla e ; 3 3 ; ; 3 No hs eema s o-egave f a oly f all he paal oelao oeffes elog o he eval eg aho [6] emma e ; ; ; 3 5 O he ohe ha he algohm geeao of val oelao maes has ee p o qe eely a hallegg polem A fs osve esve opmzao algohm fo hs s ese Be e al [6] A mpoa mpoveme s aho [6] oollay 4 ho oas esve eave expl lose fom os fo he oelao oeffes e hey o o eqe a po ay opmzao poee hese os offe a moe saghfoa mplemeao ha he opmzao algohm y Be e al [6] A geeal expl paameezao has fally ee oae aho [9] gle ey Mlfao MM 4 4 Afe havg sfe he ee fo mlfao IBOR mae moels he ooy eo e pove he pese a ex eos spefaos of some sgle a mlple ey mlfao IBOR mae moels Oe assmes a me sezao eo se a se he IBOR foa aes eqal o { }/{ } h he pe of a zeo-opo o e e a val oelao max fo he IBOR foa aes a ose a sale -a x Volme 3 Isse Noveme 4 se ese Ue eave ommos Ao BY max B sh ha B B h he foa aes of a -fao MM he ollg foa s-eal ol follo he sohas ffeeal eqaos es [7] eqao 7 a eo 7 3 max{ : } 3 hee ae epee saa Wee poesses e s loo explly a some lo fao moels Oe-fao MM he ase oespos o he ma eqao 33 o-fao MM he x max B a e hose sh ha eg heoem 3 Reoao [53] os s 3 34 Wh B B oe oas he oelao oeffes as os 35 hoemaes a offey [6] eo se 35 o llsae ha lo fao moels have s polems o mah saaeos oelaos eee foa IBOR aes ealsally see also Reoao [5] hoemaes a offey [59] Remas 3 I s emaale ha hs spefao oes h he exeme pos of a o he ompa a ovex se of all oelao maes I fa he pa yels a solo of he a o haaezg eqaos Ya [63] heoem see eo 7 fo fhe ssso hee-fao MM Usg he hype-sphee eomposo 6 he x 3 max B sh ha B B a e hose sh ha os os s 36 3 s s 4 hs moel a e fe o a mae oelao se m m sh ha he mea sqae eo m ape I: O459 46

7 Ieaoal oal of ee a Reseah IR IN Ole: Impa ao : 3358 Volme 3 Isse Noveme 4 se ese Ue eave ommos Ao BY 9 s mmze o example Reoao [53] fs sessflly a oelao se of ype gve y 5 3 max exp 37 Alexae a vov [] a m [4] epo exelle esls he fg he oelao sfae h a hee-fao MM A ee mase hess o hs op s amme [6] 4 Mlple ey Mlfao MM ollog es [7] hape 4 a Kaa a äel [37] e expa o he spefao o a mlple ey mlfao IBOR mae moel 4 oss ey mlfao moels ose he ase of a o ees MM h sohas faos eah fo he omes a foeg IBOR foa aes a oe fao fo he foeg/omes spo exhage ae he omes ey s eoe y a he foeg ey y he spo exhage ae efes o / e he me of s of omes ey ha eqal oe of foeg ey We assme ha fo he hose méae ha hee exss a oespog eqvale magale mease e hh he moelle qaes ae log-omal poesses 4 ll-ra Moels We eg h he fll-a moel h Wee poesses fo he omes a foeg foa aes a oe Wee poess fo he spo ae Wh he sal efos a oaos oe has he sohas ffeeal eqaos omes IBOR foa aes W 4 oeg IBOR foa aes W 4 po exhage ae W 43 ase : f ems e he omes ollg foa s mease o spo IBOR mease W E W 44 W E W W E W 45 ; ; l 46 A mahemaal evao of hese fomlas s fo es [7] eo 4 ase : f ems e he omes emal mease 47 W E W 48 W E W 49 hese fomlas a he epeaos ae fo Kaa a äel [37] eo Rema 4 Oe oes ha he fs ase o o epe o he oss ey oelaos a o he omes ey o oelaos o seveal pposes pala Moe alo smlao s moe ovee o expess he sohas ffeeal eqaos 4-43 as eg ve y a veo of epee saa Wee poesses o oa a fomlao of he ype sho 3 a 3 o hs oe oses he x -max ompose of he e ey omes a foeg oelaos he oss ey oelaos a he ey o oelaos efe y ape I: O459 46

8 Ieaoal oal of ee a Reseah IR IN Ole: Impa ao : 3358 Volme 3 Isse Noveme 4 se ese Ue eave ommos Ao BY 4 e 4 e a val oelao max ose s speal eomposo B B a efe he - mesoal volaly veos 4 hh sasfy he follog pese elaoshps 4 Ue hs ohogoal asfomao he sohas ffeeal eqaos 4-43 ea hee he fs a e ee ompa fom as follos ase : ; ; l 48 ase : Ree-a moels Gve ae omes a foeg foa aes a se A hs sage s easy o sae he eea moel h m Wee poesses fo he omes a foeg foa aes a oe Wee poess fo he spo ae I osss of he eqaos hh ae ve y a veo of epee saa Wee poesses he fs ase a 49-4 ase ae efe ems of -mesoal volaly veos 4 sasfyg 4 I ohe os he -a x max B flfls y oso he eea speal eomposo B B We oe ha he speal ase oes h he fll-a moel of eo 4 4 Mlple ey Mlfao Moels Wh o a ommoes Ies I he moel of eo 4 fo he ase of he omes ollg foa s mease s o geealze o he ase of moe ha oe foeg ey he pesee of o- ve payg so es O geealzao ses esls fom es [7] eo 43 We eoe he omes ey y UR he foeg ees y UR a he spo exhage aes y / UR UR o he h ey hee ae gve IBOR foa aes a o-ve payg so a ommoy I es he h IBOR foa ae of he h ey s eoe y he h spo exhage ae y a he h o-ve payg I of he h ey s eoe y he me efe y s eqal o he meso of he oelao max eee he Iô poesses assoae o he sohas vaales a We ose he ase of a mley IBOR mae moel h m sohas faos eah fo he omes a foeg IBOR foa aes a oe sohas fao eah fo he foeg/omes spo exhage aes a he o-ve payg I s Wh he appopae efos a oaos he mlple ey mlfao MM a e smmaze y he follog sohas ffeeal eqaos hh ae ve y a veo of N : ape I: O

9 Ieaoal oal of ee a Reseah IR IN Ole: Impa ao : 3358 Volme 3 Isse Noveme 4 se ese Ue eave ommos Ao BY epee saa Wee poesses N IBOR foa aes 43 po exhage aes 44 No-ve payg I s 45 I emas o spefy hese eqaos he N -mesoal volaly veos a he f ems o oveee e ose he follog ex fos hee y oso s he meso of he oelao max he he volaly veos hh ae paameeze y N N N 48 ms y oso of he eomposo B B sasfy he ovaae elaoshps 49 he fs ae gve y 43 ; ; l 43 ; l 43 hee ; ae zeo o pes he h ey hemoe f oe hooses fo ; he epolao efe y : ; 433 he he f ems smplfy o l 434 l 435 We oe ha he N -a xn max B 48 flfls y he oso 49 he ee-a speal eomposo B B I s sve o loo explly a he follog o speal ases gle ey mlfao MM he hoe yels he - fao sgle ey MM efe y 3-3 oss ey mlfao MM he hoe m yels he -fao oss ey MM h ase fs eo 4 a es [7] eo 4 ape I: O

10 Ieaoal oal of ee a Reseah IR IN Ole: Impa ao : 3358 Volme 3 Isse Noveme 4 se ese Ue eave ommos Ao BY Eqy hy mlfao MM he hoe yels he - fao sgle ey eqy hy MM osee eale es [7] eo 4 oss ey eqy hy mlfao MM he hoe m yels he -fao oss ey eqy hy MM osee eale es7 eo 43 5 Gees fo Mlple ey Mlfao MM We ae ees smlao mehos ha a e esge fo he ase of soos payoffs a eve aay pah-epee payoffs I a es ase e have sessvely apple he RM algohm of Glassema a hao [] o he allao of elas fo gal a oo aple ap ees ae opos h he fameo of he sgle ey fll-a MM I he pese eo e popose o exe he se of he RM algohm o aay pahepee payoffs fo he mlple ey mlfao IBOR mae moel xg a ey a omg he oespog ppe es he yams of he h IBOR foa ae he omes ey a e expesse he fom 5 I he follog he hoe of a eo se h osa a M smlao sep h s mae I a ex sep he assmpo h ms e valae o eee y esg he aay of he MM agas vaos applaos I hs smplfe fameo he Ele sheme of he logahm poess 5 s gve y hee e he oveo l l } { l l 5 hee he s ' ae epee N -mesoal saa omal veos a fom 43 foeg ey 53 Ove a sgle me sep he ovaae max of emes 5 has a N I ase N hs max s sgla Oe oses he so of he emes ove mlple me seps y eg 5 as l l 54 hee he o veos have ee oaeae o a sgle veo of legh N a he olm veos have ee sae o a olm of he same legh epeg o he ype of mlple ey mlfao MM s possle o f a ege sh ha he N x max 55 s assme o have a eve f N ml eglay moel assmpo I hs ase he max s vele a he RM meho [] eo 4 a e apple ppose ha has fll a o allae he ela of a payoff fo x g h espe o he IBOR foa ae e apply he eqaly g E g E val a Gassa seg y mag he follog oespoees l l h } { l hee s he foa-f appoxmao o 53 popose [] eo 33 hh s efe y foeg ey h ape I: O

11 Ieaoal oal of ee a Reseah IR IN Ole: Impa ao : 3358 Volme 3 Isse Noveme 4 se ese Ue eave ommos Ao BY e o g e he smlae pah-se vale of a aay payoff he oe has he follog RM ela pah-se esmao g 56 hee have ee efe aove Mag se of he Ele sheme 54 oe sees ha 57 Wh 57 he seo expesso 56 smplfes o 58 hee he s ae he ees of he vese max of 6 Applao: ey Opo es a elas o llsae e es he M smlao of Eopea ey opo pes a elas fo he oss ey mlfao MM of eo 4 ase agas he asympo expaso fomlas y Kaa a äel [37] he payoff of a Eopea all opo h may ae a se ae K s gve y K hee eoes he spo exhage ae a me Oe oseves ha he spo exhage ae a e expesse ems of a foeg exhage foa foex foa ae h he same may ae ha s oe has hee eoes he me vale of he foex foa ae h may ae We ll mae se of he aage-fee elaoshp eee he foex spo ae a he foex foa ae gve y 6 hee a eoe he me vales of omes a foeg zeo opo os h may espevely he goal s o eeme he all opo pe a me K E 6 a he pe sesvy o ela 63 I 6 he oao E sas fo he expeao h espe o he omes emal mease heefoe o eeme pe a ela he so of e he omes emal mease ms e evalae o hs e se he oss ey mlfao IBOR mae moel as spefe eo 4 ase I hs seg he spo exhage ae sasfes he sohas ffeeal eqao 45 h f 4 a volaly veo eeme y 4 he foex foa ae efe y 6 s a magale mease e he omes emal mease Applyg Iô s fomla o 6 oe sees ha sasfes he follog sohas ffeeal eqao e hs mease } { 64 h as 4 a 65 ose o he asympo expaso [37] Ise a peao paamee o eqao 64 o ge 66 h 67 hee he pee IBOR foa aes sasfy he eqaos 68 h 69 ape I: O

12 Ieaoal oal of ee a Reseah IR IN Ole: Impa ao : 3358 Volme 3 Isse Noveme 4 se ese Ue eave ommos Ao BY a 6 h 6 Oe oes ha he f ems of he foa aes ae appoxmae eemsally sg al foa aes as a oseqee of he Iô-aylo expasos a 6 hs s smla o he foa-f appoxmao aleay apple eo 5 a ogally popose y Glassema a hao [] eo 33 I maes he evao a he eslg fomlas smple hle hey ema aae Applyg a aylo sees expaso o he foex foa ae 66 oe oas asympo expasos of oe 3 efe y! 6 s-oe asympo expaso A saghfoa allao shos ha 63 fom hh follos ha ~ N 64 s omally se h mea zeo a vaae sh ha 65 o pove he eal of he lae osa s sefl o oe he follog axlay qaes efe he me eghs of he al IBOR foa aes y 3 66 a he egae emal ovaaes eee he sohas poesses y 67 he oe has } { } { 68 As a esl leg follos ha he fs oe omal asympo expaso of he all opo pe 6 s gve y K K K 69 hee x s he saa omal so a ' x x e he yams of he foex foa ae s lose o a log-omal so s vey aal o appoxmae fhe he fs oe omal asympo expaso y he follog log-omal so N l ~ 6 } / l{ } { K K B 6 hogh ffeeao oe ges he fs-oe asympo expaso Bla-holes ela K B 6 ape I: O

13 Ieaoal oal of ee a Reseah IR IN Ole: Impa ao : 3358 eo a h-oe asympo expaso ale 4: peal eomposo max B hogh smla moe omplae allaos s possle o eeme seo a h oe omal asympo expasos of he all opo pe as ell as oespog Bla-holes appoxmaos eals of he h-oe asympo expaso ae fo Kaa a äel [37] fomlas A a 3 o llsae he se of hese fomlas a es he aay agas M smlao e have allae a sgle smple oee example ase o he p aa of ales a ale : haaess of opo oa se 9 ae me o may 4 ale : Mae aa omes/foeg foa aes & so faos foa se & ae foa aes volaly foa aes volaly R 47598% 4% R 47598% 4% R3 5489% 4% R3 5489% 4% R % 4% R % 4% so fao so fao oa 9 se oa ae volaly % oelao max e ey oelaos e ey oelaos oss ey oelaos ρ ρ ρ Nmeal evalao of he fomlas 69 6 a 6 as ell as he oespog h-oe asympo expaso fomlas [37] s smmaze ale 3 o M smlao of he foex foa ae aog o 64 s fs eessay o allae he speal eomposo of he oelao max 4 hee B B as gve ale 4 ale 3: Aalyal asympo expaso fomlas Appoxmao fomla Vaae Opo e Opo ela s em of 3 oe om- al a asympo expaso s oe omal asympo expaso a 3 oe log-omal Bla-holes s oe log-omal Bla-holes o ompaso e have ompe M smlaos h o ffee Ele shemes amely h he fs sal oe h h h h h 63 exp h h h h hee 3 W W W W W W he seo Ele sheme s a smple appoxmao of he fs oe hh s oae y applyg foa-f appoxmaos sh ha h h hee exp h 66 h h oes e he assmpo of osa volales h 65 he sezao sep s h a elas ae oae h he pah-se fe ffeee o meho h a paallel shf he me M ' of smlae pahs osss of ahes of smlao pahs eah Resls ae smmaze ale 5 Volme 3 Isse Noveme 4 se ese Ue eave ommos Ao BY ape I: O

14 Ieaoal oal of ee a Reseah IR IN Ole: Impa ao : 3358 ale 5: mlao of pe a ela s Ele sheme Ele sheme e ela e ela M vales a me s oe Bla-holes appox M elave evaos 3% 36% 48% 35% 3 oe Bla-holes appox M elave evaos 5% 6% 68% 5% he opo pes fom he s oe Bla-holes appoxmao ae lose o he M vales ha hose fom he 3 oe Bla-holes appoxmao he oppose s e fo he ela vales Moeove he fs Ele sheme yels lose appoxmaos of he opo pes o he M vales ha he seo oe B o sgfa ffeee eee he Ele shemes s oseve oeg ela 7 olsos a Oloo Alhogh a qe geeal IBOR mae moel has ee pesee a lo of sses ema ope fo vesgao eale mplemeao sses fhe applaos a aefl esg ae lef o he paoe I he follog some moe heoeal oes ae ase lealy he Gassa seg of he geeal IBOR mae moel a e qesoe a hee s a ease ema o go eyo Hoeve he oso of mlvaae o- Gassa poesses fo faal pposes s a omplex op o mae osse valao oe ees o eflae hese poesses h sae pe eflaos A possle appoah o mve omplexy a e ase o omal vaaemea mxes as pesee aho [3] Hee e ol le o emphasze moelg h he fameo of oplas ha ae easgly se all aeas of mahemaal sees eg he e al [8] fo faal applaos o ma sses ae ssse ompaly of oelao Maes Exsee a oso of Uvesal oplas I geeal oe says ha a -mesoal opla s - vesal f evey -mesoal val oelao max a e ealze as a a oelao max e hee exss a -vaae fom so h hs a oelao se lealy hee exs qe a lo of -vesal oplas he so-alle ompehesve o lsve oplas see eg Nelse [43] Alhogh he exsee of 3-vesal oplas has ee sele y seveal ahos eg oe [33] Exese 47 pp Koa a ooe [38] eo 446 p evoye a ea [3] he effeve oso of 3-vesal oplas s moe ffl he aho oas [3] a aalyal 3-vesal opla ha s ase o he vaae lea la opla elma a Welle [46] he lae opla seems o have ee epeely oae y Koa e al [39] e he amg ellpal opla Aog o ea [4] he lea la opla s a speal ase of poaly sos se y Gaspe [9] I s emaale ha hs opla a e se o sele he exsee qeso fo a o exemal oelao maes By heoem 7 elo geealzes o he opla fameo he o-fao MM of Reoao [53] see he Remas 3 he oso hh mms esseally ea [4] s elemeay e R he ee sqae ose he lea la opla esy h fom [-] mags U V efe y v B p U V v v 7 v B emma 7 -vesal a o exeme lea la opla Gve s he exeme oelao max of a o of he fom B e he s a os he aae appoxmao Ĉ of Example sffes fom aohe aa h he opla seg I s a smple example of a posve sem-efe oelao max hh s o he a oelao max of a mlvaae omal opla I fa geeal ompaly oos e hh a o omal so h a spefe a oelao max a e ealze have oly ee eve qe eely aho [6] oollay 4 Oe a as fo o-omal vaae sos fo hh Ĉ s ompale om [6] eo s o ha Ĉ s o ompale h a Beoll mxe vaae eo moel Noe ha he ompaly of hee vaae sos éhe spaes has ee se se all Aglo [] [] see also Rüsheof [56] [57] oe [33] eo 3 a ae e al [5] he ompaly polem leas o o ex sse he hee exs a aom veo h fom [-] mags a a o oelao max oof ose he aom veo efe y os U s V hee he aom pa U V has he lea la opla esy 7 lealy he vaales ae fom [-] aom vaales Moeove hogh applao of he aoa asfomao meho oe sees ha he poaly esy of s gve y Volme 3 Isse Noveme 4 se ese Ue eave ommos Ao BY ape I: O

15 p x y x y E x y x 7 v E Ieaoal oal of ee a Reseah IR IN Ole: Impa ao : 3358 hee he sppo x y x y xy E s he e of a ellpse a os oes h he oelao oeffe of he pa eg Koa e al [39] elma a Welle [46] aho [3] eo 3 heoem 7 -vesal a o opla Gve a a o oelao max hee exs a aom veo h fom [-] mags a a o oelao max oof hs follos hogh applao of he heoem of aahéooy [7] a ez [6] Ay val oelao max of a o s a fe ovex omao of exeme oelao maes of a o e he esl hols fo he exeme oelao maes of a o y emma 7 he esl follos Beses he exsee of a sgle ey o-fao MM of aay meso h aay mags heoem 7 also seles he exsee qeso fo -vesal oplas 345 hs follos ease oelao maes of mesos 34 5 have maxmm a o eg Ya 985 heoem lealy fo smlao a aalyal pposes hee s a ee fo aleave moe expl esls o he vaae ase oe a se he aalyal esls [3] heoem 53 a heoem 6 ally a ompaoal appoah o almos -vesal oplas hh s ase o oly sohas maes a he heeoa opla has ee popose he eos 4 of aaos e al [47] [48] e eseahes shol appeae he hallege o exe lassal IBOR mae moels o a opla seg Refeees [] H A Wllams pal ompoe aalyss WIREs ompaoal ass 4 pp [] Alexae vov asal popees of IBOR foa aes IMA ssso apes 3 [Ole] Avalale: hp://maeea/pf/ssso/3-3pf [Aesse: Nov 4] [3] Aese Aease ao epeee of Bema sapos: fa o fo? oal of aal Eooms 6 pp 3-37 Volme 3 Isse Noveme 4 se ese Ue eave ommos Ao BY [4] Bla he pg of ommoy oas of aal Eooms 3 pp [5] Bgo A oe o oelao a a eo [Ole] Avalale: amaogo/oelpf [Aesse: Nov 4] [6] M Be Haavas Hoffma O he geeao of oelao maes Apple Mahemas E-Noes 8 pp [7] aahéooy Üe e Vaaläseeh e oeshe Kosae vo posve hamoshe oe Re el olo Maem alemo 3 pp [8] U he E ao W Vehao opla Mehos ae Wley ae ees Ne Yo 4 [9] M hesey o g Eopea ey opos: a ompaso of he mofe Bla-holes moel a a aom vaae moel aal a Qaave Aalyss 4 pp [] Q a K gleo em se yams heoy a ealy Reve of aal es 63 pp [] G all Aglo lla ompalà elle fzo epazoe oppa Re Ma 8 pp [] G all Aglo éhe lasses a ompaly of so fos I: ymposa Mahemaa vol I pp Aaem ess oo [3] evoye G ea oplas hee mesos h pese oelaos [Ole] Avalale: hp://axvog/as/4346v [mah] [Aesse: Nov 4] [4] esse Klaasse B Meleeg he pefomae of ml-fao em se moels fo pg a hegg aps a sapos of aal a Qaave Aalyss 383 pp [5] ae E Kleme Qesaa-Mola oplas: ompaly a éhe lasses ep 7 [Ole] Avalale: av:749v [mah] [Aesse: Nov 4] [6] R a A Gpa Rhe Hegg he possle pesee of spae sohas volaly: evee fom sapo maes of ae 585 pp [7] h es Mahemaal ae - heoy Moelg a Implemeao Wley Ne Yo 7 [8] B ly ommo ppal ompoes a elae mlvaae moels Wley 988 [9] G Gaspe Baah algea fo ao sees a posvy of a eel Aals of Mahemas 95 pp [] Glassema hao as Gees y smlao foa o moels oal of ompaoal ae 3 pp [] I Gs R eesz Effe a eo of oelao maes ea Algea a s Applaos 4 pp [] A Gpa M ahmayam g a hegg ees ae opos: evee fom ap-floo maes of Bag a ae 93 pp [3] Heso A lose-fom solo fo opos h sohas volaly h applaos o o a ape I: O459 47

16 Ieaoal oal of ee a Reseah IR IN Ole: Impa ao : 3358 ey opos Reve of aal es 6 pp [4] N Hgham ompg he eaes oelao max a polem fom fae IMA oal of Nmeal Aalyss pp [5] W Hülma A exeso of he Bla-holes a Magae fomlas o a mlple s eoomy Apple Mahemas 4 pp 37-4 [6] W Hülma ompaly oos fo he mlvaae omal opla h gve a oelao max oee oal of heoeal a Apple ass 3 pp 7-86 [7] W Hülma Opo pg he mlmesoal Bla-holes mae h Vase ees aes Mah ae e 3 pp -8 3 [8] W Hülma Opo pg he mlmesoal Bla-holes-Meo mae h Gassa Heah- ao-moo ees aes: he pasmoos a osse Hll-Whe moels of Vase a Nelso- egel ype Mah ae e Vol 4 Ale I 3 4 [9] W Hülma aesa a pola ooaes fo he -mesoal ellpope heoeal Mahemas & Applaos [3] W Hülma Nomal vaae-mea mxes III Opo pg hogh sae-pe eflaos of Mahemaal ees: Avaes a Applaos 5 pp [3] W Hülma A lose-fom vesal vaae paopla oal of asal sos a Applaos 7 5p 4 [3] äel Kahl osve sem-efe oelao max ompleo 9 [Ole] Avalale: hp://paeelespaevgmeaom/osv eemefeoelaomaxompleopf [Aesse: Nov 4] [33] H oe Mlvaae moels a epeee oeps Moogaphs o ass a Apple oaly vol 73 hapma & Hall oo 997 [34] I olffe pal ompoe Aalyss e pge-velag Ne Yo [35] Kahl Moelg a smlao of sohas volaly fae h hess Begshe Uvesä Wppeal a ABN Amo 7 [36] Kahl M Gühe omplee he oelao max 5 [Ole] Avalale: hp://mahppeale/~ahl/plaos/ompleeheoelao Maxpf [Aesse: Nov 4] [37] A Kaa äel A asympo opo fomla he oss ey IBOR mae moel Wllmo Mah Mah 7 [38] Koa MR ooe Ueay aalyss h hgh mesoal epeee moellg Wley hhese Egla 6 [39] Koa MRooe Msez Ellpal oplae I: GI hëlle & paos Es oeegs Moe alo mlao pp 9-4 sse: Balema [4] G ea he ovex se of oelao maes a ao polyomals ofeee eseao Ohogoal olyomals a Hypegops olose e 8 4 Volme 3 Isse Noveme 4 se ese Ue eave ommos Ao BY [4] M m alag a 3-fao IBOR mae moel Eole olyehqe 4 [Ole] Avalale: hp://homepagesye/~ml36/fhes/appoepf [Aesse: Nov 4] [4] ogsaff aa-laa E haz hog aay a llo ollas: he os of sopmal exese saeges he sapos mae oal of aal Eooms 6 pp [43] RB Nelse A oo o oplas ee Noes ass vol 39 eo pge-velag Ne Yo 6 [44] Olve Ohogoal ases a he QR algohm Uvesy of Mesoa 8 [Ole] Avalale: hp://mahme/~olve/ams_/qpf [Aesse: Nov 4] [45] N aham oelao paameezao a alao fo he IBOR mae Moel Ms hess B ae & Maageme af am Ma 5 [46] M elma A Welle qag he le a g he sphee: la a spheal oplas ymmey 3 pp [47] aaos G Hole W Bola Mahg he gae oelao oeffe sg a opla h maxmm soe Is Ma Op 3 pp [48] aaos G Hole W Boe oplas h maxmm eopy Opm e 6 pp 99-5 [49] R eesz Goee Ra eo of oelao maes y maozao Qaave ae 46 pp [5] R eesz A elsse A ompaso of sgle Maov-foal a mlfao mae moels Reve of evaves Reseah 3 pp 45-7 [5] Rapsaa Bgo Meo aameezg oelaos: a geome epeao IMA oal of Maageme Mahemas 8 pp [5] R Reoao Iees-ae Opo Moels Wley Ne Yo 996 [53] R Reoao O he smlaeos alao of mlfao logomal ees ae moels o Bla volales a o he oelao max omp ae pp [54] R Reoao Volaly a oelao e Wley Ne Yo 4 [55] R Reoao äel he mos geeal mehoology o eae a val oelao max fo s maageme a opo pg pposes oal of Rs pp 7-7 [56] Rüsheof Bos fo so h mlvaae magal I: ohas Oes a eso e Rs IM ee Noes Moogaph ees vol 9 Ise Mah as Haya 99 [57] Rüsheof éhe os a he applaos I: G all Aglo Koz G ale Es Avaes oaly sos h Gve Magals Kle oeh 99 [58] R höel h ohas volaly h a Ose Uhlee poess: a Exeso Eopea ae Reve 3 pp ape I: O459 47

17 Ieaoal oal of ee a Reseah IR IN Ole: Impa ao : 3358 [59] hoemaes offey IBOR ae moels elae evaves a alao WIA ep No 48 Weesass Ise Bel 999 [6] hoemaes offey ysema geeao of paame oelao ses fo he IBOR mae moel I heo Apple ae 65 pp [6] E ez Beg ovegee Rehe ovexe yseme oal Ree Age Mah 43 pp [6] amme g of osa may spea opos he eems IBOR mae moel fameo Ms hess ampee Uvesy of ehology la 8 [63] B Ya Exeme pos ovex ses of symme maes oeegs of he Amea Mahemaal oey 954 pp [64] G Yle M Keall A Ioo o he heoy of ass 4h e hales Gff & ompay Belmo alfoa 965 [65] hag W Opmal lo-a appoxmao o a oelao max ea Algea a s Applaos 364 pp Aho ofle Wee Hülma has se mahemas a physs a EH hee he oae hs h 98 h a hess hghe algea Afe posooal felloshps a Yale Uvesy a a he Max la Ise Bo he eame 984 a aay a Weh fe a esos a seo aay a Ao Re a IRMG zela 3-6 a seo osla sess expe a po maage a IRI RGloal zela a Woles Kle aal eves zela üh eee 6-4 He has ee vsg assoae pofesso aaal see a he Uvesy of ooo g he aaem yea Afe ealy eeme Apl 4 he os as a Iepee hola vaos fels of ees maly Algea a Nme heoy oaly a ass Volme 3 Isse Noveme 4 se ese Ue eave ommos Ao BY ape I: O459 47