Abstract. Lorella Fatone 1, Francesca Mariani 2, Maria Cristina Recchioni 3, Francesco Zirilli 4

Transcription

1 Joural of Appled Maheacs ad Physcs Publshed Ole Jue 4 cres. hp:// hp://dx.do.org/.436/jap.4.76 The ABR Model: Explc Forulae of he Moes of he Forward Prces/Raes Varable ad eres Expasos of he Traso Probably Desy ad of he Opo Prces Lorella Faoe Fracesca Mara Mara Crsa Reccho 3 Fracesco Zrll 4 Dpareo d Maeaca e Iforaca Uversá d Caero Va Madoa delle Carcer 9 Caero Ialy Dpareo d ceze Ecooche Uversá degl ud d Veroa Vcolo Capofore Veroa Ialy 3 Dpareo d Maagee Uversá Polecca delle Marche Pazza Marell 8 Acoa Ialy 4 Dpareo d Maeaca G. Caseluovo Uversá d Roa La apeza Pazzale Aldo Moro Roa Ialy Eal: lorella.faoe@uca. fracesca.ara@uvr..c.reccho@uvp. zrll@a.uroa. Receved 9 March 4; revsed 9 Aprl 4; acceped 8 Aprl 4 Copyrgh 4 by auhors ad cefc Research Publshg Ic. Ths work s lcesed uder he Creave Coos Arbuo Ieraoal Lcese (CC BY. hp://creavecoos.org/lceses/by/4./ Absrac The ABR sochasc volaly odel wh β-volaly β є ( ad a absorbg barrer zero posed o he forward prces/raes sochasc process s suded. The presece of (possbly ozero correlao bewee he sochasc dffereals ha appear o he rgh had sde of he odel equaos s cosdered. A seres expaso of he raso probably desy fuco of he odel powers of he correlao coeffce of hese sochasc dffereals s preseed. Explc forulae for he frs hree ers of hs expaso are derved. These forulae are egrals of kow egrads. The zero-h order er of he expaso s a ew egral forula coag oly eleeary fucos of he raso probably desy fuco of he ABR odel whe he correlao coeffce s zero. The expaso s deduced fro he fal value proble for he backward Kologorov equao sasfed by he raso probably desy fuco. Each er of he expaso s defed as he soluo of a fal value proble for a paral dffereal equao. The egral forulae ha gve he soluos of hese fal value probles are based o he Hakel ad o he Koorovch-Lebedev rasfors. Fro he seres expaso of he probably desy fuco we deduce he correspodg expasos of he Europea call ad pu opo prces. Moreover we deduce closed for forulae for he oes of he forward prces/raes varable. The oe forulae obaed do o volve egrals or seres expasos ad are ex- How o ce hs paper: Faoe L. e al. (4 The ABR Model: Explc Forulae of he Moes of he Forward Prces/Raes Varable ad eres Expasos of he Traso Probably Desy ad of he Opo Prces. Joural of Appled Maheacs ad Physcs hp://dx.do.org/.436/jap.4.76

2 pressed usg oly eleeary fucos. The opo prcg forulae are used o sudy syhec ad real daa. I parcular we sudy a e seres (of real daa of fuures prces of he EUR/UD currecy's exchage rae ad of he correspodg opo prces. The webse: hp:// coas aeral cludg aaos a eracve applcao ad a app ha helps he udersadg of he paper. A ore geeral referece o he work of he auhors ad of her coauhors aheacal face s he webse: hp:// Keywords ABR ochasc Volaly Models Opo Prcg pecral Decoposo FX Daa. Iroduco Le us cosder he ABR sochasc volaly odel. Ths odel has bee roduced aheacal face by Haga Kuar Lesewsk Woodward [] o descrbe he e dyacs of forward prces/raes ad s wdely used he facal arkes. + Le be respecvely he ses of real ad of posve real ubers ad le be a real varable ha deoes e. The ABR odel descrbes he dyacs of wo varables: he forward prces/raes varable x > ad he sochasc volaly varable v >. The varables x v > are real sochasc processes ha sasfy he followg syse of sochasc dffereal equaos: where [ ] β dx = x vd W > ( dv = vd Q > ( β ad > are real paraeers. The paraeers β ad of ( ( are called respecvely β -volaly ad volaly of volaly. The choces β = ad β = defe respecvely he oral ad he logoral ABR odels ad are o cosdered here. The oral ad logoral ABR odels have bee wdely suded he scefc leraure (see for exaple []-[9]. I hs paper we resrc our aeo o he sudy of he case β. The sochasc processes W Q > are sadard Weer processes such ha W = Q = dw dq > are her sochasc dffereals ad we assue ha: E dwdq = ρd > (3 ( where E ( deoes he expeced value of ad ρ s a cosa called correlao coeffce. The Equaos ( ( are equpped wh he al codos: x x v = (4 v = (5 where x ad v are rado varables ha we assue o be coceraed a po wh probably oe. For splcy we defy hese rado varables wh he pos where hey are coceraed. Moreover we assue x v >. The assupo v > wh probably oe ad Equao ( ply ha v > wh probably oe for >. I s kow ha whe β he sochasc volaly odel ( ( wh he codos (3 (4 (5 s uderspecfed (see [9] [] []. I fac whe β he org of he forward prces/raes varable x > s accessble fro x > ad he org of he forward prces/raes varable equao ( has o a uque soluo. I order o guaraee he uqueess of he soluo of ( ( (3 (4 (5 ad he o arbrage codo we pose a absorbg barrer zero o he forward prces/raes sochasc process x > (see [] [] for deals. Ths eas ha he pahs of he sochasc process x > ha reach zero are o loger cosdered he e evoluo. The absorbg barrer zero posed o he forward prces/raes varable s oly oe of he codos dscussed he scefc leraure ha ca be used o guaraee uqueess of he soluo of he al value proble ( ( (3 (4 (5. For exaple reflecg barrers ad xed barrers zero have bee suggesed as codos ha guaraee uqueess. We sudy he odel wh 54

3 he absorbg barrer jus for splcy. The resuls obaed here for hs odel ca be exeded o several odels wh oher uqueess codos. The absorbg barrer zero posed o he forward prces/raes process ples ha he e evoluo defed by he odel equaos ( ( does o coserve probably. Despe hs fac we coue o call probably desy fuco he fudaeal soluo of he backward Kolokorov equao assocaed o ( ( ha sasfes he hoogeeous Drchle boudary codo whe he forward prces/raes varable s zero. Ths boudary codo posed o he probably desy fuco correspods o he absorbg barrer zero posed o he forward prces/raes varable. The ABR odel suded hs paper s defed by he equaos ( ( (3 (4 (5 by he codos x > ρ > ad by he absorbg barrer zero posed o he forward prces/raes varable. The pracce of he facal arkes has show ha ay crcusaces hs ABR odel fs sasfacorly he pled volaly curves assocaed o he observed opo prces ad s able o capure he dyacs of he pled volaly sle. Moreover yelds sable hedges of eleeary porfolos bul wh he asse uderlyg he forward prces/raes varable ad s dervave producs (see for exaple [] []. These facs jusfy he use of he ABR odel by he pracoers ad he eres he ABR odel of he research couy. oe approxae expressos of he probably desy fuco of he ABR odel of he correspodg Europea opo prces ad of he pled volaly assocaed o he opo prces are avalable he scefc leraure. These forulae have bee obaed usg several aheacal ehods such as sgular perurbao heory ad hea kerel asypocs (see [] [3] [4]. For exaple a explc forula (volvg a v β oe desoal egral for he raso probably desy fuco of he ABR odel whe β = or β = ad ρ ( has bee obaed [4]. lar resuls are coaed [5] whe β = ρ ad [7] for a odfed ABR odel. I [6] a opo prcg proble s suded. Le = be he curre e T > be he aury e of he opos cosdered ad T be he oal volaly of volaly. The ABR odel for [ T ] s suded ad s derved a seres expaso powers of he oal volaly of volaly of he raso probably desy fuco of he varables x v > of he ABR odel ( ( (3 (4 (5 β [ ] ρ ( whe o codo zero s posed o he forward prces/raes varable [6]. The ers of he expaso powers of T are obaed scalg he varables of he odel ad usg a rasforao of he bvarae oral fuco. Explc forulae are gve for he frs hree ers of he expaso powers of T of he probably desy fuco ad of he correspodg expasos of he Europea opo prces. The dea of posg a absorbg barrer zero o he forward prces/raes varable of he ABR odel s dscussed [3]. I parcular [3] order o prce log daed opos he ABR odel s suggesed he dea of copleg he probably desy fuco deered posg he absorbg barrer zero o he forward prces/raes varable addg a er proporoal o a Drac s dela suppored o he absorbg barrer. The choce of he Drac's dela er resores he probably coservao durg he e evoluo. I hs paper for he prevously specfed ABR odel we deduce a seres expaso powers of he correlao coeffce ρ of he raso probably desy fuco. Explc expressos of he frs hree ers of hs expaso are derved. These ers are egrals of kow egrads. I parcular he zero-h order er of he expaso s a oe desoal egral whose egrad s expressed usg oly eleeary fucos. Ths s a ew forula of he probably desy fuco of he ABR odel whe ρ =. Prevously hs probably desy fuco was kow oly hrough a forula cossg a oe desoal egral of a expresso volvg o eleeary rascedeal fucos [9]. Relaed forulae have bee derved by several auhors. For exaple [7] a forula for he argal dsrbuo of he forward prces/raes varable of he ABR odel whe ρ = s preseed. The ers of he expaso of he probably desy fuco preseed hs paper are egrals of he produc of a fuco depedg o he forward prces/raes varable ad he egrao varable es a fuco depedg o he sochasc volaly varable ad he egrao varable (see for exaple forula (34. The egrao varable geeral s a vecor valued varable ad he corres- podg egral s a uldesoal egral. Furherore we show ha for he -h order er of he expaso powers of ρ of he probably desy fuco of he ABR odel ca be wre as he covoluo of he zero-h order er wh a forcg fuco. The ers of he expaso powers of ρ of he probably desy fuco of he ABR odel are he soluos order by order perurbao heory of he fal value proble for he backward Kologorov equao sasfed by he probably desy fuco of he odel. The paral dffereal operaor ha appears he fal value probles sasfed by he ers of he expaso ca be dagoalzed usg a procedure based o a chage of varables ad o he Hakel ad he Koorovch-Lebedev rasfors [8] [9]. Ths dago- 54

4 alzao procedure akes possble o oba egral forulae for he expaso ers. I parcular he dagoalzao procedure shows ha he zero-h order er of he expaso s a kd of covoluo bewee wo kerels oe depedg fro he rasfored forward prces/raes varable ad he oher depedg fro he sochasc volaly varable. Ths las kerel has already bee used [4] o express he raso probably desy fuco of he ABR odel whe β = or β = ad ρ ( ad [9] [5] o sudy respecvely a odfed ABR odel whe β [ ] ρ = ad whe β = ad ρ (. Prevously he sae kerel has bee used he sudy of he raso probably desy fuco of he e egral of a geoerc Browa oo (see [5] []. Despe he fac ha he ABR odel wh β ad he absorbg barrer eoed above does o coserve probably s coo pracce o use he rsk eural approach o prce opos he ABR odel fraework as expeced values of he dscoued payoff fucos. We follow hs pracce ad we exed he ehod used o derve he expaso powers of ρ of he raso probably desy fuco o deduce he correspodg expasos of he Europea call ad pu opo prces he ABR odel. The ers of hese expasos are egrals of kow egrads. The egrads are expressed as he produc of a fuco depedg fro he forward prces/raes varable ad he egrao varable es a fuco depedg fro he sochasc volaly varable ad he egrao varable. oe of hese egrals are doe aalycally hs guaraees ha (order by order perurbao heory he opo prces ca be obaed evaluag uercally egrals of he sae deso ha hose ha us be evaluaed o oba he raso probably desy fuco. Moreover hese egrals due o he specal srucure of her egrads ca be copued usg ad hoc quadraure rules. The develope of hese ad hoc quadraure rules s beyod our purposes hs paper. Fally we sudy he oes of he forward prces/raes varable. For hese oes we oba closed for forulae ha do o coa egrals or seres expasos. These forulae are polyoals he correlao coeffce ρ. The coeffces of hese polyoals are closed for expressos coag oly eleeary fucos of he reag quaes defg he odel. I [5] ad [6] slar oe forulae have bee obaed for he oral (.e. β = ad for he logoral (.e. β = ABR odels. oe uercal experes o syhec ad o real daa are dscussed. I parcular usg he opo prcg forulae eoed above we sudy he daly values of he fuures prce of he EUR/UD currecy s exchage rae havg aury epeber 6h ad of he daly prces of he correspodg Europea call ad pu = + = 8. The prces K = 8 are expressed UD. More specfcally we sudy he daly closg prces of hese coracs observed a he New York ock Exchage he e perod gog fro epeber 7h o July 9h. The uercal experes dscussed show wo facs. The frs oe s ha whe he ABR odel wh he absorbg barrer zero s cosdered he uercal evaluao wh he Moe Carlo ehod of opo prces ca be copuaoally expasve. I fac he ABR odel he loss of probably durg he e evoluo s a fuco of β ad ρ ad creases whe β creases ad/or ρ decreases. As a cosequece whe β creases ad/or ρ decreases he sze of he Moe Carlo saple used o evaluae opo prces wh a gve accuracy us crease o copesae he probably loss durg he e evoluo. For exaple eco 5 opos wh expry dae epeber 9h ad srke prces K.375.5( s show ha whe β =.6 ρ =.5 a es case for a opo wh e o aury T =.5 years o ge hree correc sgfca dgs he uercal approxao of s prce s ecessary o cosder a Moe Carlo saple of 6 pos. Ths saple s geeraed copug 6 rajecores of ( (. Ths us be copared wh he fac ha he accuracy of he opo prces obaed usg he seres expasos powers of ρ derved hs paper depeds fro ρ ad fro he quadraure rule used he uercal evaluao of he egrals coaed he coeffces of he seres expasos bu s subsaally depede of β. A es case shows ha he e requred o evaluae oe opo prce wh hree correc sgfca dgs o a Cero Iel Core Duo CPU T64 processor s a few es of secods usg he seres expasos derved here. The evaluao wh he Moe Carlo ehod of he sae prce wh he sae accuracy requres abou 5 secods ad he use of a saple geeraed copug 4 rajecores of ( (. The secod fac s ha he ABR odel erpres sasfacorly he e seres of real daa suded ha s he e seres of fuures prces of he EUR/UD currecy s exchage rae ad of he correspodg opo prces. I fac he e perod cosdered ha goes fro epeber 7h o July 9h he calbrao he ABR odel usg as daa he closg values of a day of a se of opo prces o he fuures prces of he EUR/UD currecy s exchage rae observed a he New York ock Exchage shows ha a uque se of paraeer values explas he ere daa se 543

5 cosdered. Moreover he paraeer values resulg fro he calbrao ad he opo prcg forulae are used o forecas opo prces. The coparso bewee forecas opo prces ad opo prces acually observed he arke cofrs he valdy of he odel ad of he calbrao procedure used. The webse: hp:// coas soe auxlary aeral cludg aaos a eracve applcao ad a app ha helps he udersadg of hs paper. A ore geeral referece o he work of he auhors ad of her coauhors aheacal face s he webse: hp:// The reader of he paper s orgazed as follows. I eco we derve he expaso powers of ρ of he raso probably desy fuco assocaed o he ABR odel ( ( (3 (4 (5 wh he prevously specfed absorbg barrer. I eco 3 usg he rsk eural approach we derve he correspodg expasos powers of ρ of he Europea call ad pu opo prces. I eco 4 we derve closed for forulae for he oes of he forward prces/raes varable x >. Fally eco 5 we use he seres expasos of he opo prces derved eco 3 o sudy uercally e seres of syhec ad real daa.. The eres Expaso of he Probably Desy Fuco Le us sudy he raso probably desy fuco of he sochasc processes x v > plcly defed by ( ( (3 (4 (5 ad by he absorbg barrer zero posed o x >... The Ial Value Probles asfed by he Expaso Ters Le us defe he sochasc process: β x ξ = > β [. (6 β Fro Equaos ( ( ad Io's lea follows ha ξ v > sasfy he followg syse of sochasc dffereal equaos: β dξ = v d+ vd W > (7 β ξ The al codos (4 (5 becoe: dv = vd Q >. (8 x β ξ = ξ = (9 β v v. = ( A absorbg barrer zero s posed o he sochasc process ξ >. The barrer posed o ξ > follows fro he aalogous barrer posed o x >. Le p ( xv x v x x v v > > be he raso probably desy fuco of odel ( ( (3 (4 (5 wh he prevously specfed absorbg barrer posed o x > ha s le p ( xv x v x x v v > > be he probably desy fuco of havg x = x v = v gve he fac ha we have x = x v = v whe >. Le p( ξ v ξ v ξ ξ v v > > be he raso probably desy fuco of odel (7 (8 (3 (9 ( wh he absorbg barrer prevously specfed posed o ξ > ha s le p( ξ v ξ v ξ ξ v v > > be he probably desy fuco of havg ξ = ξ v = v gve he fac ha we have ξ = ξ v = v whe >. We have: β β x x p ( xv x v dx d v = p( ξ v ξ v dξ d v ξ = ξ = β β ( xx vv > >. Forula ( shows ha he seres expaso powers of ρ of expaso powers of ρ of p. p ca be easly deduced fro he seres 544

6 Le us deduce he seres expaso powers of ρ of p. The fuco p s he soluo of he backward Kologorov equao assocaed o (7 (8 ha s: wh fal codo: ad boudary codo: p v p v p p β v p = + + ξ v ξ v β ξ ξ ρv ξ v > > > ( p ξ v ξ v = δ ξ ξ δ v v ξξ vv > (3 ( ξ p v v = ξ vv > > > (4 where δ s he Drac's dela. The Drchle boudary codo (4 poses o he fuco p he codo ha correspods o he absorbg barrer zero posed o he sochasc process ξ >. Noe ha p does o deped fro ad separaely depeds oly fro s = >. Le us roduce he fuco p ( s ξ v ξ v = p( ξ v ξ v where s = ξξ vv >. Fro ( (3 (4 follows ha p s he soluo of he paral dffereal equao: wh al codo: ad boudary codo: Le us assue ha: p v p v p p β v p = + + ρ v ξ vs > ξ v ξ v β ξ ξ (5 p ξ v ξ v = δ ξ ξ δ v v ξξ vv > (6 ( ξ = >. (7 p s v v ξ vv s = p s ξ v ξ v = p s ξ v ξ v ρ ξ ξ v v s > (8 where he fucos p = do o deped fro ρ. ubsug he seres (8 (5 (6 (7 dffereag (8 er by er ad equag he coeffces of he ers of he sae degree ρ we oba he followg probles: ad p v p v p β v p = + ξ vs > ξ v β ξ ξ p ξvξ v δ ξ ξ δ v v ξξ vv (9 = > ( ( ξ = > ( p s v v ξ vv s p v p v p β v p p = + + v ξ vs > = ( ξ v β ξ ξ ξ v p ξ v ξ v = ξξ vv > = (3 ( ξ Moreover fro (9 ( ( ad ( (3 (4 we have: p s v v = ξ vv s> =. (4 s p p ( s ξ v ξ v = dτ d d vp ( s v v v ( v v ξ τξ ξ τξ ξ ξ v ξξ vv s> =. (5 545

7 Forula (5 s oe of he forulae aouced he Iroduco. I fac for forula (5 gves as he covoluo of he zero-h order er of he expaso p wh he forcg fuco p ξ v... The Zero-h Order Ter of he Expaso Le be he se of coplex ubers be he agary u ad le = ( ( β β [ ha whe β [ we have [. Noe. The forulae ha follow uless dversely specfed hold for >. Le us sudy proble (9 ( (. The fuco p soluo of (9 ( ( ca be wre as follows: p s ξ v ξ v = ξ d λj λξ C s λ v ξ v ξ ξ v v s > (6 where J s he frs kd Bessel fuco of dex (see [] pag. 358 ad C s a fuco o be deered. I s easy o see ha whe he egral coaed (6 ad s egrad are well behaved he fuco p gve by (6 sasfes he boudary codo (. I fac whe ξ = we have ξ J ( λξ = λ >. I order o deere he fuco C of (6 le us pose equao (9 uder he egral sg (6. We have: C v v v C ξ J ( λξ = ( ξ J( λξ C + ( ξ J ( λξ C + ξ J ( λξ ξ ξ ξ v ξ vs > λ >. Usg [] page 36 forula 9..5 Equao (7 becoes: ad fro (6 we have: C v v C ξ J ( λξ = λ ξ J ( λξ C + ξ J ( λξ ξ vs > λ > v C v v C = λ C + vs > λ >. v Fro [] page 374 forula 9.6. ad (9 follows ha we have: p (7 (8 (9 C ca be wre as a Laplace rasfor ha s s 8 ω s λ C s λ v ξ v = e dωe ω v K ( ω v D λωξ v (3 ξ vv s> λ > where s he agary u K ω s he secod kd odfed Bessel fuco of dex ω also kow as Macdoald fuco (see [] pag. 374 ad D s a fuco o be deered. Fro (6 ad (3 we have: s 8 ω s λ p s ξ v ξ v = e vξ dλj d e (. λξ ω ωk ω λ ω ξ ξ ξ v D v vv s > (3 λωξ ξ v > λω > we pose o To do hs we recall he followg forulae (see [] eco.: To deere he fuco D ( v ad (see [4] [3] ( ξ δ ξ ξ J p he al codo (. = d λλ J λξ λξ ξ ξ > (3 δ ( v v = dωω sh ( π ω Re π K ω β ω β v K v v v > β β > v where Re( s he real par of he coplex uber ad sh deoes he hyperbolc se fuco. Fro (6 (3 (3 (33 we have: p s ξ v ξ v = d λl λξξ V s λ vv ξξ vv s> (34 (33 546

8 where L ad V are gve by: ξ L λ ξ ξ = λ J λξ J λξ ξ ξ > λ > ξ (35 v V s vv u su y v v vv s> λ > y v v vv λ + ycosh( u v y v ( λ = d Θ ( de e e (36 where cosh deoes he hyperbolc cose fuco ad Θ s gve by: π ( s s 8 e u ( s πu Θ ( su = e e sh ( u s us. > π π s s Fro [4] page 46 forula 5 we have: λ K v v vv cosh ( u v v λ + + V ( s λ vv = d u ( su v Θ (38 v + v + vv cosh u vv s> λ >. Copug explcly he egrals he λ ad y varables coaed respecvely forulae (34 ad (36 we have: where ad ( ξ ξ p s v v ξξ vv s> ( q( u ξ v ξ v + ( u ξ v ξ v ( ( ξ ξ + ( ξ ξ v v = ξ ξ d u ( su ( u v v ξ ξ v Θ (39 q u v v u v v q u ξ v ξ v = v + v + vv cosh u + ξ + ξ ξξ v v u > (4 4 u ξ v ξ v = q u ξ v ξ v 4 ξ ξ ξ ξ v v u >. (4 I s easy o see ha he fuco 4 (37 q u ξ v ξ v 4ξξ s posve for u ξ ξ v v >. Moreover he fuco p defed (?? sasfes he boudary codo (. I fac whe ξ = he er ξ s zero ad he fucos q ad are bouded v v for vv > ξ u (.e. he fucos q ad are well behaved. Forula (39 of p expresses he probably desy fuco of he ABR odel whe ρ = usg oly eleeary fucos. Ths las fac akes he uercal evaluao of (39 easy ad effce. Prevously oly a forula of p as a oe desoal egral of a egrad volvg o eleeary rascedeal fucos was kow [9]. Noe ha usg ( fro forula (39 ha gves p expressed he varables ξ v s easy o deduce he correspodg forula of he probably desy fuco of he ABR odel whe ρ = expressed he varables x v. Tha s forula (39 ad he aalogous forula he varables x v > are he forulae ha have bee aouced he Iroduco for he probably desy fuco whe ρ =. For laer coveece he fuco V defed (38 s rewre as follows: where Θ s gve by (37 ad V s λ vv = d uθ su M u λ vv vv s> λ > (4 547

9 λ K v v vv cosh ( u λ v v + + M ( u λ vv = v v + v + vv cosh u vv > u λ >. V Moreover le us defe D = vv s> λ >. Usg he dees sasfed by he Mcdoald v fucos (see [] pag. 376 forula we have: where D ( s λ vv = V ( s λ vv = d uθ ( ( su M uλ vv v v 3 = d uθ ( su M ( u vv M ( u vv vv s λ + λ > λ > v λ K v v vv cosh ( u λ v v + + M ( u λ vv = ( v vcosh ( u + v v + v + vv cosh ( u vv > u λ >. Forulae (44 (45 wll be used laer..3. The Frs ad ecod Order Ters of he Expaso Le us cosder he fucos p p defed (8. Proceedg as eco. ca be show ha chagg he egrao order (5 whe = we have: ( ξ ξ = dλ d λ ( ( λ λ ξ ξ λ λ (46 p s v v L V s vv ξξ vv s> where he fucos L ad V are gve by: ad ( λ λ ξξ ξ ( ( λ ξξ λ ξξ ξξ λ λ L = d L L > > ξ s ( λ λ = τ ( τλ ( τλ V s vv d d vvv s vv V vv v (48 vv s> λ λ >. larly chagg he egrao order (5 whe = we have: ( ξ ξ = dλ dλ d λ ( ( λ λ λ ξ ξ λ λ λ (49 p s v v L V s vv ξξ vv s> where he fucos L ad V are gve by: ad ( λ λ λ ξ ξ = ξ ( λ ξ ξ ( λ λ ξ ξ L d L L ξ (5 ξξ > λ λ λ > s ( λ λ λ = τ ( ( τλ ( τλ λ V s vv d d v v V s vv V v v v vv s> λ λ λ >. (43 (44 (45 (47 (5 548

10 ad Usg (35 ad he properes of he fucos J (see [] p. 36 forula 9..7 we have: where ad ξ L λ λ ξ ξ λ J λξ J λ ξ Q λ λ ξ ξ > λ λ > (5 = ξ ξ L ( λ λ λ ξ ξ = λ J ( λξ J ( λξ Q ( λ λ Q ( λ λ ξ ξξ > λ λ λ > (53 Q λη = λ d ξξj ηξ J λξ λη >. (54 Usg (43 (44 we ca rewre he fucos V V defed (48 (5 as follows: s ( λ λ = τ ( τλ ( τλ V s vv d d vvv s vv D vv vv s> λ λ > s ( ( τ λ λ λ = τ τ λ τ ( τ τ λ ( τ λ V s vv d d v v V s vv d d vvd v v D vv (56 vv s> λ λ λ >. The forulae (46 (49 for p p are ew. lar forulae ca be deduced for he hgher order ers of he expaso ha s for he fucos p 3. These forulae becoe ore ad ore volved whe creases. Noe ha gve s > o approxae he egrals (55 (56 usg a quadraure rule s suffce o evaluae he fucos V ( vv D τλ vv o a grd of he se τλ ad ( τλ vv ( τλ vv [ s ] [ [ [ { }. Ths eas ha explog he srucure of he egrads of (55 (56 ad hoc quadraure rules ca be bul o evaluae effcely he egrals (55 (56. We do o cosder he proble of buldg hese quadraure rules here. 3. The eres Expaso of he Opo Prces To prce Europea call ad pu opos he ABR odel we use he o arbrage prcg heory. Le us assue ha he rsk free eres rae s cosa e. Ths hypohess guaraees ha he forward prces/raes varable x > s a argale uder he rsk-eural easure (see [5] Proposo 3.. Tha s hs case he rsk eural easure used o copue he opo prces cocdes wh he physcal easure used o descrbe he dyacs of x v > defed plcly by ( ( (3 (4 (5 wh he absorbg barrer zero posed o he varable x >. Le C ad P be respecvely he prces a e = of a Europea call ad pu opo havg aury e T > ad srke prce E >. Uder he assupo of cosa rsk free eres rae he o arbrage heory ples ha: rt + C x v E T = e dx x E dvp x v T x v x v E T > (57 rt + where ( = ax ( ad r s he rsk free eres rae. + P x v E T = e dx E x dvp x v T x v x v E T > (58 Usg he chage of varable (6 ad forula ( he prces rewre as follows: C ad (55 P defed by (57 ad (58 ca be 549

11 where E = E. rt e C ( x v E T = dξ ξ E d vp T ξ v ξ v ( ( + ξ v ET > ( rt e P ( x v E T = dξ E ξ d vp T ξ v ξ v ( + ξ v ET > I aalogy wh he aalyss of eco. le us deduce he frs hree ers of he expaso powers of ρ of C ad P. We beg cosderg he expaso powers of ρ of P. ubsug (8 o Equao (6 ad egrag er by er he resulg seres we oba he followg forula: where e P x v E T P v E T rt = ρ ξ = x v ET > ( ( ( ξ = ξ( ξ ( ξ ξ P v E T d E d vp T v v + ξ > = v E T Le us recall he followg forulae (see [] pag. 484 forulae pag. 486 forula.4.7: x yy J y x J x x (59 (6 (6 (6 d = > Re > (63 x yy J ( y x J x x Γ + ( ( d + = Re. > > (64 ubsug (34 (35 (6 ad usg forulae (63 (64 whe = we have: ( ξ ξ dλλ ( λξ = E J ( λξ ( = P v E T J ξ ( ( ( dξ E ξ d vv T λ v v ξ d λj λξ B λ E W T λ v v E T > where ( ad E ξ = E ad he fucos B ad W are gve by: E λ + B ( λ E = E ( E J ( λe ( E J ( λe λ E + + > Γ ( W Tv λ = d vv T λ v v Tv > λ >. (67 ubsug (46 (5 (6 whe = we have: ( ξ ξ dλ λ ( λ ξ P v E T = J ( λ λ Q d λ B λ E W Tv λ λ ξ v E T> (68 λ (65 (66 55

12 where W s gve by: W Tv λ λ = d vv T λ λ v v Tv > λ λ >. (69 Fally subsug (46 (53 (6 whe = we have: ( = ( ( ( λ λ Q P v E T J Q B E where W s gve by: ξ ξ dλλ λξ d λ λ λ d λ λ λ W Tv λ λ λ ξ v E T > W Tv λ λ λ = d vv T λ λ λ v v Tv > λ λ λ >. (7 Le us deduce he expaso powers of ρ of he Europea call opo prce C correspodg o he expaso (6 of he Europea pu opo prce P. ubsug (8 (59 ad egrag er by er he resulg seres we oba he followg forula: where I s easy o see ha: C x e v E T C v E T rt = ρ ξ = ξ v ET > ( ( ( = ξ( ξ ( ξ ξ C x v E T d E d vp T v v + ξ v ET > = (7 (7 (73 rt e C ( x v ET P( x v ET = dv d ξ ξ E p T ξ v ξ v ( ( x v ET >. (74 Relao (74 s he aalogous he ABR odel coex of he well kow pu-call pary relao of aheacal face. ubsug he expasos (8 (6 (7 (74 ad posg (74 order by order powers of ρ we oba he followg forulae: ( ξ ( ξ v ξ( ξ E p( T ξ v ξ v ( T ξ v E ( T ξ v C v E T P v E T = d d = ξ v E T > = where he fucos = are gve by: ( ξ = ξ ( ξ ξ T v dv d p T v v T ξ v > = ( ξ = ξξ ( ξ ξ T v dv d p T v v T ξ v > =. Forula (75 s sply he pu-call pary relao (74 wre order by order powers of ρ. Fro forulae (34 (35 (36 [4] forula (9 pag. 9 forula (34 pag. 79 ad [6] forula (.3 we have: (75 (76 (77 55

13 ( ξ ξ ( v v vvcosh ( u ξ v + v + vv cosh ( u T v = v v d uθ Tu d v T ξ v > ( where Θ s gve by (37. Fro forulae (34 (35 (36 [4] forula (8 pag. 97 forula (8 pag. 46 forula 37 pag. 9 ad [6] forula (.3 we have: (78 T ξ v = ξ T ξ v >. (79 ubsug forulae (46 (49 forulae (76 ad (77 follows ha he fucos ad = sasfy he followg recursve relao: T j j ( T ξ v = dτ d d vv p( T v v ( v ξ τξ ξ τξ ξ v T ξ v > j = =. ubsug (79 (8 whe j = we have: Forulae (79 (8 ad (8 ply ha: ( ξ (8 T v = T ξ v > =. (8 ( = > ( dv d ξξ p T ξ v ξ v ξ T ξ v ρ. (8 Fro (78 (8 (8 (65 (68 (7 ad (75 s possble o oba forulae for C = aalogous o he forulae (65 (68 (7 obaed for P =. For he ers P C 3 expressos aalogous o he oes obaed for he ers wh = ca be deduced. These forulae becoe ore ad ore volved whe creases ad are oed for splcy. 4. The he Forward Prces/Raes Moe Forulae. Le Le us cosder he oes of he forward prces/raes varable x > sx v ρ s x v > > ρ ( [ be he -h order oe of he forward prces/raes varable x > = ha s: Fro ( ad (8 we have: where ( s x v ρ = v xx p ( x v x v > > ρ ( [ = d d sx v. sx v s v ( ( ρ = ρ ξ = β x s ξ = x v > > ρ ( [ = β ( ξ = ξ( ξ ( ξ ξ s v dv d p s v v β x s ξ = x v > = =. β Recall ha eco 3 we have already cosdered he fucos ad = ad ha hese fucos have bee expressed wh he forulae (78 (79 (8 (8. Fro Equaos (9 ( ( ( (3 (4 we oba he followg probles: (83 (84 (85 55

14 for he fucos = we have: ( v v v = + ξ v ξ ξ s ξ v > = (86 ξ v = ξ ξ v > = (87 for he fucos = = we have: I s easy o see ha: s v = s> v = (88 ( v v v = + + v ξ v ξ ξ ξ v s ξ v > = = ( ξ (89 v = ξ v > = = (9 s v = sv > = =. (9 s ξ v = ξ s ξ v (9 s he soluo of proble (86 (87 (88 whe =. ubsug (9 he Equao (89 we oba: ( ξ ubsug forulae (9 (93 (84 whe = we have: β Recall ha ξ x ( β ( s v = s ξ v =. (93 ( sx v ρ = x sx v > > ρ ( [. (94 ( = = x. We seek = 3 soluo of proble (86 (87 (88 he followg for: where for 3 of x s zero whe ( ξ = ξ j ( j ξ ξ > = 3 (95 j= s v R sv vs = he dex ( x =. I s easy o see ha f ( = s a eger such ha he fuco expressed as a fuco = = 3 sasfes he equaly: he fuco = 3 gve by (95 s zero whe ha sasfes (96 s: where [ ] deoes he eger par of. Moreover fro (87 follows ha: > = 3 (96 x =. The larges eger = 3 = = + = 3 (97 R v R v v j (98 = = > = = 3. j ubsug (95 o equao (86 ad equag he coeffces of he powers of ξ of he sae degree we oba he followg al value probles: for he fucos R j = = 3 we have: j R R = vs > = 3 (99 v v R v = v > = 3 ( 553

15 ad R R R R v = + ρ ( v vs > v v ( = 3 R v = v > = 3 ( j v j v = + ( j+ ( j+ R j+ v vs> = j= 3 3 R j (3 R v = v > = 3 j = 3. (4 The soluos of he probles (99 ( ad (4 ( are respecvely: R sv = vs > = 3 (5 R sv = vs > = 3 (6 ad he soluos of he probles (3 (4 are: s R ( sv = ( j+ ( ( j+ dτ d v ( s τ vv ( v R ( τ v j j Ψ + (7 vs > = 3 j= 3 where he fuco: s 8 ( l( v l( v s v e Ψ ( svv = e vv s> (8 v v π s s he soluo of he followg proble: Ψ v Ψ = vs > (9 v δ For laer coveece oe ha a eleeary copuao gves: Ψ vv = v v vv >. ( q q s 8 s ( q 8 d v Ψ svv v = v e e vs > q=. ( Usg equaos (5 (6 (7 we oba he followg forulae: s R ( sv = ( j+ ( ( j+ dτ d v ( s τ vv ( v R ( τ v j j Ψ + ( vs > j= 4 = 3 ad R sv = vs > = 3 j = 3 +. j (3 Forula ( reduces o (9 whe =. I fac fro ( whe = ad j = we have R sv = ad hs las forula ples ha R ( sv = v s > j = 3 4. j Usg forulae ( (5 (6 ad ( we have: 554

16 where he fucos ad recursvely: j j = ( + (( ( j ( R sv j j v f s vs j f j > = = 3 (4 R sv = vs > j = = 3 j (5 j = ca be copued by recurso. I fac we have: ( f s = s > > (6 s ( f s = e s > > (7 5 6 s s 6 e e s f4 ( s = e + s > > ( j j s s j j τ j = τ τ j f s e d f e s > > j = 3 4. Noe ha gve (8 he egral o he rgh had sde of (9 ha defes recursvely f j j = 3 4 s a eleeary egral. However s easy o see ha hs egrao becoes cubersoe whe j creases. I hs case sybolc egrao sofware ools ca be used o copue he egral of (9. Fro (84 (95 (5 (6 ( we oba he followg forula: ( j x j j j= j ( [ [ ( sx v = v f s j+ j x v s > > = 3. Noe ha he oe forulae ( we have β x ρ ρ We defe he real varable z = v = ξ v x ξ v > ad we express he oe β ( sxv ρ (8 (9 ( ρ =. Le us deduce he oe forulae for ρ. = 3 usg he varables z v > sead of he varables xv > used up o ow. For 3 c = le ( szv ρ z vs > > ρ ( [ be he oes wre usg he varables s z v ρ. Fro ( follows ha he fucos = 3 sasfy he followg equao: wh al codo: ad boudary codo: ( c c c c v v v = ( ρ + z v ρ z z+ v ρ z> vvs > > ρ ( [ = 3 c ρ ρ ( zv ρ = z+ v z> v v > > ρ = 3 [ c ( ( 555

17 c ρ s vv ρ = vs > > ρ = 3. [ c The boudary codo (3 raslaes o he fucos = 3 he codo posed o he varable x > prescrbg he absorbg barrer zero. Noe ha we do o cosder he case = because (94 we have already show ha ( sxv ρ = x sxv > > ρ ( [. We seek he soluo of proble ( ( (3 he followg for: c ( ρ j j= j ρ ρ s z v = z+ v R s v z+ v ρ z> vvs > > ρ ( [ = 3 where (4 as already posed (97 he sudy of he case ρ = we have = ( = ( + = 3. I s easy o see ha fro ( follows ha he fucos R j j = = 3 sasfy he al codos: (3 (4 R v = v > = 3 (5 Rj v = v > j = = 3. (6 ρ ubsug (4 equao ( ad equag he coeffces of he powers of z+ v of he sae degree we deduce ha he fucos R j j = = 3 sasfy he followg al value probles: R R = vs > = 3 (7 v v R v = v > = 3 (8 ad R R R v = + ρ ( v vs > = 3 (9 v v R v = v > = 3 (3 R j v R R j j+ v = + ρ ( j+ v + ( j+ ( j+ R j+ v v vs > j= 3 = 3 (3 R v = v > j = 3 = 3. (3 j Usg (8 ad ( s easy o see ha he fucos R R R R = 3 are gve by: R sv = vs > = 3 (33 R sv = vs > = 3 (

18 s R ( sv e ( ( 3 v = vs > = ( s s 3 s v e e e R ( sv = 8ρ( ( vs > = 3. = we have ples ha Noe ha whe R sv = v s > ad ha due o he recursve relao (3 hs R sv = v s > j = 3 4. j Fally subsug (33 (34 (35 (36 (3 we have: where he fucos b j j ( ρ j j (36 R sv = vb s vs > j = 3 4 = (37 j = = [ are defed by he followg recursve relao: ( ρ = > > ρ ( [ b s s = ( ρ = > > ρ ( [ b s s = s e b ( s ρ = ( ( s > > ρ = [ 3 3 s s s e e e b3 ( s ρ = 8ρ( ( s > > 3 3 ρ = s ( ρ = ( + ( ( + τ ( τ ρ s + ρ j+ j τb j τ ρ [ j j s j j τ bj s j j e d b j e j j s j j τ e d e s > > ρ j = 45 =. The egrals coaed (4 are eleeary egrals ha ca be copued usg forula (. The copuao of hese egrals s cubersoe ad ca be doe coveely usg sybolc egrao sofware ools. Noe ha he fucos bj ( s ρ s > > ρ ( [ j = = are polyoals ρ. Fro (4 ad (37 we have: ( j j ( sx v = x v b s x j j= ( ( ( j ρ ρ [ x v s > > ρ = = + = 3. I parcular whe s a posve eger forula (43 reduces o: ( + j + j ( sx v = v b ( s x j j= [ ( j ( ρ ρ x v s > > ρ = 3. Forulae (94 (43 (44 are he oe forulae aouced he Iroduco. These forulae are fe sus of eleeary fucos parcular are polyoals ρ ad are easy o copue. They ca be (38 (39 (4 (4 (4 (43 (44 557

19 used ay crcusaces for exaple [5] [6] slar forulae have bee used o sudy calbrao probles for he oral ad logoral ABR odels. 5. oe Nuercal Experes I he uercal experes preseed hs eco we use he dpo quadraure rule o approxae he egrals coaed he forulae deduced ecos ad 3. Le us beg choosg he paraeer values of he uercal quadraures doe he experes. Le N u N λ be posve egers ad u ax λ ax be posve cosas le us defe: uax u = ( = Nu (45 N u λax λ = ( = Nλ. (46 N λ N λ The pos u = Nu ad λ = defed (45 (46 are respecvely he odes of he dpo quadraure rule wh N u ad N λ odes appled o he ervals [ u ax ] ad [ λ ax ]. Le us defe he fucos as follows: Θ p d u pu p Θ = Θ > (47 pλ ( ξ + ξ ( 4 p ξξ ( p ξξ = dλλj ( ξλ J ( ξλ e = e I ξξ p > (48 p p where I ( s he frs kd odfed Bessel fuco of order. We have Θ ( p = p > > ad we choose: p = p =.5 = ξ = ˆ ξ = ξ = ξ f j =.5( j j =. We evaluae he fucos Θ ( p ad ( ˆ p ξξ f j = ˆ ξ = j = approxag (47 (48 usg he dpo quadraure rule he erval [ u ax ] he egral he u varable (.e. (47 ad he erval [ λ ax ] he egral he λ varable (.e. (48. We deoe hese approxaos of Θ ( p ˆ ξξ a a respecvely wh ˆ ξξ = ˆ ξ = j =. The ( p f j uber of odes u a Θ p guaraee ha Θ p ad ( p f j N N λ ad he cosas u ax λ ax of he uercal quadraure are chose order o a ˆ ξξ = ˆ ξ = j = have a leas sx correc ad ( p f j sgfca dgs. Table shows he quay: Θ = Θ Θ = a E p p (49 as a fuco of. The values of E Θ for =..4.6 have bee copued choosg u ax = 6π ad N u = 5 (see Table. Recall ha for > we have Θ ( p = = hs akes easy o deduce fro he value of E Θ show Table he uber of correc sgfca dgs of he =. approxao of Θ p Θ p Table. E Θ versus ad N u. E ( N u = Θ E ( N u = 5 Θ E ( N u = Θ

20 Le us cosder he quay defed (48 whe 3 = we have ( ˆ p ξξ f j > = ˆ ξ = j = oreover he uercal evaluao of (48 shows ha he axu value of ( ˆ p ξξ f j = ˆ ξ = j = s approxaely equal o 5 3. Table shows he quay: E ( ˆ a p ( ˆ ξξ f j p ξξ f j ( p ˆ ξξ f j = = j= 3 for = ad has bee copued choosg λ ax = N λ = he uercal quadraures. I he experes dscussed he res of hs eco we choose u ax = 6π N u = 5 λ ax = N λ =. We prese hree experes. The frs expere vesgaes how he oal probably of he ABR odel depeds fro he e s = whe = ad fro he paraeers β ad ρ. The secod expere vesgaes he covergece of he seres expaso powers of ρ of he Europea pu opo prce ad verfes uercally oe of he oe forulae deduced eco 4. The hrd expere roduces a calbrao procedure for he ABR odel based o he forulae of he opo prces deduced eco 3 ad sudes a e seres of real daa. I he frs expere we beg assug ρ =. Iegrag forula (39 wh respec o he varables ξ v he doa [ [ ad choosg = ξ = ξ v = v we have he followg expresso for he oal probably of he ABR odel: ( ξ = dξ d ( ξ ξ p s v vp s v v M β Θ ( su ( ( = 4v v du d v v s > ξ ξ v ηuv v ηuv v + ξ ( where s = = ad η ( u v v = v + v + vv cosh ( u v v > u. (5 I he asse prce odels where he probably s coserved durg he e evoluo he quay aalogous o pm β s decally equal o oe. However he ABR odel cosdered here due o he absorbg barrer zero posed o he varable ξ > he probably s o coserved durg he e evoluo. I fac he loss of oal probably s ooocally creasg e ad depeds fro ξ v. I parcular gve v he loss of probably creases whe ξ goes o zero. Furherore forula (5 ad he relao = ( ( β ply ha he loss of oal probably creases whe β goes o oe. Fgure shows p as a fuco of M β ( ξv whe ( ξ v [ ] [ ] for β =..5.9 s = year 5 years years ad =.6. I parcular Fgure shows ha he loss of oal probably he ABR odel wh ρ = ad he absorbg barrer zero posed o he varable ξ > whe β s close o oe s eglgble oly for very sall e values. Le us cosder ow he oal probably of he ABR odel he case ρ ad deoe wh pm ( s ρβ ξ v he oal probably of he ABR odel as a fuco of ρ ( β s ξ v >. To copue he oal probably pm ρβ whe ρ we copue he egral ha gves pm ρβ wh he Moe Carlo ehod. Ths s doe egrag uercally he sochasc dffereal equaos (7 (8 wh he al codos (9 (. Due o he absorbg barrer zero posed o he varable ξ > he uercal copuao of a rajecory of (7 (8 (9 ( s sopped whe he varable ξ > hs zero. The uercal egrao of (7 (8 (9 ( s repeaed he uber of es eeded o buld he Moe Carlo saple ecessary o approxae he egral ha gves he oal probably. The loss of probably s easured usg he quay L ρ defed as follows: Lρ = ( pm ( x v ρβ (53 = (5 (5 559

21 Table. E versus. β ( ( = E Fgure. Loss of probably. where x =.5 +. = v =.5. Table 3 shows L ρ as a fuco of ρ whe β =.5 =.6 = year. The prevous aalyss shows ha praccal crcusaces he use for large e values of he ABR odel wh he absorbg barrer ca lead o erroeous judgees. To address hs po several auhors have suggesed he dea of addg o he probably desy fuco suded eco soe exra ers suppored ξ = o resore probably coservao. I [3] [7] [5] he large e asypoc properes of he ABR odel ad of several odels relaed o he ABR odel are suded ad he dea of resorg probably coservao addg a er suppored ξ = s vesgaed. The secod expere sudes he behavour of he seres expaso of he prce a e = of a Europea T =.5 years ad srke prce E = whe he values of he pu opo havg e o aury half a year 56

22 Table 3. Loss of oal probably as a fuco of ρ whe β =.5 =.6 s = year v =.5. ρ L ρ forward prces/raes varable are geeraed egrag uercally he odel (7 (8 wh =.8 v =.5 β =..6 ρ =.5.5 ad ξ = ξ =.5 +. =. We use he frs hree ers of he seres expaso powers of ρ of he Europea pu opo prce P ha have bee derved eco 3. We deoe he approxae opo prces obaed hs way wh e P ( ξ v E T = ρ P ( = ξ v E T = where P = are gve (6. We e copare he prces P ( v E T ξ = wh he prce PMC ρ copued evaluag (6 usg he Moe Carlo ehod. The rajecores of he ABR odel used o saple he varables ξ T v T he Moe Carlo copuao of he opo prces are obaed egrag uercally he sochasc dffereal equaos (7 (8 wh he codos (9 ( usg he explc Euler ehod wh varable sep-sze. Noe ha due o he absorbg barrer zero posed o he forward prces/raes varable he copuao of a sulaed rajecory of he ABR odel s sopped whe he varable ξ > hs zero. We choose he sze N MC of he Moe Carlo saple usg forula (8 as a es case. Tha s we copue a T ξ v T ξ v > he resul he egral (8 usg he Moe Carlo ehod ad we deoe wh a obaed hs way. Noe ha for > ρ ( [ he quay ( T v T v > ξ approxaes ξ T ξ v = ξ T ξ v > (see (79. Le us defe he quay: E MC = ( a ξ ( ξ ( T ξ v T v T v T ξ v >. (54 MC 3 Table 4 shows he saple sze N MC ha akes E saller ha whe ξ = ξ =.5 +. = v =.5 T =.5 =.6 β =..6.9 ad ρ =.5.5. I parcular Table 4 shows ha he accuracy of he Moe Carlo copuao depeds srogly fro he values of he paraeers β ad ρ. Ths s due o he loss of oal probably ha akes place durg he e evoluo ad o he fac ha hs loss s parcularly severe whe β s close o oe ad/or ρ s close o us oe. As suggesed Table 4 o copue he prce P a e = of he prevously specfed opo we choose he Moe Carlo saple sze N MC = 4. Ths choce guaraees ha he Moe Carlo approxaos of he opo prces obaed he expere have a leas hree correc sgfca dgs whe he opo prces cosdered are greaer ha 5 ad a leas wo correc sgfca dgs whe he opo prces cosdered are saller ha 5. The evaluao of oe of he Europea pu opo prces cosdered above usg a Cero Iel Core Duo CPU T64 processor ad he Moe Carlo ehod wh a saple of sze N MC = 4 akes abou 5 secods whle he evaluao of forula (6 wh he sae processor usg he dpo quadraure rule (wh he prevously chose values of he uercal egrao paraeers akes 44 56

23 Table 4. Moe Carlo saple sze N MC requred o have MC E 3. β ρ N MC secods o produce he approxae prce of he pu opo obaed sug he frs hree order ers of he seres expaso (6 8 secods o produce he prce obaed sug he frs wo order ers ad secods o produce he prce obaed usg oly he zero-h order er. Noe ha whe ρ o ge wo or hree correc sgfca dgs he pu opo prce suded s ecessary o use he frs wo order ers or he frs hree order ers of he seres expaso (6 depedg fro he value of ρ. Le us defe he relave errors: e ρ e P PMC ρ = =. (55 P MC ρ Fgure shows e ρ e ρ e ρ for =.6 ρ =.5.5 ad β =.6. Noe ha order o guaraee ha he Moe Carlo ehod gves a leas hree correc sgfca dgs of he opo prces cosdered we use a Moe Carlo saple sze N MC = whe ρ =.5 ad N MC = 6 whe ρ =.5. Fgure shows ha he frs wo order ers of he expaso powers of ρ of he Europea pu prce (.e. he approxae prce P e already gves a sasfacory approxao of he pu prce whe ρ =.5 ad ρ =.5. More specfcally he expere shows ha he ea relave errors (57 obaed usg he zero-h order er he frs wo order ers ad he frs hree order ers of he expaso are respecvely whe ρ =.5 ad whe ρ =.5. Fgure shows ha he frs few ers of he expaso powers of ρ of he pu prce provde hgh qualy approxaos whe he pu opo s he oey or a he oey (.e. he forward prces/raes varable s saller ha or equal o he srke prce E ( E = he expere. I fac hs case we have ha he ea relave errors are whe ρ =.5 ad whe ρ =.5. Noe ha whe ρ =.5 he effec of he secod order er o he copued pu opo prce s eglgble. Fally le us ur our aeo o oe of he oe forulae deduced eco 4. Table 5 shows he ea E s β v ρ sx v ρ relave error ( M bewee he heorecal secod order oe ( sx v > > ρ ( [ of he forward prces/raes varable x > sulaed oe MC ( sx v ρ obaed usg he Moe Carlo ehod wh a saple sze N MC = 4 whe s =.5 =.6 v =.5 x ˆ = x = where ˆ x ξ ξ =.5 +. = = ( ( β β =.5.6 ha s: MC ( sx v ρ ( sx v ρ ( ρ EM ( s β v ρ =. sx v = ad he = where Table 5 cofrs he valdy of he oe forula deduced eco 4 I he hrd expere we use he ABR odel o erpre real daa. Ths s doe frs calbrag he ABR 5 odel usg a e seres of real daa ad he usg he calbraed odel o forecas opo prces. Le R be 5 he fve desoal real Eucldea space we roduce he vecor Θ= ( r β v ρ of he ukows of 5 he calbrao proble ad he se of he feasble pos of he calbrao proble defed as (56 56

24 Fgure. Relave errors e ρ e ρ e ρ ρ =.5.6 v =.5 =.8. β = ( ( β p e = P = ρ =.5 = aury T =.5 years E = Table 5. Coparso bewee sulaed ad heorecal secod order oes. follows: = ( ( ρ E ( sx v ρ M β { ( r β v ρ r β v ρ } = Θ= > < < > < <. (57 The equales coaed (59 ha defe are he aural cosras pled by he eag of he copoe of Θ he odel equaos. Recall ha sce he facal arkes v cao be observed us be regarded as a paraeer o esae he calbrao procedure ad ha for splcy also he rsk free eres rae r s regarded as a ukow of he calbrao proble. We use as daa of he calbrao proble suded a se of opo prces observed a a gve e ad we forulae he calbrao proble as a olear cosraed leas squares proble. Le P C be posve egers be he observao e ad x be he forward prces/raes observed a e =. The quaes C ( x TC EC = C P ( x TP EP = P are ~ respecvely he observed prces a e = of he Europea call opos havg aury e T C ad srke prce E C = C ad of he Europea pu opos havg aury e T P ad srke prce E P = P. We assue < T C = C ad < T P = P. I hs expere o 563

25 ephasze he depedece of he opo prces fro he paraeers coaed he vecor Θ we chage he oao used o represe he Europea call ad pu opo prces obaed usg he perurbave expaso Θ C x T E = C powers of ρ roduced eco 3. I fac we deoe wh ( C C Θ P ( x TP EP = P he prces as a fuco of M Θ of he Europea call ad pu opos obaed usg he frs wo ers of he expaso powers of ρ of eco 3 wh he dscou facors assocaed o he aury es τ C = TC = C or τ P = TP = P ad we choose x = x. We use oly he frs wo ers he power seres expaso o evaluae he prces cosdered because he prces volved hs expere are of he order of agude of ces of UD. For prces of hs order of agude he approxae prces obaed wh he frs wo ers of he seres expaso have wo correc sgfca dgs. The calbrao proble for he ABR odel s forulaed as follows: F ( Θ (58 where he objecve fuco F ~ ( Θ s gve by: Θ ( ( Θ Θ C C x P TC EC C x TC E C P x TP EP P x TP E P ( Θ = + = C x TC EC = P x TP EP F Θ. Proble (6 s he a olear cosraed leas squares proble. Ths proble s solved uercally usg a varable erc seepes desce ehod. Deals abou he uercal soluo of proble (6 ca be foud [7]. The daa used he calbrao expere are he daly values of he fuures prce o he EUR/UD currecy exchage rae havg aury epeber 6h ad he daly prces of he correspodg Europea call ad pu opos wh expry dae epeber 9h ad srke prces EC = EP = E = ( = 8. The srke prces E = 8 are expressed UD. The fuures prce o he EUR/UD currecy exchage rae ad he Europea call ad pu opo prces are observed he e perod ha goes fro epeber 7h o Deceber h. The observaos are daly observaos ad he values observed are he closg prces of he day a he New York ock Exchage. Fgure 3 shows he fuures prce of he EUR/UD currecy exchage rae (cker YTU Curcy (sold le ad he EUR/UD currecy's exchage rae (dashed le as a fuco of e. Fgure 4 ad Fgure 5 show respecvely he prces ( UD of he correspodg Europea call ad pu opos wh aury e epeber 9h ad he prevously defed srke prces E = 8 as a fuco of e. pecfcally we choose = epeber 7h ad = j ex radg day afer he day + j j = 59 wh hese choces we have 6 = Deceber h. We calbrae he ABR odel solvg proble (6 every (radg day durg he perod ha goes fro = epeber 7h o 6 = Deceber h usg he prces of he Europea call ad pu opos show Fgure 4 ad Fgure 5 whe = j j = 6 C = P = 8 ad usg he values of he forward prces/raes varables show Fgure 3. Fgure 6 shows he rsk eural paraeers obaed usg he calbrao procedure descrbed above. Recall ha he aury e of he opos cosdered s epeber 9h ad ha he e o aury show he abscssa of Fgure 6 Fgure 7 s he aury e (.e. epeber 9h us he curre e expressed (radg days. The paraeer values resulg fro he calbrao are show Fgure 6 ad are approxaely cosas as fucos of he e o aury. Nex we use he values of he paraeers show Fgure 6 o forecas opo prces oe day ahead. Tha s we use he paraeer values obaed calbrag he odel wh he daa of = j o copue he opo prces a = j + obaed usg x = x j + j = 59. The forecas opo prces are obaed evaluag he Europea call ad pu opo prces wh he frs wo order ers of he expasos (7 ad (6. Of course forulae (7 ad (6 afer beg rucaed us be adaped o he specfc feaures of he daa suded. Fgure 7 shows he observed ad he forecas values of he Europea call ad pu opo prces for fve dffere values of he srke prces: E 3 (Fgure 7(a E 5 (Fgure 7(b E 7 (Fgure 7(c E 9 (Fgure 7(d E (Fgure 7(e (59 564

26 Fgure 3. YTU (sold le ad EUR/UD currecy s exchage rae (dashed le versus e. Fgure 4. Call opo prces o YTU wh srke prce E.375.5( = + = 8 ad expry dae T = epeber 9h versus e. Fgure 5. Pu opo prces o YTU wh srke prce E.375.5( = + = 8 ad expry dae T = epeber 9h versus e.. I Fgure 7 he opo prces expressed UD are ploed o he vercal axs ha s arked wh V ad he horzoal axs shows he e o aury expressed days. The e u of he horzoal axs s he sae as ha of Fgure 6. The average relave errors over he e erval epeber 7h Deceber 7h o he forecas values of he Europea call ad pu opo prces whe copared wh he correspodg prces observed he facal arke are respecvely 7 % ad 5 %. These perceages reduce o 4% ad 3% he case of a he oey opos. Noe ha hs expere we have used oly oe calbraed ABR odel o forecas boh call ad pu prces. More accurae resuls ca be obaed calbrag he ABR odel wce usg respecvely oly pu prces ad oly call prces o forecas respecvely pu ad call prces. The wo calbraed odels gve respecvely beer forecass of pu ad call opo prces ha he forecass obaed wh he odel calbraed usg boh call ad pu prces. We coclude ha he hrd expere shows ha he ABR odel 565

27 Fgure 6. Paraeer values esaed he perod epeber 7h Deceber 7h versus e o aury expressed days. The u of easure of v s years / ad he u of r s years. The paraeers β ρ ad µ are desoless. Fgure 7. Observed ad oe day ahead forecas call ad pu opo prces ( UD for fve dffere srke prces: ((a E 3 = E 3 = E3 =.385 (b EC5 = EP5 = E5 =.395 (c EC7 = EP7 = E7 =.45 (d EC9 = EP9 = E9 =.45 (e EC = EP = E =.45 versus e o aury expressed days. erpres sasfacorly he daa suded sce he values of he paraeers resulg fro he calbrao are sable (Fgure 6 ad he forecas opo prces are accurae (Fgure 7. The webse: hp:// coas auxlary aeral cludg aa- C P 566

28 os a eracve applcao ad a app ha helps he udersadg of he paper. Refereces [] Haga P.. Kuar D. Lesewsk A.. ad Woodward D.E. ( Maagg le Rsk. Wlo Magaze epeber hp:// [] Che B. Ooserlee C.W. ad va Weere. ( Aalycal Approxao o Cosa Maury wap Covexy Correcos a Mul-Facor ABR Model. Ieraoal Joural of Theorecal ad Appled Face hp://dx.do.org/.4/94969 [3] Dous P. ( No-Arbrage ABR. The Joural of Copuaoal Face [4] Faoe L. Mara F. Reccho M.C. ad Zrll F. (3 oe Explcly olvable ABR ad Mulscale ABR Models: Opo Prcg ad Calbrao. Joural of Maheacal Face 3-3. hp://dx.do.org/.436/jf.3.3 [5] Faoe L. Mara F. Reccho M.C. ad Zrll F. (3 The Use of ascal Tess o Calbrae he Noral ABR Model. Joural of Iverse ad Ill-Posed Probles hp://dx.do.org/.55/jp--93 [6] Faoe L. Mara F. Reccho M.C. ad Zrll F. (3 Closed For Moe Forulae for he Logoral ABR Model Varables ad Applcao o Calbrao Probles. Ope Joural of Appled ceces hp://dx.do.org/.436/ojapps [7] Forde M. ( Exac Prcg ad Large-Te Asypocs for he Modfed ABR Model ad he Browa Expoeal Fucoal. Ieraoal Joural of Theorecal ad Appled Face hp://dx.do.org/.4/ [8] Haga P.. Lesewsk A.. ad Woodward D.E. (5 Probably Dsrbuo he ABR Model of ochasc Volaly. hp://lesewsk.us/papers/workg/probdsrforabr.eps [9] Islah O. (9 olvg ABR Exac For ad Ufyg I wh LIBOR Marke Model. hp://papers.ssr.co/sol3/papers.cf absrac d=48948 hp://dx.do.org/.39/ssr [] Aderse L. ad Adrease J. ( Volale Volales. Rsk [] Aderse L. ad Perbarg V.V. (7 Moe Explosos ochasc Volaly Models. Face ad ochascs 9-5. hp://dx.do.org/.7/s [] Wes G. (5 Calbrao of he ABR Model Illqud Markes. Appled Maheacal Face hp://dx.do.org/.8/ [3] Johso. ad Noas B. (9 Arbrage-Free Cosruco of he wapo Cube. hp://papers.ssr.co/sol3/papers.cf absrac d=33869 [4] Obloj J. (8 Fe-Tue Your le: Correco o Haga e al. hp://arxv.org/abs/ [5] Forde M. ad Pogud A. (3 The Large-Maury le for he ABR ad CEV-Heso Models. Ieraoal Joural of Theorecal ad Appled Face 6 Arcle ID: [6] Wu Q. ( eres Expaso of he ABR Jo Desy. Maheacal Face hp://dx.do.org/./j x [7] Aoov A. ad pecor M. ( Advaced Aalycs for he ABR Model. hp://papers.ssr.co/sol3/papers.cf absrac d=635 [8] Yakubovch.B. ( The Hea Kerel ad Heseberg Iequales Relaed o he Koorovch-Lebedev Trasfor. Coucaos o Pure ad Appled Aalyss hp://dx.do.org/.3934/cpaa [9] Yakubovch.B. (9 Beurlg s Theores ad Iverso Forulas for Cera Idex Trasfors. Opuscula Maheaca [] Ishyaa K. (5 Mehods for Evaluag Desy Fucos of Expoeal Fucos Represeed as Iegrals of Geoerc Browa Moo. Mehodology ad Copug Appled Probably hp://dx.do.org/.7/s [] Abraowz M. ad egu I.A. (97 Hadbook of Maheacal Fucos. Dover New York. [] Arfke G.B. ad Weber H.J. (5 Maheacal Mehods for Physcss. 6h Edo Acadec Press a Dego. [3] zykowsk R. ad Belsk. ( Coe o he Orhogoaly of he Macdoald Fucos of Iagary Order. Joural of Maheacal Aalyss ad Applcaos hp://dx.do.org/.6/j.jaa [4] Erdely A. Magus W. Oberheger F. ad Trco F.G. (954 Tables of Iegral Trasfors. Vol. McGraw- Hll Book Copay New York. 567