Comparative Study Analytic and Numerical Methods for Solving Non-Linear Black-Scholes Equation with European Call Option

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1 Ieraioal Joral of Scieific a Iovaive aemaical Researc IJSIR olme 3 Isse ebrary 5 PP 6678 ISSN 34737X Pri & ISSN Olie wwwarcjoralsorg Comparaive Sy Aalyic a Nmerical eos for Solvig NoLiear BlackScoles Eqaio wi Eropea Call Opio ila Gülkaç eparme of aemaics acly of Sciece Kocaeli Uiversiy Umepe Camps438 Kocaeli TURKEY vglkac@kocaelier Absrac: I is work we apply He s varioioal ieraio meo for obaiig aalyic solios o oliear BlackScoles eqaio wi boary coiios for Eropea opio pricig problem Te aalyical solio of e eqaio is calclae i e form a coverge power series wi easily compable compoes Te powerfl I meo is capable of alig bo liear a oliear eqaios i irec maer A ree approimae merical meos of e o liear BlackScoles eqaio wi Eropea call opio are efie sig fiie iffereces fiie ifferece eqaios wi aleraive erivaio a Eler meo of fiie ifferece eqaios wi aleraive erivaio Te resls obaie epilici fiie ifferece meo a fiie ifferece meo wi aleraive erivaio a Eler meo of fiie ifferece meo wi aleraive erivaio a e resls gave a goo agreeme wi e previos meos [4 5 6 ] Keywors: He s ariaioal ieraio meo BlackScoles eqaio Eropea call opioree boary problem iie ifferece Eler meo INTROUCTION iace is oe of e mos rapily cagig a fases growig areas i e corporae bsiess worl Becase of is rapi cace moer fiacial isrmes ave become eremely comple As sock prices all over e worl ramaically rise a fall ivesors are coially i searc for fiacial isrmes o rece e variabiliy of eir porfolio vales Coseqely e volailiy i e marke receives mc ieres from marke paricipas a researcers New maemaical moels are esseial o impleme a rice ese ew fiacial isrmes Te worl of corporae fiace oce maage by bsiess se is ow corolle by maemaicias a comper scieiss iacial secriies ave become esseial ools for corporaios a ivesors over pas few ecaes Opio pricig eory as mae a grea leap forwar sice e evelopme of e Black Scoles opio pricig moel by iser Black a yro Scoles i [] a previosly by Rober ero i [] Recely may scieiss ave pai more aeio o ew meos for solvig opio valaio Carao [3] a Wilmo e Al [4] se fiie ifferece meos for opio valaio Baroe Aesi [5] BaroeAesi a Ellio [6] Geske a Joso [7] cilla [8] BaroeAesi a Waley [9] Gülkaç [] evelope a accrae aalyical approimaio meo ay aors ave applie several iffere meos o varios applicaios [] I is paper we will se He s variaioal ieraio meo I propose by He [8] is oe of e meos wic as receive mc aeio I as bee sow by may aors o be a powerfl maemaical ool for solvig varios kis of fcioal eqaio [935] A ree approimae merical solio of e BlackScoles eqaio are efie I e prese meos firs a approimae merical solio of BlackScoles eqaio is efie sig epilici fiie ifferece eqaio a e mari meo of aalysis of e sabiliy of e meo is also ivesigae Seco a approimae merical solio of e BlackScoles ARC Page 66

2 ila Gülkaç eqaio is efie sig fiie ifferece eqaios wi aleraive erivaio a covergece of e meo is also ivesigae Tir a approimae merical solio of e problem is efie sig Eler eo for fiie ifferece eqaios wi aleraive erivaio BLACKSCHOLES EQUATIONS Te BlackScoles moel is oe of e mos impora coceps i moer fiacial eory I was evelope i 973 by iscer Black a yro Scoles [] a Rober C ero [] a is sill wiely se oay a regare as oe of e bes ways of eermiig fair prices of opios I e fiace e syle or family of a opio is a geeral erm eoig e class io wic e opio falls sally efie by e aes o wic e opio may be eercise Te vas majoriy of opios are eier Eropea or America opios A Eropea opio may be eercise oly a e epiraio ae of e opio ie a a sigle preefie poi i ime A America opio o e oer a may be eercise a ay ime before e epiraio ae or bo e pay offwe i occrs is via: a[sk] for a call opio a[ks] for a p opio were K is srike price a S is spo price of e erlyig asse amos liear Black Scoles eqaio S ss rss r S T S T as S 3 S TmaSK 4 Were S : S > a provies bo a opio pricig formla for a Eropea opio a a egig porfolio a replicaes e coige claim assmig a [] e volailiy of e erlyig asse; K e eercise srike price; T e epiry; r e riskfree ieres rae I is easy o imagie a e qalificaory spposiios meioe i e liear BlackScoles eqaio are ever flfille i realiy e o rasacio coss large ivesor prefereces a icomplee markes ey are likely o become realisic a e classical moel resls i srogly oliear I is paper we will be ierese i oliear BlackScoles eqaio for Eropea opios wi a cosa re a o cosa moifie volailiy fcio; ~ : ~ S S SS Accorig o ese circmsaces eqaio becomes e followig oliear BlackScoles eqaio wi e ermial a boary coiios: ~ S s ss S SS rss r 5 Were S S ~ SW S > T 6 Ieraioal Joral of Scieific a Iovaive aemaical Researc IJSIR Page 67

3 Comparaive Sy Aalyic a Nmerical eos for Solvig NoLiear BlackScoles Eqaio wi Eropea Call Opio S S K for S < 7 for T 8 S r T S Ke as S 9 I orer o able o solve eqaio 5 wi ermial a boary coiios we perform e followig variable rasformaio [36 4]: S S l T K e K Sice S Ke a S iffereiaio yiels: S S S 3 S S SS S S 4 S Sbsiig ese erivaives io eqaio 5 leas o S ~ S rs rs 5 A a fial mliplicaio by gives S ~ 6 r ~ Were a ~ T epes o e volailiy moel R a T Now solves 6 o e rasforme omai R T ~ sbjec o e followig iiial a boary coiios reslig from e for R 7 as 8 e as 9 3 ETHOS 3 He s ariaioal Ieraio eo I Cosier e iffereial eqaio L N g L a N are respecively liear a oliear operaors a g a kow aalyical fcio I [ 8] He propose e variaioal ieraio meo were a correcio fcioal for eqaio ca be wrie as Ieraioal Joral of Scieific a Iovaive aemaical Researc IJSIR Page 68

4 ila Gülkaç Ieraioal Joral of Scieific a Iovaive aemaical Researc IJSIR Page 69 λ g N L ~ Were λ is a geeral Lagrage mliplier [37] wic ca be ieifie opimally via variaio eory a ~ is cosiere as a resrice variaio ie ~ δ I is meo we firs eermie e Lagrage mliplier λ a will be ieifie opimally via iegraio by pars Te sccessive approimaios of e solio will be reaily obaie po sig e eermie Lagrage mliplier a sig e iiial approimaio Coseqely e solio is give by lim 3 Te BlackScoles Eqaio wi I To clarify e basic ieas of He s variaioal ieraio meo we cosier eqaio 6 wi e iiial a boary coiios Accorig o variaioal ieraio meo I we erive a correc fcioal as follows: λ ~ ~ ~ 3 ~ Were akig e above correcio fcioal saioary we ave δ δ λ δ 4 is yiels e followig saioary coiios: λ 5 λ 6 λ 7 Sbsiig is vale io Eq 3 resls e iera formla: e [ ] e 8 We obai e followig sccessive approimaios: 9 3 e e 3 3 e 33

5 Comparaive Sy Aalyic a Nmerical eos for Solvig NoLiear BlackScoles Eqaio wi Eropea Call Opio Te I amis e se of lim e 34 Tis gives e eac solio Obaie po sig e Taylor epasio of e 4 APPROXIATE NUERICAL ETHOS 4 irs eo of Solio 35 Usig e sal forwar a ceral ifferece approimaio for e ime a spaial erivaives i eqaio 6 akes e followig form: i 36 j r i j r r i j r r i j ~ were i jk a a r k i i 4 Sabiliy of e irs eo To ivesigae e sabiliy aalysis of eqaio 36 i is coveie o se mari aalysis meo [38] Eqaio 36 ca be wrie e followig form: j A j Were 37 r r r r r rr r r A r r r r r O r r r is of orer N Te eigevale μ of A 38 Braer s Teorem: Le iagoal eleme a e circles ss ss P s be e sm of e mole of e erms alog e s row eclig e Te every eigevale of a μ a P [38] ss s A lies isie or o e boary of a leas oe of Te fiie ifferece eqaios will be sable we e mole of every eigevale of A oes o ecee oe a is we μ r r r r r 39 μ r r 4 Every posiive a every posiive for r> μ erefore e eqaios are coiioally sable as μ for all vales of a r 4 Seco eo of Solio Cosier e eqaio 6 were saisfies e iiial coiio g X a as kow boary vales a a X > If e eriveive a is replace by Ieraioal Joral of Scieific a Iovaive aemaical Researc IJSIR Page 7

6 ila Gülkaç { O } a cosiere as a cosa eq 6 ca be wrie as e oriary iffereial eqaio [38] { } { } 4 I e follows a e vales i approimaig i will be eac solios vales of e sysem of oriary iffereial eqaios { } { } 3 3 Ieraioal Joral of Scieific a Iovaive aemaical Researc IJSIR Page 7 { } Or is eqaio sysems ca be wrie as i i i i 4 Were a are kow boary vales Tese ca be wrie i mari form as i e as b A 43 e solio of e oriary scalar iffereial eqaio b A A b a were are iepee a g saisfies e iiial coiio easily sow by e meo of seperaio of variables o be A } {ep b A g A b A ep A b g A b 44 45

7 Comparaive Sy Aalyic a Nmerical eos for Solvig NoLiear BlackScoles Eqaio wi Eropea Call Opio k A b {ep k A} g A b 46 k A b {ep A}{ep ka} g A b 47 k A b {ep ka} A b 48 if all boary vales are zero see eq 8 k {ep ka} 49 Te boary vales ca always be elimiae if we are cocere more say wi sabiliy a wi a pariclar merical solio 4 Sabiliy of Seco eo To ivesigae e sabiliy aalysis of eqaio 43 i is coveie o se mari aalysis meo [38] Te eigevale μ of A By Gerscgori s eorem e mols of larges eigevales cao ecee e larges sm of moli of erm alog ay row or colm of A [38]; ece μ ma 5 μ ma provig a is meo coverges for all posiive a k vales 43 Tir eo of Solio Eqaio 49 ca be wrie as Eler meo if all boary vales are zero i i f i j were f i j A 5 i i Ai 53 Tis gives e ieraive solio [ I A] i i Sabiliy of Tir eo To ivesigae e sabiliy aalysis of eqaio 5 i is coveie o se mari aalysis meo [38] I A ma By Gerscgori s eorem if ξ larges eigevale is of mari IA 5 Ieraioal Joral of Scieific a Iovaive aemaical Researc IJSIR Page 7

8 ila Gülkaç ξ ma 55 ξ 56 ma Provig a is ieraio coverges for all posiive vales of a k 5 CONCLUSION I is paper He s variaioal ieraio meo as bee applie sccessflly for solvig oliear BlackScoles eqaio wi Eropea call opio He s variaioal ieraio meo sccessflly worke o give eac solio o is problem Also is meo provies e solio i a rapily coverge form He s variaioal ieraio meo gives several sccessive approimaios rog sig e ieraio of e correcio fcioal I is meo ere is o specific ee o ale oliear erms He s variaioal ieraio meo provies a efficie meo for alig is oliear beavior igre igre igre 3a igre 4 illsraes call opio vales of o liear BlackScoles eqaio wi variaioal ieraio meo igre 5 illsraes call opio vales of o liear BlackScoles eqaio fiie ifferece wi aleraive erivaio igre 6 illsraes call opio vales of oliear BlackScoles Eqaio wi epilici fiie ifferece a igre 7 illsraes call opio vales of oliear BlackScoles Eqaio wi Eler meo Te seco meo is approimae merical solio of e oliear BlackScoles eqaio wi Eropea call opio efie sig eplici fiie ifferece meo a aalysis of e sabiliy of e seco meo is also ivesigae a eplici fiie ifferece eqaios were fo o be coiioally sable for all a r> Te ir meo is efie by aleraive erivaives Compig e procere of is meo is very effecive a aalysis of e sabiliy of is ieraive meo is also ivesigae a ieraive meo was fo o sable for all posiive vales of a k Te for meo is efie by Eler eo for fiie ifferece meo wi aleraive erivaives Compig e procere of is ieraive meo is also ivesigae a ieraive meo was fo o sable for all posiive vales of a k All of ese meos ave several avaages irs ey ca evalae opio posiios wi e same mariy for esseially all possible asse prices simlaeosly Seco ey meos are believe o be aapive o oer opios vale problem A ir ey solve e opimm eercise boary ogeer wi opio prices wio era eergy REERENCES [] Black a Scoles Te pricig of opios a corporae liabiliies J Poli Eco [] ero R C Teory of raioal opio pricig Bell J Eco [3] Corao G A more accrae fiie ifferece approimaios for e valaio of opios J ia Qar Aal [4] P Wilmo S Howiso a J ewye Te maemaics of fiacial erivaives Cambrige Uiversiy press New York 995 [5] BaroeAesi G Te Saga of e America p J Bakig iace [6] BaroeAesi G a Ellio R Approimaios for e vales of America opios Socasic Aal Appl [7] Geske Ra Joso H Te America p opios vale aalyically J iace [8] acilla W A aalyical approimaio for e America p prices Av res Opios Res [9] BaroeAesi G a Waley R Efficie aalyic approimaio of America opio vales J iace Ieraioal Joral of Scieific a Iovaive aemaical Researc IJSIR Page 73

9 Comparaive Sy Aalyic a Nmerical eos for Solvig NoLiear BlackScoles Eqaio wi Eropea Call Opio [] Gülkaç Te omoopy perrbaio meo for e Black Scoles eqaio J of Saisical Comp A Simlaio [] Boyle P a ors T Opio replicaio i iscree ime wi rasacio coss J iace [] A Jgel as kleie iieelemeeskrip Uiversiä aiz [3] Hll J a Wie A Te pricig of opios o asses wi socasic volailiies J iace [4] Geske Ra Roll R O valig America call opios wi e BlackScoles Eropea formla J iace [5] Ha H a W X A fas merical meo for BlackScoles eqaio of America opios SIA J Nmer Aal [6] Paii R a Srivasav RP Opio pricig wi elli rasform a Comp oellig [7] eyer G H a vaer Hock J Te valaio of America opios wi e meo of lies Av res opios res [8] Zao J aviso a Corless R Compac fiie ifferece meo for America opio pricig J Comp Appl a [9] Zager Z Covergece of a leassqares oecarlo algorim for boe approimaig ses Appl a iace 35 9 [] Primbs JA a Raiam Traer beavior a is effece o asse price yamics Appl a iace 58 9 [] He J H Some asympoic meos for srogly oliear eqaios Iera J oer Pys B [] J H He Noperrbaive meos for srogly oliear problems isseraio eerlag im Iere GmbH Berli 6 [3] He J H Approimae aalyical solio for see page flow wi fracioal erivaives i poros meia Comp eos Appl ec Egg [4] He J H ariaioal ieraios meo for aoomos oriary iffereial sysems Appl a Comp 4 53 [5] He J H A ew approac o oliear parial iffereial eqaios Comm Noliear Sci Nmer Siml [6] He J H A variaioal ieraio approac o oliear problems a is applicaios ec Appl [7] He J H ariaioal ieraio meo A ki of oliear aalyical eciqe: Some eamples Iera J NoLiear ec [8] He J H A geeralize variaioal priciple i micromorpic ermo elasiciy ec Res Comm [9] Ablwafa E Abo A a amo A A Te solio of oliearcoaglaio problem wi mass loss Caos Solios racals [3] Wazwaz A Te variaioal ieraio meo for raioal solios for K K Brgers a cbic Bossiesq eqaios J Comp Appl a [3] Cagzeg Q Eac solios o oliear iffsio eqaios obaie by a geeralize coiioal symmery meo IA J Appl a [3] Biazar J Golami P a Hosseii K ariaioal ieraio meo for solvig okker Plack eqaio Joral of e rakli Isie [33] omai S a Abasa S Applicaio of He s variaioal ieraio meo o Helmoz eqaio Caos Solios racals [34] Wazwaz A A compariso bewee e variaioal ieraio meo a Aomia ecomposiio meo J Comp Appl a Ieraioal Joral of Scieific a Iovaive aemaical Researc IJSIR Page 74

10 ila Gülkaç [35] Abo A a Solima A A ariaioal ieraio meo for solvig Brger a cople Brger eqaio J Comp Appl a [36] ürig B orier a Jügel A Hig orer compac fiie ifferece scemes for a oliear BlackScoles eqaio I J Appl Teor iace [37] Ioki Sekie H a ra T Geeral se of e Lagrage mliplier i oliear maemaical pysics ariaioal mmmeos i e ecaics of Solis Pergamo Press New York [38] G Smi Nmerical Solio of Parial iffereial Eqaios Clerao PressOfor 993 igre Call opio vales of oliear BlackScoles Eqaio wi varioioal ieraio meo 8 5 igre Call opio vales of oliear BlackScoles Eqaio wi varioioal ieraio meo 5 Ieraioal Joral of Scieific a Iovaive aemaical Researc IJSIR Page 75

11 Comparaive Sy Aalyic a Nmerical eos for Solvig NoLiear BlackScoles Eqaio wi Eropea Call Opio igre 3Call opio vales of oliear BlackScoles Eqaio wi varioioal ieraio meo 5 igre4call opio vales of oliear BlackScoles Eqaio wi varioioal ieraio meo 5 Ieraioal Joral of Scieific a Iovaive aemaical Researc IJSIR Page 76

12 ila Gülkaç igre5 Call opio vales of oliear BlackScoles Eqaio fiie ifferece wi aleraive erivaio 5 igre6 Call opio vales of oliear BlackScoles Eqaio wi epilici fiie ifferece 5 Ieraioal Joral of Scieific a Iovaive aemaical Researc IJSIR Page 77

13 Comparaive Sy Aalyic a Nmerical eos for Solvig NoLiear BlackScoles Eqaio wi Eropea Call Opio igre7 Call opio vales of oliear BlackScoles Eqaio wi Eler meo 5 AUTHOR S BIOGRAPHY r ila Gülkaç is workig as Associae Prof a Kocaeli Uiversiy sciece a ars facly eparme of maemaics Her fiels of ieres icle merical solios for free a movig boary problems Ieraioal Joral of Scieific a Iovaive aemaical Researc IJSIR Page 78