Valuation of European Call Options via the Fast Fourier Transform and the Improved Mellin Transform

Transcription

1 Joural of Mahmaal Fa, 6, 6, Publshd Ol May 6 SRs. h:// h://dx.do.org/.436/mf.6.68 Valuao of Euroa Call Oos va h Fas Fourr rasform ad h Imrovd Mll rasform Suday Emmaul Fagba, Chuma Rahal Nwozo * Darm of Mahmaal Ss, Ek Sa Uvrsy, Ado Ek, Ngra Darm of Mahmas, Uvrsy of Ibada, Ibada, Ngra Rvd 8 Marh 6; ad 8 May 6; ublshd 3 May 6 Coyrgh 6 by auhors ad Sf Rsarh Publshg I. hs work s lsd udr h Crav Commos Arbuo Iraoal Ls (CC BY. h://ravommos.org/lss/by/4./ Absra hs ar osdrs h valuao of Euroa all oos va h fas Fourr rasform ad h mrovd Mll rasform. h Fourr valuao hqus ad Fourr vrso mhods for dsy alulaos add a vrsal ool o h s of advad hqus for rg ad maagm of faal drvavs. h Fas Fourr rasform s a umral aroah for rg oos whh ulzs h harars fuo of h udrlyg srum s r ross. h Mll rasform has h ably o r omlad fuos by ralzao of s may rors. Mll s rasformao s losly rlad o a xdd form of ohr oular rasforms, arularly h Lala rasform ad h Fourr rasform. W osdr h fas Fourr rasform for h valuao of Euroa all oos. W also xd a framwork basd o h Mll rasforms ad show how o modfy h mhod o valu Euroa all oos. W oba a w gral quao o drm h r of Euroa all by mas of h mrovd Mll rasform. W show ha our gral quao for h r of h Euroa all oo rs o h Blak-Shols-Mro formula. h umral rsuls show ha h rmdous sd of h fas Fourr rasform allows oo rs for a hug umbr of srks o b valuad vry radly bu h damg faor or h grably aramr mus b arfully hos s orols h sy of h fluuaos ad h magud of h fuoal valus. h mrovd Mll rasform s mor aura ha h fas Fourr rasform, ovrgs fasr o h Blak- Shols-Mro modl, rovds aura omarabl rs ad h aroah a b rgardd as a good alrav o xsg mhods for h valuao of Euroa all oo o a dvdd ayg sok. ywords Blak-Shols Paral Dffral Equao, Euroa Call Oo, Fas Fourr rasform, * Corrsodg auhor. How o hs ar: Fagba, S.E. ad Nwozo, C.R. (6 Valuao of Euroa Call Oos va h Fas Fourr rasform ad h Imrovd Mll rasform. Joural of Mahmaal Fa, 6, h://dx.do.org/.436/mf.6.68

2 Imrovd Mll rasform, Mll rasform. Iroo I h as dads, oo rg has bom o of h maor aras modr faal hory ad ra. S h roo of h lbrad Blak-Shols oo-rg modl, whh assums ha h udrlyg sok r follows a gomr Browa moo (GBM, hr s a xlosv growh radg avs o drvavs h worldwd faal marks []. [] dvlod a fas Fourr mhod o omu oo rs for a whol rag of srks. hs mhod maks us of h harars fuo of h udrlyg ass r. h us of h fas Fourr rasform mhod s movad by h followg rasos: h algorhm has sd advaag. hs abls h Fourr rasform algorhm o alula rs auraly for a whol rag of srks. h harars fuo of h log-r s kow ad has a sml form for may modls osdrd lraur whl h dsy s of o kow h losd form. Oo valus a b alulad umrally by mullyg a ayoff fuo wh raso dsy of a udrlyg ass, h akg s dsoud xao wh rs o a quval margal masur (s [3] ad [4]. hs mhod of margal rg s of omud wh rs o h sa of h ass, ds of osg mor hallgg. I, [5] ord h mhod of usg h Mll rasform o solv h assoad Blak-Shols aral dffral quao for a Euroa all oo. Mll rasforms oo hory wr also rod by [6]. hy drvd gral rrsaos for h r of Euroa ad Amra bask u oos usg Mll rasform hqus. [7] drvd gral rrsaos for h rs of Euroa ad Amra u oos o a bask of wo-dvdd ayg soks usg gral mhod basd o h doubl Mll rasform. hy showd ha by h domoso of h gral quao for h r of Amra bask u oo, h gral quao for h r of Euroa bask u oo a b obad drly. h Mll rasform mhod for h valuao of h Amra owr u oo wh o-dvdd ad dvdd ylds rsvly was osdrd by [8]. hy usd h Mll rasform mhod o drv h gral rrsaos for h r ad h fr boudary of h Amra owr u oo. hy also xdd hr rsuls o drv h fr boudary ad h fudamal aaly valuao formula for rual Amra owr u oo. For mahmaal bakgrouds, sorad alaos of rasform mhods faal oxs s [9]-[8] us o mo a fw. I hs ar, w osdr h valuao of h Euroa all oo va h fas Fourr rasform ad h mrovd Mll rasform o a dvdd ayg sok. h rs of h ar s orgazd as follows. So rss ovrvw of h Blak-Shols-Mro modl. So 3 osdrs som fudamal rors of h Fourr rasform ad h fas Fourr rasform mhod for h valuao of Euroa oos. I So 4, w rs h Mll rasform, som bas rors ad h alao of h mrovd Mll rasform h hory of Euroa all oo valuao. So 5 rss som umral xamls ad dsusso of rsuls. So 6 oluds h ar.. Blak-Shols-Mro-Lk Valuao Formula W osdr a mark whr h udrlyg ass r S, s govrd by h sohas dffral quao of h form ds ( r q S d + σ S d W, < S < ( whr σ s h volaly, r s h rsklss rs ra, q s h dvdd yld ad W s a o-dmsoal Wr ross. Sadard arbrag argums show ha ay drvav u( S, wr o S mus sasfy h aral dffral quao [9]. horm [5]: L S do h r of h udrlyg ass, σ > h volaly, r h rsklss rs ra, h owr of h oo, q h dvdd yld ad W h Wr ross. If h udrlyg r of h ass S follows a radom ross (Gomr Wr ross ds ( r q + ( σ S d + σs dw ( 339

3 h h xl formula for h voluo of h udrlyg r of h ass s gv by Proof: L Dffrag (4 w hav x σ σ S S r q + W (3 ( S S ( S ( S Rall from Io s lmma ad usg ( for ay drvav u( S, From (, w a wr for u u S, l S (4 u S, u S, u S,,, w hav g ( S,,, u S, d (, u S u S u S + + u S f d + g dw S S ( σ, f r q + S g σs (7 Subsug (4, (5 ad (7 o (6 ad rarragg h rms, w hav hrfor, S r q S ( + ( d l σ d S + + ( S σ S d σs d W S d ( l d d σ S + σ r q W (9 r ad vara aramr σ. o d- hus, l S s a Browa moo wh drf aramr σ rv a xl formula for h voluo of h sok r, w gra (9 from o o oba Equao ( a also b wr as S l σ r q + σ W S (5 (6 (8 ( x σ σ S S r q + W ( x σ σ S S r q + Z ( whr Z N(,. hrfor h sok dyam follows a log-ormal dsrbuo. For, ( boms x σ σ S S r + Z (3 Equao (3 shows ha la valla oo follows a log-ormal dsrbuo. 34

4 horm : L h udrlyg ass r ross S follows a logormal radom walk (gomr Wr ds ( r q + ( σ S d + σs dw usg h Io s lmma, udr h sadard arbrag argum h Blak-Shols aral dffral quao for u S, o S for valla owr oo whh ays dvdd yld s gv by ay drvav ( (,,, σ ( ( σ u S u S u S + + r q S + S ru S, S S Rmark For, (4 s kow as h rgular Blak-Shols-Mro aral dffral quao whh s gv by If u( S, u ( S, u S, u S,, σ u S r q S S ru S S S (, + + s a valla all oo, h (5 boms h Blak-Shols-Mro aral dffral quao for a valla all oo gv by 3 h soluo o (6 s obad as whr u S, u S, u S, r q S S ru S (, + + σ S S ( q( r( (4 (5 (6 u S, S N d N d (7 σ S σ S l + r q+ ( l + ( r q d, d d σ σ σ ( ad N (. s h ommuav dsrbuo fuo for h sadard ormal dsrbuo Euroa Oos Dfo Euroa s a oo ha a b xrsd oly a h xry da wh lar ayoff. Euroa oo oms wo forms amly Euroa all ad u oos. Dfo A Euroa all oo s a oo ha a b xrsd oly a xry ad has a lar ayoff gv by h dffr bw udrlyg ass r a maury ad h xrs r. Dfo 3 A Euroa u oo s a oo ha a b xrsd oly a xry ad has a lar ayoff gv by h dffr bw h xrs r ad udrlyg ass r a maury. For a Euroa oo o h udrlyg r of h ass S wh xrs r ad m o xry, w hav h ayoff for h Euroa all oo as (, ( h ayoff for h Euroa owr u oo s gv as + E S S 8 (, ( + E S S (9 Sg u( S, E( S, (5, h w hav h Blak-Shols-Mro aral dffral quao for h r of Euroa all oo gv by 34

5 wh boudary odos ad fal m odo gv by E S, E S, E S, r q S S re S (, + + σ S S (, as o [, ( E S S ( (, as o [, E S S ( (, ( + o [, E S S g S (3 3. Fas Fourr rasform Mhod for h Valuao of Euroa Call Oos hs so rss som fudamal rors of Fourr rasform ad h fas Fourr rasform mhod for h valuao of Euroa oos. h fas Fourr rasform was frs roosd by []. I surs ha h Fourr rasform of h all r xs by h luso of a damg faor. Morovr, Fourr vrso a b aomlshd by h fas Fourr rasform hs as. h rmdous sd of h fas Fourr rasform allows oo rs for a hug umbr of srks o b valuad vry radly. 3.. Fourr rasforms Dfo 4 Suos f ( x s absoluly grabl (,,.. f ( x s dfd as d f x x <, h h Fourr rasform of ( kx d F f x k f k f x x (4 h Fourr rasform s a gralzao of h omlx Fourr srs. Dfo 5 If f ( x s squar grabl, h h vrs Fourr rasform of f ( k s dfd as ( kx F f k f x f k dk π (5 3.. Som Fudamal Prors of Fourr rasforms L h Fourr rasform of f ( x b dfd as F( f ( x ( k f ( k h h followg fudamal rors hold as follows: Salg Prory Shfg/raslao Prory 3 Fourr rasform of Drvavs ( kx k F f υx f υx dx f (6 υ υ k( z+ x d d kx kx ( ( ( F f x x k f x x x f z z kz kx f z dz f k (7 34

6 d F f x k k f x x kf k dx hs ross a b rad for h h drvav o yld kx d d F f x k k f x x k f k dx kx d hus, a dffrao ovrs o mullao Fourr sa. 4 Covoluo Prory whr 5 Lar Prory 3.3. h Fas Fourr rasforms kx ( ( F f x g x k f x g x x dx dx kx k( x x ( f ( x dx ( g( x x dx kx k( x x ( f ( x dx ( g( x x dx f ( k g ( k ( d (8 (9 (3 f x g x f x g x x x (3 ( + ( + kx F af x ag x k af x ag x dx kx kx a f x dx+ a g x dx a f ( k a f k + h fas Fourr rasform (FF s roosd by []. h fas Fourr rasform s a ff algorhm for omug h dsr Fourr rasform of h form; N π m( ρ x ρ x, ρ,, 3,, N (33 N whr N s yally a owr of wo. Equao (33 rs h umbr of mullaos h rqurd N summaos from a ordr of N o ha of ( Nlog N, a vry osdrabl ro. L ad b wr as bary umbrs.. ρ ρ + ρ ad + wh,, ρ, ρ {,}, h (33 boms π m( ρ, ρ x ( ρ + ρ( + x(, N (34 ( ρ+ ρ( + ρ ( + ρ A x A A x (, (, h fas Fourr rasform a b dsrbd by h followg hr ss as Rmark ρ ( ρ+ ρ m ( ρ, x A, m ( ρ,, m ρ, A, m(, m ( ρ, ρ (3 (35 343

7 [] showd ha was fa ossbl o hav h dsr Fourr rasform valuad wh O( Nlog N arhm oraos. hs O( Nlog N algorhm s alld h fas Fourr rasform. As a mar of fa, ff mhods for valuag h dsr Fourr rasform hav alrady b dvsd as log ago as 85 by Gauss. Howvr, h world was dorma ul 965. Fgur llusras h hug dffrs bw O N ad O( N N. log h bas da of h fas Fourr rasform s o dvlo a aaly xrsso for h Fourr rasform of h oo r ad o g h r by mas of Fourr vrso. By mas of a hag of varabls x,dx d, s obsrvd ha h Fourr rasform bars a L M do h wo-sdd Lala ad Mll rasforms rsvly, h w hav srkg rsmbla o h Lala ad h Mll rasforms. I arular, f (. ad (. a ( F f, β L f, ω M f x, ω 3.4. h Charars Fuo h Doma of h Blak-Shols Modl h dyams of h sok r S a rsk-ural Blak-Shols world follows gomr Browa moo wh a o-dvdd yld s of h form ds rs d + σ S d W, < S < Ulzg h Io s formula w a xlly solv for from whh w a s ha log S w hav ha S S ; (.5σ log ( r + S+ σw S s logormally dsrbud. H for h harars fuo ( v of ϕ ( v (( σ.5 + log.5σ r S v v ϕ Fgur. Hug dffrs bw O( N ad ( log O N N. 344

8 3.5. Alao of h Fas Fourr rasform for h Valuao of Euroa Call Oos h Fourr rg hqus ad Fourr vrso mhods for dsy alulaos add a vrsal ool o h s of advad hqus for rg ad maagm of faal drvavs. h Fas Fourr rasform mhod s a umral aroah for rg oos whh ulzs h harars fuo of h udrlyg srum s r ross. hs aroah was rod by []. h Fas Fourr rasform mhod assums ha h harars fuo of h log-r s gv aalyally. Cosdr h valuao of Euroa all oo. L h rsk ural dsy of harars fuo of h dsy s gv by vs s log S b f ( s. h ϕ v : f s ds (36 h r of a Euroa all oo udr h rsk-ural valuao wh maury ad xrs r dod by C ( s gv by + C E S E r + r S ( + r S r S f s ds f s ds (37 whr s h log of h srk r.. Subsug (38 o (37 ylds log (38 S ( ( C x r f s ds (39 whh h xao s ak wh rs o som rsk-ural masur. S lm lm S h gral rrsao gv by (39 s o squar grabl,.. C ( L as ( d o zro for. W osdr a modfd vrso of h all r (39 gv by a C dos o C, a > (4 Equao (4 s squar grabl ovr h r ral l. Usg (4 ad (5, w hav ha v ( F v v d (4 v ( F v v d (4 π Subsug (4 o (4 w oba a w all valu h Fourr rasform doma as Subsug (43 o (39 w hav ha v a v C d (43 v a r ( s r v a ( s v f s dd s f s dd s 345

9 S for a >, r v s+ a + a ( v f s dd s s r v s+ a + a f ( s ( dd s r s ( a+ v ( a+ v+ f ( s ( dd ( a++ v s ( a++ v s r f ( s ds a v a v ( v + a ( v + a + ( a + lm lm lm. hrfor w hav ha ( v r ϕ ( ( r v a+ ϕ v a+ a + v a + + v a + a v + a+ v whr ϕ s h harars fuo of h log S gv by (36. Rmark 3 A suff odo for ( o b squar-grabl s gv by ( bg f. hs s quval E S <. [] sablshd ha f h grably aramr a, h domaor of (44 vashs Q a+ o ( wh, g a sgulary h grad. S h fas Fourr rasform valuas h grad a, h us of h faor a s rqurd. Now, w oba h dsrd oo r rms of ( v usg Fourr vrso of h form: Subsug (44 o (45 ylds v a v ( π π (44 a C R v dv R v dv (45 ( v ( a+ a r ϕ v C ( R d π v a a v a v By rogzg ha h all r s ral (v ral ar, odd magary. Du o h odo a, (46 s wll dfd. Afr dsrzg ad usg Smso s 3 h fas Fourr rasform as: ( a( λ ( u b N ( ( u (46 rul, (46 a b omud umrally by mas of x π η C x + 3 ( δ u bv + π v 3 N Nλ π v η, b, λ η N ad δ s h rokr dla fuo dfd as whr f δ ohrws H, h grao (46 s a alao of h summao (33. W formalzd rmark 3 h followg rsul blow. horm 3 L a >. h Fourr rasform of ( xss f ES a+ < a Proof: W o ha ( ES + < < s (47 346

10 whr From (4, w wr ha ( ( ( + ϕ a ES r r a+ a + a a + a ( a+ ( ( a + log S ϕ + ES a E ( ( v dv Combg hs wh ( < omls h roof. Rmark 4 For h Blak-Shols modl h grad (46 rs o whr ( σ v + a σ + as + ar + σ a + s x BS g a r s v a + a + a v + a + av + v + v (,,,, σ,, ( ( (,,,, σ,, ( + os ( σ σ ( + s ( σ σ g a r s v a a v a s r v v a a s r v From (48b, w a g mor fluuag grad by rasg ay of h aramrs σ, as,, ad r. h maguds of hs fluuaos g largr whh a b s from h xoal rm (48a. Rmark 5 A hs o s uavodabl o omm o h ho of h grably aramr a. A small valu of a s favourabl s hs rs boh h osllaos ad h maguds hrof. Howvr hoosg a oo small a ur h grad o a sor of muls fuo, whh s o raabl a all from a umral grao o of vw. hs follows from h fa ha h org v, h Blak-Shols grad (48a boms ( BS akg h lm of (48 as a ylds lm ( BS a a Smlarly (48 ds o fy as a O h ohr had, for > ( a σ + as + ar + σ a + s( a + a x a + a + a ( a σ + as + ar + σ a + s( a + a x.5.5 lm 4 3 a + a + a ( a σ + as + ar + σ a + s( a + a x.5.5 lm ( BS lm a a 4 3 a + a + a v ad by lg a, h grad (48a boms: x(.5σ v +.5a σ + as + ar +.5σ a + s lm ( BS lm g ( a,, r, s, σ,, v a a a + a + a v + a + av + v + v x(.5σ v + s ( v os (( m v vs (( m v 4 v + v wh m σ a + s + r +.5σ. Equao (48f drass vry fas as a fuo of v baus of h xoal rm. h followg rsul shows how h Blak-Shols grad aas s maxmum a v (48a (48b (48 (48d (48 (48f 347

11 horm 4 L v. h Blak-Shols grad r ϕ ( ( v a+ ( BS R + + ( + a a v a v v, whr (.5σ + log.5σ ϕ v r S v v aas s maxmum a Proof: From (48 w hav ha ( BS W s ha (48 s quval o hs follows s whr Morovr, v ( a σ + as + ar + σ a + s( a + a x a + a + a ( a σ + as + ar + σ a + s( a + a x.5.5 v R( ϕ ( v, for v 4 3 a + a + a Usg (48 ad (48k, w hav ha ϕ hs omls h roof. ( v v ( ϕ v ϕ ϕ (48g (48h R v v v (48 ϕ ( v ( r ( v ( a+ ( ϕ a + a v + a+ v ((.5σ ( ( +.5σ ( + (48 ϕ + r v a v v a v a (48k ( σ v + a σ + as + ar + σ a + s x a + a v + a+ v ( ( σ v + a σ + as + ar + σ a + s ( v ( a + ( v a ( a σ + as + ar + σ a + s x x.5.5 a a ( + 4. Mll rasform Mhod for h Valuao of Euroa Call Oos I hs so, w rs h Mll rasform, som bas rors ad h alao of h mrovd Mll rasform h hory of oo rg. 4.. Mll rasforms Dfo 6 h Mll rasform s a omlx valud fuo dfd o a vral sr h ω-la whos boudars ar drmd by h asymo bhavour of f(x as x + ad x. h Mll rasform of h fuo f(x s dfd as ω ( ω f f x x dx (49 348

12 s alld h Mll rasform of f ( x. I gral, h gral dos xs oly for omlx valus of ω + d suh ha ( UV,, whr u ad v dd o h fuo f ( x o rasform. hs ros wha s alld h sr of dfo of h Mll rasform ha wll b dod by ( UV,. I som ass, hs sr may b xdd o half-la ( U or ( V h fuo f ω Dfo 7 If f ( x s a grabl fuo wh fudamal sr ( UV,, h f suh ha U < < V ad f ( ω : ω + d, R( ω s grabl, h qualy { } + ω f ( x f ( ω x dω π (5 holds almos vrywhr. Morovr, f f ( x s ouous, h h qualy holds vrywhr o (,. If f ( x s dfd o h osv ral axs x (, ad M f ( x, ω f ω, h h followg rors of h Mll rasform hold. a Shfg Prory b Salg Prory h Mll rasform of Drvavs whr h symbol ( ω υ υ ω ( ω ( ω+ υ M x f( x, x f( x x dx f o ( U, V (5 ω ω M f( υx, ω f( υx x dx υ f ( ω o ( U, V (5 d M f( x, ω ( ( ω f( ω o ( U V, (53 dx s dfd for k gr by; ( ω ( ω ( ω + ( ω Equaos (5 ad (53 a b usd varous ways o fd h ff of lar ombao of dffral oraor suh ha d ω dω m, m, grs. h mos rmarkabl rsuls ar whr s a osv gr ad ω ω( ω+ ( ω+ d Covoluo Prory d M x f ( x, ω ( ω f ( ω d x d M x f ( x, ω ( ( ω f ( ω d x. (54 y f g x f g x d x x x ( Igral Rrsao for h Pr of Euroa Call Oos va h Mll rasform W obsrv ha 349

13 + O( as S E( S, OS as S h Mll rasform for h all oo dos o xs ad h gral fals o ovrg. I hs ar w us h mrovd vrso of h Mll rasform for h valuao of h Euroa all oo wh h varabl hag ω ω (57 Wh hs hag h usual oao for h rasform s rsrvd. hs s o sur h Mll rasform of h Euroa all ayoff fuo xss for som fudamal sr. h mrovd Mll rasform for h r of h Euroa all oo s dfd as Covrsly h vrso formula for (58 s gv by ω ( ω ( ω (56 M E S,, E, E S, S ds (58 + ω E (, S ( ω, d ω, < < ω π E S akg h mrovd Mll rasform of ( ylds E(, (, (, (, σ S E S E S M + S + r q re S S S Equao (6 boms E ( ω, σ + ( ω αω α E ( ω, ( ω αω α (59 (6 (6 E ( ω, E ( ω, (6 ( r q r α, whr ( ω, Equao (6 s h gral soluo of (6, wh α ad σ σ E s a osa ha dds o h fal m odo/ rmal odo whh s of h form E ω, g ( ω σ ω αω α (63 whr g S S S ω ( S S ds + ω ( ω ( ω d, R > ω ω ω ( Equao (64 s h mrovd Mll rasform of h fal m odo. Subsug (64 o (63 w hav ha Usg (6 ad (65, w hav ha ( ( σ ω αω α (64 ω E ( ω, (65 ω ω ω ( (, σ ω αω α E ω (66 ( ω ω akg h vrs mrovd Mll rasform of (66, h gral rrsao for h r of Euroa all oo s obad as 35

14 + ω σ ( ω αω α( π ( ω ω ω E S, S dω wh ( S, (, [,, (, s a osa ad ω :<R( ω <. { } horm 5: h boudary valu roblm for h Blak-Shols-Mro Equao ( for h r of h Euroa all oo wh xrs r, sub o h boudary odo (3 has a uqu soluo of h form S β ( l α (, x u E S S α σ π ( σ ( u β ( σ π Proof: h dffral quao gv by (6 of h form whr a also b wr as E (, S l u α S x α + ( u σ ( ω σ ω αω α ω (, wh h fal m odo gv by h soluo o (69 s obad as Smlfyg (7 furhr, ylds whr ( E +, ( r q α ad α r σ σ E ω σ σ + ω ω r + q r ( ω, E ( ω, ( ω (67 (68 (69 E g (7 σ σ E ( ω, g( ω x ω ω r + q r ( (7 r r E σ σ ( ω, g ( ω x ( ω + αω g ( ω x ( ( ω + α α (7 σ σ α r σ q σ akg h vrs mrovd Mll rasform of (7, w hav ha σ r E( S, g( S M x ( ( ω + α α, ω σ σ r σ x + x +, σ (73 α ( g( S M ( ( ω α ω (74 35

15 Sg Equao (74 boms L wh σ r σ α + β ( σ (, x ( ( ω + α, ω β E S g S M (76 Usg h salg rory (5 ad h ovoluo rory (55, w hav ha If w mos h fal m odo Equao (79 boms β ( ( (75 ω u α (77 S l u α σ ( S E( S, g( u σ π u u β ( ( β ( ( β ( ( ( ( + + (78 g S S g u u (79 S E( S, ( u σ π u u S ( u σ π u u S σ π u ( S l u α σ ( + S l u α σ ( S l u α σ ( β ( S σ π u u S l u α σ ( S l S l u u σ σ β β α S α ( α α+ σ π u σ π u E S, S σ r r q σ σ σ β α + ad α H, (68 s sablshd. horm 6: h boudary valu roblm for h Blak-Shols-Mro quao for h r of h Euroa u oo wh xrs r gv by sub o h boudary odo E S, E S, E S, r q S S re S (, + + σ S S (8 35

16 ad fal m odo has a uqu soluo of h form wh (, as o [, E S S (8 r ( [ E S, as S o, (8 (, ( + o [, E S S h S (83 S β ( l α (, x u E S S α + σ π ( σ ( u β ( σ π S l u α S x α ( u σ ( σ r r q σ σ σ β α + ad α Proof: Usg salg rory (5, ovoluo rory (55, h fal m odo (83 ad h u-all ary. Equao (84 follows from horm 5. horm 7: h xrssos (67, (68 ad Blak-Shols-Mro modl q( r( (84 E S, S N d N d (85 for h Euroa all oo ar aalyally quval. Proof: From (67 w hav ha whr wh Smlfyg (87 furhr ylds S l σ + r q + ( d σ ( d d σ ( + ω σ ( ω αω α( ω ( π ω( ω + σ ( ω αω α( ω ( ω dω π g S E S, S dω ( β α + ad α ( r q ω r g ( ω, α ad α ω ω σ σ σ r σ α + ( ω α + + σ ω ( ω E S, g S dω (88 σ r r q σ σ σ (86 (87 353

17 Now, g ( ω s h Mll rasform of ( ω Usg quao (7.. [] whh s of h form W g σ ( ( ω+ α ω g g S S ds (89 ( l S ωθ 4θ ω d, R( θ πθ α β ( S g ( S σ π S S (9 ( l S σ By mas of ovoluo rory of h Mll rasforms (s []. h r of h Euroa all oo boms From (8 w wr ha S E( S, g( u g u u (9 (9 α l S u σ β ( u S g u σ π S (93 Combg (79, (9 ad (93 w hav ha whr β ( S E( S, ( u σ π u u ( ( u β ( u S σ π u β( β( ( ( S l u α σ + S l u α σ S l u α σ β( α S ( β( α S S σ π u σ π α S I I A B σ π σ π A σ π B σ β ( α S β ( π I ( S α ( I α ( u S l u α + σ (94 (95 (96 354

18 I u S l u α σ (97 ad I u S l u α + σ o valua h frs ad sod grals (97 ad (98, w us h rasformaos ( S y l σ ( ( α σ u S y l σ ( α σ u ( rsvly. Fally, w oba h frs ad sod ars of (94 as whr d ad d ar gv by (86. Subsug ( ad ( o (94 ylds hs omls h roof. q (98 (99 ( A S N d ( r B N d ( q( r( E S, S N d N d 5. Numral Examls ad Dsusso of Rsuls I hs so, w rs som xamls o omar h rsuls obad by h fas Fourr rasform ad h mrovd Mll rasform wh h valus of h Blak-Shols-Mro modl. Examl W osdr h rg of h Euroa all oo o a dvdd-ayg sok va fas Fourr rasform ad h Mll rasform wh h followg aramrs Paramrs Valus Sok Pr, S Srk Pr/Exrs Pr, 8, 9,,, Rsk fr rs ra, r 5% Volaly, σ 5% Dvdd yld, q 5% m o xry,.8 Sz of grao grd, N 4 Igrably, a Fss, η 5% Cosa, 355

19 h ox of h Blak-Shols-Mro modl. h oo valus, P, absolu rror ad log absolu rror for h wo rasforms agas h valus of h Blak-Shols-Mro modl ar show Fgurs -4. Examl W osdr h valuao of Euroa all oo wh Fory-Egh mohs o go ul xrao o h Sadard ad Poor s 5 dx (S & P 5, wh a urr r of $, a srk r of $, a ouously omoudd rs ra of 5%, a volaly of 35% ad varyg a osa aual dx dvdd smad a q { %, %,3%, 4%,5% } usg fas Fourr rasform, mrovd Mll rasform, Mo Carlo mhod h ox of h Blak-Shols-Mro modl. h flu of dvdd yld o h rsuls grad s show Fgur 5. Dsusso of Rsuls Fgurs -4 show ha h fas Fourr rasform ad h mrovd Mll rasform rform br ad agr wh h valus of h Blak-Shols-Mro modl. W a also s ha h mrovd Mll rasform rovds a los aroxmao o h Blak-Shols-Mro formula. Fgur 5 shows ha h valus of h mrovd Mll rasform, Mo Carlo mhod ad h Blak-Shols-Mro modl od. Also h hghr 5 FF BSM Oo Valu, P Srk Pr, Fgur. h omarav rsul aalyss of h fas Fourr rasform (FF ad h Blak-Shols-Mro modl (BSM. 5 M BSM Oo Valu, P Srk Pr, Fgur 3. h omarav rsul aalyss of h mrovd Mll rasform (M ad h Blak-Shols-Mro modl (BSM. 356

20 5 FF M BSM Oo Valu, P Srk Pr, Fgur 4. h omarav rsuls aalyss of h fas Fourr rasform (FF, h mrovd Mll rasforms (M ad h Blak-Shols-Mro modl (BSM FF M MC BSM Oo Valu, P Fgur 5. h flu of dvdd yld o h valuao of Euroa all oo va h fas Fourr rasform (FF, h mrovd Mll rasform (M, h Mo Carlo mhod (MC ad h Blak-Shols-Mro modl (BSM. h dvdd yld, h smallr h valus of h mhods. h umral rsuls show ha h rmdous sd of h fas Fourr rasform allows oo rs for a hug umbr of srks o b valuad vry radly bu h damg faor or h grably aramr mus b arfully hos s orols h sy of h fluuaos ad h magud of h fuoal valus. h mrovd Mll rasform rovds aura omarabl rs ad h aroah a b rgardd as a good alrav o xsg mhods. 6. Coluso Dvdd Yld, q W hav osdrd h fas Fourr rasform ad h mrovd Mll rasform for h valuao of h Euroa all oo whh ays dvdd yld. Faal modlg h ara of oo rg volvs dald kowldg abou sohas rosss dsrbg h ass ayoffs. For sohsad r dyams, hs ar mos ovly hararzd hrough fuos mag sa. By a mag of h robably fuo from saal doma o h u rl h omlx la, xd valus of a fuur ayoff ar h avalabl h form of a gral rrsao. I hs ar, w ould gral faurs of Fourr rasform h- 357

21 qus alabl o boh h modlg of dsy fuos ad Euroa all oo rg. W also osdr som rors of h Mll rasform ad s alaos h hory of oo valuao. o mhass h graly of our rsuls, w hav show h quval of h gral rrsao for h r of h Euroa all oo va h mrovd Mll rasform o h Blak-Shols-Mro formula. From Fgurs -5, w a s ha h mrovd Mll rasform rovds a los aroxmao o h Blak-Shols-Mro modl, mor aura ha h fas Fourr rasform ad s a good alrav mhod for h valuao of h Euroa all oo o a dvdd ayg sok. Rfrs [] Blak, F. ad Shols, M. (973 h Prg of Oos ad Corora Labls. Joural of Polal Eoomy, 8, h://dx.do.org/.86/66 [] Carr, P. ad Mada, D. (999 Oo Valuao Usg h Fas Fourr rasform. Joural of Comuaoal Fa, 3, [3] Musla, M. ad Rukowsk, M. (6 Margal Mhods Faal Modllg. Srgr, Brl. [4] Duff, D. ( Dyam Ass Prg hory. 3rd Edo, Pro Uvrsy Prss. [5] Cruz-Baz, D.I. ad Gozalz-Rodrguz, J.M. ( Smgrou hory Ald o Oos. Joural of Ald Mahmas,, h://dx.do.org/.55/s757x4 [6] Pa, R. ad Srvasav, R.P. (4 Oo Prg wh Mll rasforms. Mahmaal ad Comur Modllg, 4, h://dx.do.org/.6/.mm [7] Fagba, S.E. ad Nwozo, C.R. (5 Igral Rrsaos for h Pr of Valla Pu Oos o a Bask of wo-dvdd Payg Soks. Ald Mahmas, 6, h://dx.do.org/.436/am [8] Fagba, S.E. ad Nwozo, C.R. (5 Mll rasform Mhod for h Valuao of h Amra Powr Pu Oo wh No-Dvdd ad Dvdd Ylds. Joural of Mahmaal Fa, 5, h://dx.do.org/.436/mf [9] Ma, H. (965 Adx: A Fr Boudary Problm for h Ha Equao Arsg from a Problm Mahmaal Eooms. Isral Maagm Rvw, 6, [] Busr, S.A. (986 LaPla rasforms as Prs Valu Ruls: A No. h Joural of Fa, 4, h://dx.do.org/./ b45.x [] Jodar, L.P., Svlla-Prs, P., Cors, J.C. ad Sala, R. ( A Nw Dr Mhod for Solvg h Blak-Shols Equao. Ald Mahmas Lrs, 8, 9-3. h://dx.do.org/.6/.aml...6 [] Baksh, G. ad Mada, D. ( Sag ad Drvav Sury Valuao. Joural of Faal Eooms, 55, h://dx.do.org/.6/s34-45x(995- [3] Frozak, R. (3 Sml Aalyal Aroxmaos for h Cral Sok Pr of Amra Oos. Workg Par. h://ssr.om//absra766 h://dx.do.org/.39/ssr.766 [4] AlAzm, F., AlAzm, A. ad Boyadv, L. (4 Mll rasform Mhod for Solvg h Blak-Shols Equaos. Iraoal Joural of Pur ad Ald Mahmas, 97, h://dx.do.org/.73/am.v973.3 [5] Nwozo, C.R. ad Fagba, S.E. (4 Mll rasform Mhod for h Valuao of Som Valla Powr Oos wh No-Dvdd Yld. Iraoal Joural of Pur ad Ald Mahmas, 96, h://dx.do.org/.73/am.v96.7 [6] Nwozo, C.R. ad Fagba, S.E. (4 Prforma Masur of Lala rasforms for Prg Pah Dd Oos. Iraoal Joural of Pur ad Ald Mahmas, 94, h://dx.do.org/.73/am.v94.5 [7] Zhu, J. ( Molar Prg of Oos. A Alao of Fourr Aalyss. Srgr, Brl. h://dx.do.org/.7/ [8] Nwozo, C.R. ad Fagba, S.E. (5 O wo rasform Mhods for h Valuao of Cog Clams. Joural of Mahmaal Fa, 5, 88-. h://dx.do.org/.436/mf.5.59 [9] Wlmo, P., Dwy, J. ad Howso, S. (993 Oo Prg, Mahmaal Modls ad Comuao. Oxford Faal Prss, Oxford. [] Cooly, J.W. ad urky, J.W. (965 A Algorhm for h Mah Comuao of h Comlx Fourr Srs. Mahmas of Comuao, 9, h://dx.do.org/.9/s

22 [] Erdly, A., Magus, W., Obrhgr, F. ad rom, F. (954 abls of Igral rasforms. Vol. -, MGraw- Hll, Nw York. [] Sddo, I. (97 h Us of Igral rasforms. MGraw-Hll, Nw York. 359