Pricing of Arithmetic Asian Quanto-Basket Options

Transcription

1 Helsnk Unversy of Technology Faculy of Inforaon and Naural cences Deparen of Maheacs and yses Analyss Ma-.8 Independen research proecs n appled aheacs Prcng of Arhec Asan Quano-Baske Opons Tana Eronen 6W.9.8

2 Table of conens Inroducon... The Asan Quano-Baske Opon... Prcng an Asan Quano-Baske Opon The Asan Quano-Baske Call Opon Prce... 8 Analycal Approxaons Edgeorh Expanson around he Lognoral Dsrbuon.... Edgeorh Expanson around he Inverse Gaa Dsrbuon... 5 Perforance Analyss of Approxaons Concluson... 8

3 Inroducon nce he begnnng of he 98 s here have been any nnovaons n he orld of dervaves and he ore coplex opon conracs ere developed called he exoc opons. The use of dervaves n rsk anageen acves has evolved rapdly snce he early 99 and dervaves are no he os poran ool o anage fnancal rsk bu hedgng and porfolo anageen requre ever ncreasng aoun of flexbly and effcency for he fnancal nsruens. Wh ha developen he nuber as ell as he coplexy of he nsruens proposed o anage rsk have been seadly ncreasng. The Asan opon s a fnancal opon hose value depends on he average of he underlyng asse durng a gven e nerval. Asan opons can use arhec or geoerc average bu he arhec average s coonly used. Asan opons are very frequenly raded over-he-couner dervaves. Mlevsky and Posner (998a) saed ha he ousandng volue s ranged fro 5 o bllon U.. dollars. Asan opons are one of he os coonly used pah dependen opons. Pah dependence ean ha he payoffs of he opons are deerned by he pah of he asse s prce. The payoff of baske opons depend on he value of a baske of asses and s usually a eghed arhec average of he underlyng asses. They offer he flexbly for he sulaneous anageen of exposures on vrually any nuber and any knd of asses and hence can sasfy he specfc rsk exposure hedgng needs. Because he averagng generally decreases he varance of he varable he baske opons becoe norally cheaper han a slar porfolo of sandard opons. In pracce baske opons are desgned specfcally o ee he needs of he buyer and hey are raded over-he-couner. Wh he Asan and baske opons coes he proble o prce he effcenly and accuraely. Because hey have parcular characerscs he faous prcng forula nroduced by Black and choles (97) s nadequae. The ehod of Black and choles requres he sandard lognoraly assupon and he arhec Asan and baske feaure of hs opon lead o an expresson ha canno be solved analycally. In Asan and baske opons here s a su of lognoral rando

4 varables ha s no lognoral. In such cases he generalsaon and adusen are needed o aan prcng echnques and approxaons. Asan opons are pah-dependen dervaves and hey are aong he os dffcul o prce and hedge. Also he prcng of baske opons s ore challengng han ha of sandard opons because here s no explc analycal soluon for he densy funcon of a eghed su of correlaed asses. everal approaches are proposed n he leraure o prce Asan and baske opons. They can be caegorzed as follos: nuercal ehods upper and loer bounds and analycal approxaons. Boyle (977) nroduced a Mone Carlo sulaon for opon prcng and snce hen has been dely used. There exs also ore recen developen of sulaon ehods see Broade and Glasseran (996) and Boyle Broade and Glasseran (997). The sulaon ehods are very flexble and he heory s sple bu prcng s very e-consung. In hs sudy he Asan quano-baske opon s exaned. I s soeha ore coplex han prcng baske or Asan opons. I s an Asan baske opon hch has srke n dfferen currency han he payoff n aury s pad. In ulnaonal nsuons hs conrac s useful because allos parcpaon ulnaonal porfolos hou beng affeced by he foregn exchange rsk. Baske feaure allos for he sulaneous anageen of exposures on a varey of underlyng coodes and Asan feaure provdes he sae flexbly for asse flos n e. Posons ha can be hedged h Asan quano-baske opons could be hedged as ell h a porfolo conss of plan vanlla European opons n asses and currences bu Asan quano-baske opons provde n general less expensve alernave. Ths fnancal opon hus splfes he nernaonal hedgng by onng he hedgng of exchange and ndusry rsks. There s varey of ehods n he leraure for prcng suggesed o prce he Asan quano-baske opons. The rsk anageen procedure n he fnancal ndusry hoever requres ofen o prce large nuber of fnancal opons quckly and h suffcen accuracy. The prcng ehod should also be easy o negrae o copuer and rsk anageen syse. The proble h nuercal ehods s ha adequae precson requres ore copuaon e and snce her lenghy calculaon hey are ofen dffcul o negrae he exsng rsk anageen syses hus n he fnancal ndusry he approxae analycal soluons are ofen preferred.

5 Boh Asan and baske opon values depend on a eghed arhec average of asse prce. Alhough he average s no lognoral can be odelled as such n several ays. The sples odels resor o approxang arhec average h geoerc averages (see Vors 99 and Genle 99) adusng he srke for he dfference as ell as oen-achng arhec average hn he lognoral faly (Levy 99 Mlevsky and Posner 998b). Many sudes have esed he precson of hese analycal approxaons for eher Asan or baske opon bu hey are ypcally done h sall se of paraeer values o ake he coparson. Daey Gauher and onao () exane he precson of hree dfferen ypes of analycal approxaons for Asan quano-baske opon by dong coprehensve sulaon experen perfored on a large es pool of opon conracs. They sae ha he qualy of analycal approxaon s no alays consan n a paraeer space and hey found ha he Edgeorh-lognoral and Johnson ype denses are he os accurae ehod n hole paraeer space. The an conrbuon of hs sudy s o copare o analycal approxaons: he Edgeorh expanson h lognoral dsrbuon (Turnbull and Wakean 99) and he Edgeorh expanson h recprocal gaa (Mlevsky and Posner 998a). The a s o copare he accuracy of he approxaon for dfferen averagng frequences. Daey Gauher and onao () have ade a general saeen of accuracy of hese ehods and Johnson dsrbuon (Posner and Mlevsky 998) n he hole paraeer space. In he prevous verson of her paper Daey Gauher and onao also exane o oher approxaons proposed by Vors (99) for Asan opons and Genle (99) for baske opons bu hey ere found o he nferor and ha s hy hey are no consdered n hs sudy. The nex secon explans deals he Asan quano-baske opons. econ exanes he rsk neural prcng odel hle econ proposes a heorecal fraeork for he o coonly used analycal approxaon for prsng Asan quano-baske opons ha s he Edgeorh expanson for he nverse gaa and for he lognoral dsrbuons. Afer dervng approxae valuaons for Asan quano-baske opons n econ 5 he hree analycal approxaons are valdaed by coparng he h Mone Carlo sulaon. Fnally econ 6 concludes.

6 The Asan Quano-Baske Opon European opon s he sandard opon ha gves he holder he rgh o acqure share on a specfc dae for predeerned srke prce K. Is payoff funcon s ren as Payoff a aury = ( ) ax T K;. () The Asan opon s a fnancal opon hose value depends on he average of he underlyng asse durng a gven e nerval. There exs any varaons of Asan opons lke average rae and average srke opons. In he average srke opon he payoff n aury s fxed and he srke s floang hle n he average prce opon he srke s fxed and he payoff s floang. Here afer he er Asan opon s used for an arhec average rae Asan opon. In heory can be also dsngushed beeen he connuous and dscree average calculaon anyay n pracse only dscree Asan opons are raded hereas he connuous average calculaon s used as an approxaon of raded opon n aory of research papers. Asan opon proecs he nvesor agans possble ad hoc flucuaons of he underlyng asses. On fnancal arkes he Asan opon as nroduced o proec he pares fro sock arke anpulaon of underlyng values on or near he expry dae of opon. The Asan opon uses an average value of he underlyng asses a predeerned daes deerne he payoff. Is payoff funcon s as follos... o Payoff a aury = ax = ( ) K; () and s gven n currency. Baske opon can be seen as a European opon bu he underlyng s coposed of several fnancal asses all generally expressed n sae currency. The payoff funcon of a baske opon s gven by

7 Payoff a aury = ax n = K; T () here...n represen he eghs assocaed h each asse and n = =. One of he an advanages of he baske opon s ha allos a generally less expensve alernave o cover several underlyng asses han ha of purchasng a porfolo consss of each underlyng asse o be covered. Quano opon s defned n ers of an underlyng o be ade ou n a currency oher han he currency of he payen o be ade upon expry dae. Is payen funcon s gven by ( ) ( ) ( ;) Payoff a aury = C ax T K () here C s he nuber of uns of local currency by un of foregn currency a e =. The quano opon allos ndrec nvesen n a foregn asse hou beng subeced o he foregn exchange rsk. I can be used also o hedge agans poenal flucuaon n foregn secures. Asan quano-baske opon cobnes he of Asan baske and quano opons. Is oal payen n a predeerned currency s as follos Payoff a aury = ax n = = K;. (5) The eghs nclude he fxed exchange raes. Prcng an Asan Quano-Baske Opon The Harrson and Plska (98) sugges copung he value of an opon h he rsk neural easure and he dscoun a he rsk-free rae. In he odel here exss n foregn underlyng 5

8 secures n exchange raes n rsk-free foregn bonds and one rsk-free doesc bond. They are noed as follos C B D { : } ( ) ( ) = ( ) ( ) { C : } = ( ) ( ) { B : } = ( ) ( ) { D : } = for he foregn underlyng secury (6) for he exchange rae (doesc/foregn) (7) for he foregn rsk free bond (8) for he doesc rsk free bond. (9) The underlyng secury and he exchange raes are expeced o follo he lognoral dsrbuon. Thus e apply he follong hypohess of sochasc varables ( ) ( ) ( ) ( ) d = µ d d W =... n () ( ) ( ) ( ) ( ) dc = µ C d α C dz =... n () ( ) ( r B ) d db = = n () dd = rd d = n () here ( n) W... W and Z ( n) Z are sandard Bronan oons h correlaons ( k ) ( ) ( W W ) θ k P Corr = () ( k ) ( ) ( Z Z ) λk P Corr = (5) ( k ) ( ) ( W Z ) ρ k P Corr =. (6) 6

9 We also suppose ha he underlyng secury pays a dvdend a a consan rae of δ on a connuous bass. Then hese all sochasc processes are bul on probably space ( F F P) here F s he flraon { : } F h ( ) ( ) ( ℵ Z & W : s & { n} ) Ω F = s s... (7) and ℵ s a se of zero probably evens. In order o presen he opon prcng forula e need o represen he prce of he underlyng asse usng a rsk neural probably easure denoed as Q. Fnally s found ha rsk neural easure s unque and follong sochasc dfferenal equaon defnes he requred processes d ( ) ( ) ( ) ( ) ~ = r δ α ρ dw = n (8) d here α ρ s an adusen er hch deernes he effec of he quano feaure of he ~ ~ ( n) opon and W... W are sandard Bronan oon under a Q probably easure and hey have correlaons ~ ( k ) ~ ( ) ( W W ) θ k Corr =. (9) The dfferenal equaon obaned s alos dencal o ha represenng he prce of underlyng asse h dvdend n he Black and choles forula. Acually he only dfference s he er α ρ hch reflecs he quano feaure of he opon. Wh hs er he opon prce s adused o ake no accoun he pac of he exchange rae. The Equaon (9) allos us o rea opons here he baske ncludes soe secures ade ou n doesc currency as ell as suaons here nuerous secures share he sae currency hen e can use α k = α v k = v ( k ) ( ) Z Z = and r k = r. larly f secury k s n doesc currency he odel conans one less exchange rae 7

10 ( k ) ( = vk = k = ) C α. () Because Equaon (8) s a sandard geoerc Bronan oon he soluon s ~ [ r δ α ρ W ] ( ) ( ) ( ) = exp = n. () Ths secon represens a sandard opon prcng odel ha s dely used bu neverheless ncludes eaknesses lke consan volales and neres raes. The an advanage of he odel s ha suppors opon prcng h an accepable accuracy n ers of eprcal observaons. If opons expre hn a relavely shor perod of e lke hey usually do he above-enoned eaknesses have dnshng nfluence.. The Asan Quano-Baske Call Opon Prce To prce of he quano opon n he baske can be copued h he Equaon (). No he prce of he baske can be calculaed h he follong forula V exac r [ ( A K) ] = e ax( A K ) f ( x) r Q = e E ax () here f ( x) represens a densy funcon of he arhec average A of he underlyng baske value on he predeerned daes.... The arhec average s A = ( )... T () here he process for s gven by Equaon () and here s he vecor of eghs of he secures n he baske. In Equaon () s equvalen o negrang fro negrang fro o o and snce f(x)= for any x<. There s no exac soluon for () because A s su of lognoral rando varables and negral represenng he densy funcon of a su of lognoral rando varables canno be solved analycally. 8

11 9 Lace-based ehods are coonly used for prcng opons on a sngle asse. Hoever hey are copuaonally exensve and exponenally coplcaed for opons on ulple asses. Nuercal approxaon usng a Mone Carlo or Quas-Mone Carlo ehods allo o acheve a value ha s as accurae as preferred. The draback of hese approaches s ha requres exensve calculaons alhough hey are less e-consung han lace-based approaches. The analycal approxaons offer a very fas ehod o prce Asan quano-baske opon even f a ceran aoun of accuracy s los n he process. The e-savng s an obvous advanage for real e radng. Analycal Approxaons In hs secon hree ell-knon analycal approxaons for he Asan quano-baske opon are exaned. In order o apply oen achng -based approxaons e need o calculae he frs four oens of he eghed su of he underlyng baske of opon under easure Q. Lea : Under he rsk neural easure Q he frs four oens of he arhec ean A are respecvely [ ] = = = n f r exp ρ α δ () = = = n f r r ax n exp ρ α δ θ ρ α δ (5) = = = ax n exp ed n f r r r ρ α δ θ ρ α δ θ θ ρ α δ (6)

12 = = = ax n exp n f r r r r ρ α δ θ ρ α δ θ θ ρ α δ θ θ θ ρ α δ (7) here x represen h x value of he decreasng ordnae quadruple. The lognoral dsrbuon akes possble o apply he follong deny [ ] = exp exp b a bz a E (8) here ~ N Z. The oens can be derved usng Equaon (8). For he nex approxaons e adop follong noaon. The expeced value as per he densy funcon g s = x xg g µ (9) and he cenral h k oen n ers of he densy funcon g s = x g x k g g k µ. (). Edgeorh Expanson around he Lognoral Dsrbuon Our frs analycal approxaon s based on a generalzed Edgeorh expanson around he lognoral dsrbuon. Jarro and Rudd (98) ere he frs o propose usng Edgeorh expansons o solve opon prcng probles. The basc dea of hs ehod s o replace an unknon densy funcon f h a Taylor-lke expanson around an easy-o-use densy funcon

13 denoed a ( ). Hoever he Edgeorh expansons coonly lead o a funcon hch s no a posve and unodal. To guaranee ha he approxaon obaned h a runcaed Edgeorh expanson s a rue densy funcon Baron and Denns (95) derve specal condons on he hrd and fourh oens of he unknon dsrbuon. Moreover Ju () pons ou ha he Edgeorh expanson ay dverge for soe paraeer values hch consequenly can gve ncorrec prces for hgh volaly and long aury opons. As suggesed by Turnbull and Wakean (99) and by Huynh (99) he lognoral su n Asan baske opon s approxaed by a lognoral dsrbuon and an Edgeorh expanson of he fourh order s used. f ( x) s he rue densy funcon of Asan quano-baske opon prce and ( x) approxaon of f ( x). ( x) a s chosen as an a s he lognoral densy funcon of he rando varable ( aˆ bz ˆ ) here Z ~ N( ). Jarro and Rudd (98) sho ha ( x) expanson f ( x) = a( x) ( f ) ( a) ( f ) ( ) ( f ) ( a) ( f ) ( a) d a( x) d a( x)! ( a) ( f ) ( a) ( ( ) ( ) ) ( ) d a( x)!! exp f can be ren as follong Edgeorh ε ( x) () here he cenered oens (*) are defned n Equaon () and ( x) k ε s an error er. In general no analycal feaures can be conneced o error er. We choose funcon a ( x) such ha he frs o oens are equal o hose of f ( x). Because a lognoral dsrbuon s copleely descrbed by s frs o oens Hence â and bˆ us be equal o he expresson gven n Equaon () and (5). Fnally e can rere Equaon () as follos f ( x) a( x) ( ) d a( x) ( f ) ( a) ( f ) ( a) d a x =!! ε ( x). () V lognoral s defned as

14 V lognoral e = e rt rt K = e ax rt ( x K) a( x) ax ( x K ) a( x) ( x K) f ( x) ( ) d a( x) ( f ) ( a) ( f ) ( a) ( ) d a( x) ( f ) ( a) ( f ) ( a)!! d a x d a x!! ε ( x) ε ( x).() Lke Jarro and Rudd (98) sae ha he equaon K ( x K ) d a ( x) d a( x) = ( K ) for () yelds V lognoral rt = e K ( x K ) a( x) ( f ) ( a) ( f ) ( a)! da x K ( ) d a( x) ( K )!. (5) Because he negraon s perfored h respec o he lognoral densy he frs er n Equaon (5) s he Black and choles forula. The cenered oens of he lognoral dsrbuon are easly derved fro he frs four oens of a. The hrd and fourh oens of he lognoral dsrbuon needed for he Edgeorh expanson depend only on he frs and second oens of he dsrbuon and can be gven as follos a bˆ ( f ) = exp aˆ = (6) ( a) ( f ) = exp ( aˆ bˆ ) = (7) 9 ˆ a f a = exp b ˆ = = a (8) a f

15 ( a) exp( aˆ 8bˆ ) ( a) ( ) ( a) ( ) 6 8 ( f ) ( ) ( f ) ( ) 8 6 = = = (9) here ( f ) and ( f ) are respecvely he frs o oens of A under he rsk neural easure. Those relaed h f are descrbed n lea. Thus he follong analycal approxaon s obaned as a sor of Black and choles prce adused for he excess skeness and he excess kuross fro he lognoral densy V = V e ( f ) ( a) da ( K ) e ( f ) ( a)! d a rt rt log noral! ( K ) () here V ( N( dˆ ) KN ( dˆ ) rt = e µ () dˆ = dˆ bˆ () d ˆ ( K ) ln aˆ = () bˆ ( f ) ( f ) ( ) ˆ = ln () a ln ( f ) ( f ) ( ) ˆ = (5) b ln ln and here a ( ) s he densy funcon of a lognoral dsrbuon and ( ) dsrbuon densy funcon. N s a sandardzed noral

16 . Edgeorh Expanson around he Inverse Gaa Dsrbuon The analycal approxaon derved n hs secon uses also an Edgeorh expanson bu nsead of lognoral dsrbuon he nverse Gaa dsrbuon s used o approxae he su of lognorals as proposed by Mlevsky and Posner (998a). They sho ha h soe paraeers an nfne su of correlaed lognoral rando varable converge asypocally o an nverse gaa dsrbuon. o hey sugges ha he fne su of lognorals s approxaed by nverse gaa funcon hen prcng Asan and baske opons. As he lognoral dsrbuon he nverse gaa densy funcon s defned by s frs o oens. Thus e choose he gaa nverse funcon g ~ ( x) such ha he frs o oens equals hose of f(x). Hence he approxaon ll be defned n a slar ay as he one obaned n he case of lognoral dsrbuon ha s V gaa e = e rt rt = e K rt ax ax ( x K) f ( x) ( x K) g ( x) ( x K ) a( x) R ~ ( ) d g~ ( x) ( f ) ( g ) ( f ) ( g ) ( ) d g~ ( x) ( f ) ( g ) ( f ) ( g~ )! ~! ~ d g~ x d g~ x!! ε ( x) ε ( x). (6) Defnon : The densy funcon of a gaa rando varable h paraeers ( α β ) X ~ G( α β ) s gven by g R ( x) β = x α α β x Γ e ( α ) x >. (7) Proposon : The densy funcon of he rando varable Y=/x here ~ G( α β ) X s gven by g~ ( y) ( α ) yβ α β = e y >. (8) α y Γ

17 Then s assued ha Y ll have a n nverse-gaa dsrbuon ren as ~ G( α β ) Proposon : Le X ~ G( α β ) E [ Y ] =. The oens of Y are gven by ( α )( α ) ( α ) β... X. (9) here < α. Then such nverse-gaa dsrbuon needs o be seleced ha s frs o oens ll be equal o he frs four oens of A. Le ( ) f ( ) ( f ) ( f ) α = (5) ( f ) ( ) ( f ) ( f ) β = ( f ) ( f ) (5) here nce ( f ) and ( f ) are respecvely he frs o oens of A under he rsk neural easure. ( K ) ( K ) K ( K ) ~ d g d g~ x = for (5) hen V e gaa rt = e rt K ~ ( f ) ( g )! ( x K ) g~ ( x) d g~ ( K ) e rt ~ ( f ) ( g )! dg~ ( K ). (5) Usng he change of varable y=/x he frs er can be ren as 5

18 e here ( ηλ) rt K ( ) ~ rt x K g( x) = e µ α β KG α β G (5) K K G s a gaa dsrbuon funcon h he paraeers η and λ. These paraeers are deerned by achng he frs o oens of he exac and approxae dsrbuons. The approxaon uses he frs four oens of he nverse-gaa dsrbuon. The frs and second oens are dencal o hose of Y and he hrd and fourh oens of Y are ~ ( g ) ( f ) ( f ) ( ) ( f ) ( f ) ( ) = (55) ~ ( g ) ( f ) ( f ) ( ) ( ) ( f ) ( f ) ( f ) ( ) 7( ) 6( ) =. (56) Thus he belo analycal prcng forula s obaned V = V e ~ ( f ) ( g )! dg~ ( K ) e ( f ) ( g )! d g~ rt rt gaa ~ ( K ) (57) here V = rt e µ G α β KG α β (58) K K α and β as saed n he Equaons (5) and (5). 5 Perforance Analyss of Approxaons To confr he nuercal accuracy of o analycal approxaons h dfferen volales and correlaons e conduc sulaon experens. pecfcally e copare he Asan quano-baske 6

19 opon prce obaned h he analycal approxaons o he Mone Carlo prce obaned h 75 pahs. In sulaon e use baske h hree underlyng asses and hey are equally eghed. We sulae he prces of a-he oney opons. The aury of he opons s 5 years. The dvdend rae doesc and foregn rsk-free raes as ell as correlaons are also kep fxed. These are shon n Table. Table : Fxed paraeers n sulaons Consan Paraeers Values Nuber of secures n he baske Weghs [ ] rke Prce Inal underlyng secury prce Dvdend rae. Doesc rsk-free rae.5 Foregn rsk-ree rae Correlaon beeen he secury prces and exchange raes Correlaon beeen secury prces Maures [ ] T [ ] T years.. The prced opons have onhly averagng and annual averagng. We also vary he prce volaly of he underlyng and exchange rae volaly. These are shon n Table. For hese hree paraeers e use hgh and lo values o be able o defne f he volales and averagng frequency have an effec o he nuercal accuracy of he approxaons. Table : The varous paraeers n sulaons Varable Paraeers Values Lo Hgh Exchange rae volales [ ] T.5..8 ecury prce volales [.7..9] T.5..8 Averagng Annual Monhly [ ] T [ ] T 7

20 The Asan quano-baske opon prces calculaed h Mone Carlo sulaon Edgeorh expanson usng lognoral dsrbuon and Edgeorh expanson usng nverse gaa dsrbuon are presened n Table. Table : Asan quano-baske opon prces h dfferen ehods. FX Volaly Volaly Lognoral Inverse Gaa Mone Carlo d Annual averagng hgh hgh hgh lo lo hgh lo lo Quarerly averagng hgh hgh hgh lo lo hgh lo lo The Edgeorh expanson h lognoral dsrbuon sees o ork beer n he os of he paraeer ses. The resuls are he os accurae h he sall underlyng secures volales. Especally hen he volales of he underlyng secures are hgh and he volales of he exchange raes are lo he resul s naccurae. The analycal approxaons see o be ore accurae hen he averagng frequency s hgh. Ths s because a connuous averagng s used o odel he Asan opons and he ore frequen he averagng s he ore accurae he approxaon ll be. 6 Concluson Frs can use Asan quano-baske opons o hedge her exposure o dfferen rsks such as coody rsk neres rae rsk and exchange rae rsk. Hoever snce no closed-for soluon can be derved for he su of lognoral rando varables he prcng of hese opons s no easy because hey do no have closed for soluon such as Black and choles. The an conrbuon of hs sudy s sensvy analyss of o analycal approxaons o prce he Asan quano-baske opons. We easure her sensvy o he underlyng asse prce volaly and o he volaly beeen he underlyng asse and he exchange rae. The o approxaon used are he Edgeorh 8

21 expanson around he lognoral dsrbuon proposed by Wakean (99) Edgeorh expanson around he nverse gaa dsrbuon suggesed by Mlevsky and Posner (998a). In order o asses and copare he accuracy of he approxaon e use local sensvy analyss here he paraeers of he odel are fxed arbrarly bu hree of he ge hgh and lo values. Our resuls suggess ha boh approxaons are accurae hen he volales of he secures are lo. When he averagng frequency s quarerly he approxaons ere found o be ore accurae han annual averagng. Ths s because he connuous averagng s used o odellng he Asan opons. The Edgeorh expanson usng he lognoral dsrbuon as found o be he os accurae approxaon. Ths s nlne h he prevous researches. Donne Gauher Queran and Tahan (6) copared hree dfferen analycal approxaons for heerogeneous baske opons h Mone Carlo sulaon. The approxaons ere Inverse gaa Edgeorh expanson around he lognoral dsrbuon and Johnson approxaon. They found ha he Edgeorh-lognoral and Johnson approxaon ere far ore accurae han he nverse gaa approxaon. Dealed look a he resul shos ha he ou-of he oney and hgh volaly opons have he larges relave errors. Daey Gauher and onao () exaned he precson of hree dfferen ypes of analycal approxaons for Asan quano-baske opon by dong coprehensve sulaon experen perfored on a large es pool of opon conracs. Lke hey sae ha he qualy of analycal approxaon s no alays consan n a paraeer space and hey found ha he Edgeorh recprocal gaa and Johnson ype denses are he os accurae ehod n he hole paraeer space. Exendng he approach s a prosng area for fuure research. 9

22 References Baron D. and Denns K. 95. The condons under Gra-Charler and Edgeorh curves are posve defne and unodal Boerca Black F. and choles M. 97. The prcng of opons and corporae lables. Journal of Polcal Econoy Boyle P Opons: A Mone Carlo approach. Journal of Fnancal Econocs -8. Boyle P. Broade M. and Glasseran P Mone Carlo ehods for secury prcng. Journal of Econoc Dynacs and Conrol 67-. Bröade M. and Glasseran P Esang secury prce dervaves usng sulaon. Manageen cence Daey J. Gauher G. and onao J.. The perforance of analycal approxaons for he copuaon of Asan quano-baske opon prces. The Mulnaonal Fnance Journal Donne G. Gauher G. Queran N. and Tahan N. 6. Heerogeneous baske opons prcng usng analycal approxaons. Workng paper 6- Cenre of Research on e-fnance (CREF). (ubed). Genle D. 99. Baske eavng. Rsk Harrson M. and Plska. 98. Marngales and sochasc negrals n he heory of connuous radng. ochasc Processes and her Applcaons 5-6. Hll I. Hll R. and Holder R Fng Johnson curves by oens. Appled ascs

23 Huynh C. 99. Back o baskes. Rsk Jarro R. and Rudd A. 98. Approxae opon valuaon for arbrary sochasc processes. Journal of Fnancal Econocs Johnson N. 99. yses of frequency curves generaed by ehods of ranslaon. Boerca Ju N.. Prcng Asan and baske opons va Taylor expanson. The Journal of Copuaonal Fnance Levy E. 99. Prcng European average rae currency opons. Journal of Inernaonal Money and Fnance 7-9. Mlevsky M. and Posner. 998a. Asan opons he su of lognorals and he recprocal Gaa dsrbuon. Journal of Fnancal and Quanave Analyss 9-. Mlevsky M. and Posner. 998b. A closed-for approxaon for valung baske opons. The Journal of Dervaves Posner. and Mlevsky M Valung exoc opons by approxang he PD h hgher oens. Journal of Fnancal Engneerng Turnbull. and Wakean L. 99. A quck algorh average Asan opons. Journal of Fnancal and Quanave Analyss Vors T. 99. Prces and hedge raos of average Asan opons. Inernaonal Reve of Fnancal Analyss 79-9.