Exotic Options Pricing under Stochastic Volatility

Transcription

1 Exoic Opion Pricing undr Sochaic olailiy Nabil AHANI Prliminary draf April 9h Do no quo Conac informaion: Nabil ahani HEC Monréal Canada Rarch Chair in Rik Managmn 3 Chmin d la Cô-Sain-Cahrin Monral Qubc Canada H3 A7 l: Fax: [email protected] Nabil ahani i a Ph.D. candida in financ a HEC Monréal Canada. Financial uppor wa providd by h Canada Rarch Chair in Rik Managmn.

2 Exoic Opion Pricing undr Sochaic olailiy Nabil AHANI Abrac hi papr propo a mi-analyical formula o pric xoic opion wihin a ochaic volailiy framwork. Auming a gnral man rvring proc for h undrlying a and a quar-roo proc for h volailiy w driv an approximaion for opion pric uing a aylor xpanion around an avragd dfind volailiy. h momn of h avragd volailii ar compud analyically a any ordr uing a Frobniu ri oluion o om ordinary diffrnial quaion. Pricing om xoic uch a barrir and digial barrir opion h approximaion i found o b vry fficin and convrgn vn a low aylor xpanion ordr. Réumé C aricl propo un formul mi-analyiqu pour évalur l opion xoiqu dan un cadr d volailié ochaiqu. En conidéran un procu avc rour à la moynn pour l acif ou-jacn un procu racin-carré pour la volailié on dériv un approximaion pour l opion n uilian un dévloppmn d aylor auour d un crain volailié «moynn» qui ra défini. momn d volailié moynn on calculé analyiqumn comm un oluion n éri d Frobniu d crain équaion différnill ordinair. En évaluan crain opion xoiqu comm l opion barrièr l opion barrièr digial on monr qu l approximaion convrg rapidmn qu ll rè préci.

3 Inroducion Svral papr propo pricing formula for plain vanilla opion on ock wihin diffrn ochaic volailiy framwork. Hon 993 i h fir on who propo a clod-form pric for a andard Europan call whn uing a quar-roo volailiy proc by invring h characriic funcion n a a Fourir ranform. Bakhi Cao and Chn 997 propo an mpirical prformanc udy of om alrnaiv opion pricing modl including ochaic volailiy and jump proc by driving clodform oluion in h am way a Hon 993. Schöbl and Zhu 999 and Zhu driv in a vry lgan way a modular pricing mhod which includ h quar-roo and h Ornin-Uhlnbck volailiy proc mixd vnually wih jump. For om volailiy modl howvr no clod-form oluion can b drivd and om numrical chniqu ar ud inad. Hull and Whi 987 and Sabani 3 obain an approxima oluion uing a aylor ri xpanion bad on h undrlying a diribuion condiional on h avrag valu of h ochaic varianc. Slighly lil work wa don for pricing xoic drivaiv uch a pah-dpndn opion in non-conan volailiy modl. Davydov and inky driv clodform oluion in rm of Bl and Whiakr funcion for barrir and lookback opion undr a conan laiciy of varianc diffuion modl. Hndron and Hobon pric papor opion in Hull and Whi 987 and Sin and Sin 99 ochaic volailiy framwork uing h ri xpanion chniqu. In boh ca vry impl clod-form oluion for h cnral momn of h avrag ochaic varianc ar propod. hi i mad poibl for om volailiy proc uch a h gomric Brownian moion and h man rvring diffuion. Unforunaly for ohr ochaic volailiy proc uch a h quar-roo diffuion no impl clod-form formula for h momn of h avrag varianc can b found. In ha ca ohr chniqu uch a numrical approximaion or Mon Carlo imulaion may b ud o pric drivaiv. Apl Winklr and Wyup propo a fini lmn mhod o pric plain vanilla and barrir opion undr a quar-roo ochaic volailiy modl. Sabani 3 driv an iraiv procdur o compu h momn d κθ λ d σ dw. Sabani 3 call man rvring h following volailiy diffuion

4 Many a including inr ra crdi prad [ ongaff and Schwarz 995 ahani Prigan al. and Jacob and i 3] and om commodii Schwarz 997 ar hown o xhibi a man rvrion faur. Bu hr i lil liraur on pricing drivaiv for hi yp of undrlying a. Undr ochaic volailiy modl mo of h work i don on plain vanilla drivaiv. Clwlow and Srickland 997 pric andard inr ra drivaiv undr a quarroo volailiy modl uing Mon Carlo imulaion. Auming h lar volailiy proc ahani 4 pric crdi prad opion cap floor and wap uing Gauian quadraur. Undr a conan volailiy aumpion blanc and Scaill 998 propo om pah-dpndn inr ra opion formula for h affin rm rucur modl. hi papr propo o pric om xoic opion on a man rvring undrlying a in a quar-roo volailiy modl uing a ri xpanion around wo avragd dfind volailii. h choic of hi powr ri mhod i ncouragd by h finding of Ball and Roma 994 abou i accuracy and i ay implmnaion in comparion o ohr approach. h ky hing of hi mhod i ha h pric of a coningn claim may b compud a h xpcaion of h corrponding conan-volailiy modl pric whr h volailiy and h po pric ar random variabl accouning for ochaic varianc [ Hull and Whi 987 and Romano and ouzi 997]. I rmain hough o driv h cnral momn of h avragd varianc and u hm in h ri xpanion. Bu inc h clod-form formula for h momn can only b drivd in rm of Whiakr funcion 3 which ar havy-compuaional i i prfrabl o compu hm uing a Frobniu ri oluion which i vry accura vry fa and vry ay o implmn. h nx cion prn h propod modl and inroduc h ri xpanion mhod. Scion II driv a Frobniu ri oluion o h momn of h avragd varianc. Scion III prn valuaion formula for om xoic opion. Scion I prn om numrical rul on convrgnc and fficincy. Scion will conclud. 3 In fac w can driv h momn of h avragd varianc in rm of drivaiv of Whiakr funcion w.r.. h fir and h hird argumn. 3

5 I h propod modl Following ahani 4 w conidr h wo ochaic diffrnial quaion SDE for h a variabl and i volailiy undr a rik-nural maur Q : d u µ u γu d u ρdbu ρ dwu du κθ λu d σ u dbu whr { u u } crdi prad { u } xp i h pric proc of a primiiv a uch a a ock or a u i h volailiy proc and ρ i h corrlaion bwn h a variabl and i volailiy. W and B ar wo indpndn Brownian moion on a F i h Q-augmnaion of h filraion probabiliy pac Ω FQ and { u u } gnrad by W B. h paramr µ γ κ θ λ and σ ar conan. Pricing hory allow u o wri h pric p of any Europan coningn claim on a h xpcaion undr a rik-nural maur of h dicound payoff of h conrac in ordr o g : p Q r E H { } whr r i h conan rik-fr ra h conrac mauriy H i h payoff ha could dpnd on h whol pah of h a variabl F 3. In ordr o dvlop a ri xpanion approximaion o h conrac pric w hall adap h mhodology in Romano and ouzi 997 o our man rvring proc. Rfrring o h dail in Appndix w can wri h oluion o h SDE a : whr Y µ ρ d ρ d γ d dw 4 Y db ρ ρ d 5 Dfining an ffciv a variabl ~ and om avragd varianc by : ~ Y 6 4

6 5 and : d d d d d d 3 7 lad o conclud ha h proc ~ can b viwd a a oluion o h following SDE : dw d d 3 ~ ~ ρ ρ γ µ 8 Puing r µ and γ lad o h corrponding xprion in Romano and ouzi 997 givn by : dw d r Y ρ ρ 9 d db Y ρ ρ and d dw d r d ~ ρ ρ h pric of h coningn claim givn in Equaion 3 can hu b rwrin a : { } F G ~ H E E p r Q Q whr { } u B W u : σ G i a nw σ-algbra which aum ha h movmn of h volailiy ovr h nir lif of h conrac ar known a im. In hir non-zro corrlaion modl Romano and ouzi 997 driv h pric of a andard Europan call opion in hi way a h xpcaion of h Black and Schol 973 pric whr h

7 undrlying a pric i rplacd by xp Y and h volailiy paramr i rplacd by. Bu hi i no an xplici formula inc on ill ha o compu h xpcaion which i almo impoibl in h non-zro corrlaion ca. wi noic ha vn in h ca of a zro corrlaion h pricing formula in Equaion 3 do no alway lad o an analyical oluion bcau h ingrad volailiy dniy i difficul o driv in clod-form 4. Boh Hull and Whi 987 and Sabani 3 aum zro corrlaion and driv an approximaion o h Europan call pric. aking ino accoun hi finding and inc our aim i o g om xplici mi-analyical pricing formula for xoic opion w will aum for h rmaindr of h aricl ha h corrlaion bwn h a variabl and i volailiy i zro. In h zro corrlaion ca h proc h Equaion 4 7 and 8 can b implifid a : Y givn in Equaion 5 i alway and µ dw d γ d 3 whr γ d dw µ 4 d d ; d d d 5 h pricing formula in Equaion can b implifid by fir compuing a pudo-pric undr h condiion of h wo avragd varianc : Q r p E H { } and hn aking h xpcaion of hi pudo-pric condiional on h iniial Q- augmnd filraion { u } F o g h pric of h coningn claim : u G 6 4 In wi pag 6. 6

8 p E Q p F 7 o obain an approximaion for h pric in Equaion 7 p i xpandd in a aylor ri around h xpcd valu of and. hu h xpcaion on h righ-hand id of Equaion 7 ak h following form : n m [ E E ] E E E p!! n m 8 n m n m p whr all h xpcaion ar akn undr h rik-nural maur Q condiional on W mu compu h nm-drivaiv of h pudo-pric w.r.. o h avragd n m varianc and h cro-momn E for all ordr nm 5. Sinc h diffrniaion of h pudo-pric dpnd on h coningn claim pcificaion i will b don lar and w ar by driving h cro-momn givn h quar-roo volailiy diffuion in Equaion. F. II h Frobniu oluion Dfin h cro-momn gnraing funcion o b givn by : g a b; Exp a Q whr E dno E F d b. Givn h xprion of w can aily ha h nm h cro-momn can b wrin a : d 9 and in Equaion 5 n m n m n m d d n m g a b E Conidring g a a funcion of Fynman-Kac horm allow u o wri g a h oluion o a parial diffrnial quaion PDE ha ak h following form : g g g σ κθ λ η g g 5 Hopfully w won nd o g o vry larg valu of n and m. Sabani 3 and Apl al only nd h cond and find h hird ordr of ngligibl impac. 7

9 8 whr and xp xp η b a. Following ahani 4 w aum ha g i log-linar in and can b wrin a : xp C D g whr D. and C. ar oluion o h wo following ordinary diffrnial quaion ODE : ' D D D D η λ σ 3 and ' C D C κθ 4 h xac oluion o h ODE ar givn by : ln ' σ κθ σ U C U U D 5 whr U olv h following linar homognou cond-ordr ODE : ' ' ' ' U U U U U η σ λ 6 ahani 4 provid h xac oluion o hi ODE ha involv Whiakr funcion : Ψ Φ σ λ σ λ σ λ σ λ b b a W b b a M b a U 4 xp 4 xp ; 7 whr M. and W. ar Whiakr funcion and conan 6 Φ and Ψ can b drmind uing h iniial condiion in ODE 6. o compu h cro-momn w mu compu h drivaiv of funcion D and C w.r. variabl a and b ha appar in h fir and h hird argumn in Whiakr 6 Φ and Ψ will dpnd on h variabl a and b.

10 funcion a wll a in Φ and Ψ which i vry havy. Inad w will dvlop a Frobniu 7 oluion o h ODE 6 Appndix for dail. Making h chang of variabl z xp and dfining h funcion S z by : β S z z U λ β lad o h following ri oluion : 8 S z Φ n k n n β nβ β z Ψ k n β z 9 n whr conan Φ and Ψ ar drmind uing h fac ha S and S ' β and funcion k. ar compud rcurivly by h following formula : n n k k n aσ ε k 4 ε σ ε ak ε bk ε n n n nε ; n whr k i an arbirary conan. Onc funcion D and C ar compud according o Equaion 5 h compuaion of h momn can b achivd by diffrniaing h funcion g in Equaion. W only nd o diffrnia h ri oluion in Equaion 9 by runcaing i a a fini ordr inad of daling wih Whiakr funcion in Equaion 7. h compuaion of h cro-cnrd-momn in Equaion 8 i hu raighforward. 3 h no man rvrion ca In h ca of no man rvrion i.. h funcion η in PDE i a conan and h momn gnraing funcion 8 i dfind by : g η; E xp η d 3 7 Undr om rgulariy condiion an ODE may hav a ri oluion aking h form z β k z n. n 8 I can b n a a zro-coupon pric by conidring Ingroll and Ro 985 modl. n η a h inananou inr ra in Cox 9

11 W do no nd a Frobniu ri oluion inc h momn gnraing funcion g i givn by a impl clod-form xprion [ Cox Ingroll and Ro 985] : D C g xp 3 whr ω η D ω λ ω δ ω κθ δ κθ C log ω λ 33 σ δ σ ω λ ω λ ησ ; δ ω λ Noic ha h oluion U o h ODE 6 i alo vry ay o compu by : λ ω ω λ U η ; xp λ ω xp λ ω 34 ω ω h formula will b ud lar for pricing om xoic drivaiv on quii. A hi ag w ar abl o u Equaion 8 o compu h approxima pric for any coningn claim uing a aylor xpanion a long a w can compu h drivaiv of h pudo-pric in Equaion 6 w.r.. avragd varianc ihr analyically or numrically. h nx cion will prn om andard and om xoic opion on boh man rvring and non-man rvring undrlying a. h compuaion of andard opion pric in a ochaic volailiy modl will allow u o chck for h accuracy of h ri xpanion in impl ca whr mi-clod-form oluion xi among which Hon 993 and ahani 4 modl 9. III aluaion formula for xoic opion In hi cion w will rmind om wll-known clod-form pricing formula for pah-dpndn ock opion undr a conan volailiy modl which will b ud in h ri xpanion for pricing h am pah-dpndn opion undr h quar-roo 9 h conan volailiy counrpar of Hon 993 and ahani 4 modl ar rpcivly Black and Schol 973 and ongaff and Schwarz 995 modl which will b ud o compu h pudopric in Equaion 6.

12 ochaic volailiy modl. In h ca of a man rvring a and a conan volailiy blanc and Scaill 998 propo clod-form oluion up o an invrion of Fourir ranform for om pah-dpndn opion on affin yild among which h arihmic avrag opion. hi approach will b ud o driv a pricing formula for crdi prad avrag opion. W alo will u h diribuion of h fir paag im for an Ornin-Uhlnbck proc o a boundary drivd in blanc al. o pric digial barrir crdi prad opion. Onc h pudo-pric of xoic opion ar compud w will u hm in Equaion 8 in ordr o pric h am xoic undr a ochaic volailiy modl. Barrir and digial a-or-nohing ock opion h formula ar drivd in dail in Rinr and Rubinin 99 and Haug 997. h diffuion of h a variabl undr h filraion G and h avrag varianc ar givn in Equaion. h ock pric i givn by and h pudopric of a down-ou call wih rik pric K and barrir i givn whn K by : d K d d K d r K C r r DO Ν Ν Ν Ν ψ ψ ψ ψ 35 whr ln ; d r ψ ψ 36 and Ν i h andard normal cumulaiv funcion. h pudo-pric of a digial down-ou a-or-nohing opion can b obaind by aking a rik pric qual o K in Equaion 36 : d d r Dig DO Ν Ν ψ ψ 37

13 whil h pudo-pric of an up-ou pu wih K i givn by : d K d d K d r K P r r UO Ν Ν Ν Ν ψ ψ ψ ψ 38 Digial cah-or-nohing crdi prad opion In h ca of man rvrion i.. h diffuion of h a variabl undr h filraion G i an Ornin-Uhlnbck proc givn in Equaion 4 and h avrag varianc ar givn in Equaion 5 : dw d d γ µ 39 Dfining { } : inf o b h fir hiing im h diribuion of i drivd in blanc al. : d Q ; χ 4 3 inh coh xp γ µ γ µ π A digial up-in cah-or-nohing crdi prad opion wih barrir pay off a crain amoun a mauriy if h crdi prad nvr fall blow h barrir during h opion lif. I pudo-pric i hn givn by : r UI d r Dig ;ln χ 4 In ordr o apply h ri xpanion in Equaion 8 h drivaiv of h pudopric w.r.. ar compud by diffrniaing undr h ingral ign.

14 Opion on avrag crdi prad Dnoing h avrag crdi prad by Y d whr h diffuion of i givn in Equaion 39 h pudo-pric of a call on avrag crdi prad wih rik K can b obaind by : C Av r r Y E xp K ln K y xp K Q Y dy 4 whr h dniy of Y can b compud by invring i characriic funcion and hn h ingraion in Equaion 4 will b don numrically uing Gauian quadraur [ ahani 4]. o b coninud I Numrical rul In ordr o a h fficincy and h accuracy of h propod mhodology w pric om plain vanilla and xoic opion in Hon 993 and ahani 4 quar-roo ochaic volailiy framwork. For andard opion Hon 993 and ahani 4 opion pric will b conidrd a h ru pric oward which h ri xpanion mu convrg. abl o 4 how h rul for andard call opion. I i found ha h numrical pric convrg rapidly o h ru pric; a mo w nd h 4 h ordr o achiv a good accuracy. For xoic opion inc hr ar no clod-from formula in a ochaic volailiy modl w will ak h aympoic pric a h ru pric. abl 5 o 8 how h rul for barrir and digial barrir opion. h numrical pric ar hown o h characriic funcion of d φy Y i givn by E xp φ y Q Y dy and can b compud a a zro-coupon bond pric in aick 977 modl whr h inananou ra i φ. 3

15 convrg rapidly a low xpanion ordr which prov h accuracy of h ri mhod vn for xoic opion. h ri xpanion i alo found o b vry fficin. In fac h cro-cnrd momn ar compud for a givn of paramr and hy ar ud o pric a many opion a w wan imply by adding rm in h ri xpanion which mak h compuaion im vry mall. Concluion W propo a ri xpanion pricing formula for xoic opion on ock and ohr man rvring a whn h volailiy i ochaic. h main purpo of h xpanion mhod i h compuaion of h momn of h avragd varianc in a quar-roo volailiy modl which i don uing ihr a clod-form formula if hr i no man rvrion; or uing a Frobniu ri oluion in h ca of a man rvring proc. h ri xpanion mhod i found o b vry accura and vry fficin whn pricing ock and crdi prad andard and xoic opion. o b coninud Appndix o b coninud 4

16 Bibliography Apl. G. Winklr and U. Wyup aluaion of Opion in Hon Sochaic olailiy Modl Uing Fini Elmn Mhod Forign Exchang Rik Rik Publicaion. Bakhi G. C. Cao and Z. Chn 997 Empirical Prformanc of Alrnaiv Opion Pricing Modl h Journal of Financ ol Ball C. and A. Roma 994 Sochaic olailiy Opion Pricing Journal of Financial and Quaniaiv Analyi ol Black F. and M. Schol 973 h aluaion of Opion and Corpora iabilii Journal of Poliical Economy ol Clwlow. J. and C. R. Srickland 997 Mon Carlo aluaion of Inr Ra Drivaiv undr Sochaic olailiy h Journal of Fixd Incom ol Cox J. J. Ingroll and S. Ro 985 A hory of h rm Srucur of Inr Ra Economrica ol Davydov D. and. inky Pricing and Hdging Pah-Dpndn Opion Undr h CE Proc Managmn Scinc ol Haug E.G. 997 h Compl Guid o Opion Pricing Formula McGraw-Hill. Hndron. and D. Hobon Papor Opion wih Sochaic olailiy Working papr Univriy of Bah. Hon S A Clod-form Soluion for Opion wih Sochaic olailiy wih Applicaion o Bond and Currncy Opion h Rviw of Financial Sudi ol Hull J. and A. Whi 987 h Pricing of Opion on A wih Sochaic olailii Journal of Financ ol Karaza I. and S.A. Shrv 99. Brownian Moion and Sochaic Calculu Springr rlag Nw York. Jacob K. and. i 4 Modling h Dynamic of Crdi Sprad wih Sochaic olailiy Financ and Sochaic ol

17 blanc B. and O. Scaill 998 Pah Dpndn Opion on Yild in h Affin rm Srucur Modl Financ and Sochaic ol blanc B. and O. Scaill A Corrcion No on h Fir Paag im of an Ornin-Uhlnbck Proc o a Boundary Financ and Sochaic ol wi A. Opion aluaion undr Sochaic olailiy Financ pr California. ongaff F.A. and E.S. Schwarz 995 aluing Crdi Drivaiv h Journal of Fixd Incom ol Prigan J.-. O. Rnaul and O. Scaill An Empirical Invigaion ino Crdi Sprad Indic Journal of Rik ol Rinr E. and M. Rubinin 99 Braking down h barrir Rik ol Romano M. and N. ouzi 997 Coningn Claim and Mark Compln in a Sochaic olailiy Modl Mahmaical Financ ol Sabani S. Sochaic volailiy Inrnaional Journal of horical and Applid Financ ol Sabani S. 3 Sochaic volailiy and h Man Rvring Proc h Journal of Fuur Mark ol Schöbl R. and J. Zhu 999 Sochaic olailiy Wih an Ornin-Uhlnbck Proc: An Exnion Europan Financ Rviw ol Schwarz E.S. 997 h Sochaic Bhavior of Commodiy Pric: Implicaion for aluaion and Hdging h Journal of Financ ol Sin E. and J. Sin 99 Sock Pric Diribuion wih Sochaic olailiy: An Analyic Approach h Rviw of Financial Sudi ol ahani N. Crdi Sprad Opion aluaion undr GARCH Working papr - 7 Canada Rarch Chair in Rik Managmn HEC Monréal. hp:// ahani N. 4 aluing Crdi Drivaiv Uing Gauian Quadraur: A Sochaic olailiy Framwork h Journal of Fuur Mark ol

18 aick O.A. 977 An Equilibrium Characrizaion of h rm Srucur Journal of Financial Economic ol Zhu J. Modular Pricing of Opion Working papr Ebrhard-Karl-Univriä übingn. 7

19 abl abl : Call pric undr Hon modl Ordr Pric ru pric 8675 abl prn h rul of h valuaion of a call wihin Hon 993 modl for diffrn xpanion ordr. h ru pric i givn by Hon clod-form formula. h opion paramr ar ln ;. 4 ; K ; r. 5 ;. h modl paramr ar µ r ; ; γ. 5 ; σ. ; λ 4 ; κ and θ. 5. 8

20 abl : Call pric undr Hon modl Ordr Pric ru pric 5369 abl prn h rul of h valuaion of a call wihin Hon 993 modl for diffrn xpanion ordr. h ru pric i givn by Hon clod-form formula. h opion paramr ar ln ;. 4 ; K 9 ; r. 5 ;. h modl paramr ar µ r ; ; γ. 5 ; σ. ; λ 4 ; κ and θ. 5. 9

21 abl 3 : Call pric undr ahani modl abl 3 prn h rul of h valuaion of a call wihin ahani 4 quar-roo modl for diffrn xpanion ordr. h ru pric i givn by ahani mi-clod-form formula. h opion paramr ar ln. ;. 4 ; K. ; r. 5 ;. h modl paramr ar µ. 3;. ; γ. ; σ. ; λ 4 ; κ and θ. 5. Ordr Pric ru pric 9738

22 abl 4 : Call pric undr ahani modl abl 4 prn h rul of h valuaion of a call wihin ahani 4 quar-roo modl for diffrn xpanion ordr. h ru pric i givn by ahani mi-clod-form formula. h opion paramr ar ln.5 ;. 4 ; K. ; r. 5 ;. h modl paramr ar µ. 3;. ; γ ; σ. ; λ 4 ; κ and θ. 5. Ordr Pric ru pric 66558

23 abl 5 : Down-Ou Barrir Call pric undr Hon modl abl 5 prn h rul of h valuaion of a down-ou barrir call wihin Hon 993 modl for diffrn xpanion ordr. h opion paramr ar ln ;.4 ; K 9 ; 95; r. 5 ;. h modl paramr ar µ r ; ; γ.5 ; σ. ; λ 4 ; κ and θ. 5. Ordr Pric

24 abl 6 : Up-Ou Barrir Pu pric undr Hon modl abl 6 prn h rul of h valuaion of an up-ou barrir pu wihin Hon 993 modl for diffrn xpanion ordr. h opion paramr ar ln9 ;.4 ; K 9 ; 95; r. 5 ;. h modl paramr ar µ r ; ; γ.5 ; σ. ; λ 4 ; κ and θ. 5. Ordr Pric

25 abl 7 : Digial Down-Ou A-or-Nohing opion pric undr Hon modl Ordr Pric abl 7 prn h rul of h valuaion of a digial down-ou a-or-nohing opion wihin Hon 993 modl for diffrn xpanion ordr. h opion paramr ar ln ;. 4 ; K ; 95; r. 5 ;. h modl paramr ar µ r ; ; γ. 5 ; σ. ; λ 4 ; κ and θ. 5. 4

26 abl 8 : Digial Down-Ou A-or-Nohing opion pric undr Hon modl Ordr Pric abl 8 prn h rul of h valuaion of a digial down-ou a-or-nohing opion wihin Hon 993 modl for diffrn xpanion ordr. h opion paramr ar ln ;. 4 ; K ; 8; r. 5 ;. h modl paramr ar µ r ; ; γ. 5 ; σ. ; λ 4 ; κ and θ. 5. 5