Mellin Transform Method for the Valuation of the American Power Put Option with Non-Dividend and Dividend Yields

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1 Joural of Mahmacal Fac, 5, 5, 49-7 Publshd Ol Augus 5 ScRs. h:// h://dx.do.org/.436/jmf Mll Trasform Mhod for h Valuao of h Amrca Powr Pu Oo wh No-Dvdd ad Dvdd Ylds Suday Emmaul Fadugba, Chuma Rahal Nwozo Darm of Mahmacal Sccs, Ek Sa Uvrsy, Ado Ek, Ngra Darm of Mahmacs, Uvrsy of Ibada, Oyo Sa, Ngra Emal: mmasfad6@yahoo.com, crwozo@yahoo.com Rcvd 5 May 5; accd 4 July 5; ublshd July 5 Coyrgh 5 by auhors ad Scfc Rsarch Publshg Ic. Ths work s lcsd udr h Crav Commos Arbuo Iraoal Lcs CC BY. h://cravcommos.org/lcss/by/4./ Absrac I hs ar w rs h Mll rasform mhod for h valuao of h Amrca owr u oo wh o-dvdd ad dvdd ylds, rscvly. W us h Mll rasform mhod o drv h gral rrsaos for h rc ad h fr boudary of h Amrca owr u oo. W also xd our rsuls o drv h fr boudary ad h fudamal aalyc valuao formula for rual Amrca owr u oo whch has o xry da. Numrcal xrms hav show ha h Mll rasform mhod s a br alrav chqu comard o h bomal modl BSM, rcursv mhod RM ad f dffrc mhod FDM for h valuao of h Amrca owr u oo. I gral, h Mll rasform mhod s accura, flxbl ad roducs accura rcs for h omal xrcs boudary of h Amrca owr u oo for a wd rag of aramrs. Hc h Mll rasform mhod s muually coss ad agrs wh h valus of h aalyc oo valuao formula calld h Black- Schols modl. ywords Amrca Powr Oo, Dvdd Yld, Mll Trasform Mhod, No-Dvdd Yld, Prual Powr Pu Oo. Iroduco Oo valuao has b sudd xsvly h las hr dcads. May roblms facal mahmacs al h comuao of a arcular gral. I may cass hs grals ca b valud aalycally ad How o c hs ar: Fadugba, S.E. ad Nwozo, C.R. 5 Mll Trasform Mhod for h Valuao of h Amrca Powr Pu Oo wh No-Dvdd ad Dvdd Ylds. Joural of Mahmacal Fac, 5, h://dx.do.org/.436/jmf.5.533

2 som cass hy ca b comud usg a aral dffral quao, or valud usg umrcal grao. Powr oo s dfd as a cog clam o h roduc of owrs of svral udrlyg asss. Th holdr has hr h rgh, bu o h oblgao o buy, as h cas of h owr call oo, or h ossbly o sll, as h cas of h owr u oo, a ass for a cra rc a a rscrbd da h fuur. Th dffrc bw h Amrca ad h Euroa owr oos s ha h Euroa owr oo ca oly b xrcsd a h maury or xry da whl h Amrca owr oo ca b xrcsd by s holdr a ay m bfor h xry da. Ths arly xrcs faur maks h valuao of h Amrca owr oo mahmacally challgg ad hrfor, cras a gra fld of rsarch. A rual Amrca owr oo s a oo ha has o xry da. I ohr words, hs y of owr oo vr xrs. I a scal cas of a la valla rual oo, a closd form soluo for h fr boudary ad rc of h Amrca u was drvd by []. Mll rasforms oo hory wr roducd by [], [3] xdd h rsuls obad [] ad showd how h Mll rasform aroach could b usd o drv h valuao formula for h rual Amrca u oos o dvdd-ayg socks. [4] cosdrd h Mll rasform mhod for h valuao of som valla owr oos wh o-dvdd yld. Thy drvd h fudamal valuao formula kow as h Black-Schols modl usg h covoluo rory of h Mll rasform mhod. Th aalycal valuao of h Amrca oos was cosdrd by [5]. A alrav aroach o h valuao of Amrca oos ad alcaos was cosdrd by [6]. For h mahmacal backgroud of h Mll rasform mhod drvavs valuao s [7]-[5], jus o mo fw. I hs ar, w focus o h Mll rasform mhod for h valuao of h Amrca owr u oo wh o-dvdd ad dvdd ylds, rscvly, ad s xso o owr oo whch has o xry da,.. rual Amrca owr u oo. Th rs of h ar s srucurd as follows: Sco, w rs Amrca owr oos ad h ayoffs for owr call ad u oos. Sco 3 rss h Mll rasform mhod for h valuao of h Amrca owr u oo. Sco 4 cosdrs h xso of h Mll rasform mhod o h valuao of h rual Amrca owr u oo. I Sco 5, w rs som umrcal xrms. Sco 6 cocluds h ar.. Amrca Powr Oos h Th owr oos ca b s as a class of oos whch h ayoff a xry s rlad o h owr of h udrlyg rc of h ass. Amrca owr oos ar oos ha ca b xrcsd bfor or a h xry da wh o-lar ayoff. Th Amrca owr oo coms wo forms, amly, h Amrca owr call oo ad h Amrca owr u oo. Th Amrca owr call oo s a oo wh olar ayoff gv by h dffrc bw h rc of h udrlyg ass a maury rasd o a srcly osv owr ad h xrcs rc. Th Amrca owr u oo s a oo wh o-lar ayoff gv by h dffrc bw h xrcs rc ad rc of h udrlyg ass a maury rasd o a srcly osv owr. For a Amrca owr oo o h udrlyg rc of h ass S T wh xrcs rc ad m o xry T, w hav h ayoffs for h Amrca owr call ad u oos as ad, max, A S S S c T T T, max, A S S S T T T rscvly. Rmark For, h ayoffs for Amrca owr call ad u oos ad bcom h ayoffs for la Amrca call ad u oos,.. ad rscvly., max, A S S S 3 c T T T, max, A S S S 4 T T T 5

3 3. Th Mll Trasform Mhod for h Valuao of h Amrca Powr Pu Oo Thr ar may mhods for h valuao of h Amrca owr oo ladg o dffr bu quval mahmacal formulaos. W cosdr h drvao of h gral rrsao for h rc of h Amrca owr u oo ad h gral quao o drm h fr boudary of h Amrca owr u oo va h Mll rasform mhod for h cas of boh o-dvdd ad dvdd ylds. 3.. Amrca Powr Pu Oo wh No-Dvdd Yld Cosdr h o-homogous Black-Schols aral dffral quao for h Amrca owr u oo wh o-dvdd yld gv by, S A S, A S, σ rs S A S σ S ra S f S,, h arly xrcs fuco f dfd o,,t s gv by, f S r, f < S ˆ S, f ˆ S > S. Th fal m codo gv by A, max, S T φ S S S hgh coac codo. Th ohr boudary codos ar gv by ad o [ [ 5 6, s calld h lm A S, o, T 7 S S [ lm A S, o, T 8 Th fr boudary S ˆ s drmd by h smooh asg codos gv by ˆ, ˆ A S S 9 ˆ, A S ˆ S Alyg h Mll rasform o 5, w hav ha A, σ r r A, f, σ σ Sg r α σ ad r α. Th ylds σ A, σ α α A, f, Th Mll rasform of h arly xrcs fuco s obad as ˆ Sˆ r S,, d f f S d S S r S S 3 Solvg furhr, w hav h arcular soluo of as 5

4 ˆ T r S σ α α y A,. dy sol 4 Th comlmary soluo o h lf had sd of s obad as, σ α α A c 5 com. sol c s h grao cosa obad as, σ αα c φ T φ, s h Mll rasform of h fal m codo ad s gv by 6 7, ST ST ST ST φ Usg 6 ad 7 5 w hav ha Hc h gral soluo o s gv by σ α α T, A com. sol 8 σ α α T,,, A A A com. sol. sol T ˆ y σ α α y r S Th Mll vrso of 9 s obad as dy T c σ αα c A S, S d π r π S, {, [, T }, c, c c S T ˆ S y σ α α y ad { C < R < }. dd y Rmark Th frs rm s h gral rrsao for h rc of h Euroa owr u oo sms from h mmum guarad ayoff of h Amrca owr u whch ays o dvdd yld s [4]. Th scod rm s calld h arly xrcs rmum h valu arbuabl o h rgh of xrcsg h oo arly for h Amrca owr u oo wh o dvdd yld dod by S,. Thrfor bcoms,,, 9 A S E S S T c σ αα, d c E S S π ˆ S y σ α α y r c T S, S dd y π c Sg ˆ S S ad usg h smooh asg codos gv by 9 ad, w hav h gral rrsao for h fr boudary of h Amrca owr u oo wh o-dvdd yld as 5

5 r c ˆ S y y T σ αα, dd c Sˆ E Sˆ Sˆ y π σ α α T ˆ c ˆ π c E S, S d W formalzd h rors hghlghd Rmark h followg rsuls. Thorm Th Amrca owr u oo A S, whch ays o dvdd yld sasfs h dcomoso ˆ S y σ α α y r c T A S, E S, S dd y π c r α σ ad r α σ C < R <. { }, S, {, [, T }, c, Thorm Usg h smooh asg codos gv by ˆ, ad ˆ A S S ad boudary formulao of h Amrca owr u oo wh o-dvdd yld s gv by r c ˆ S y y T σ αα, dd c Sˆ E Sˆ Sˆ y π 3.. Amrca Powr Pu Oo wh Dvdd Yld ˆ, A S Sˆ, h fr Th drvao of h gral rrsao for h rc of h Amrca owr u oo whch ays dvdd yld usg h Mll rasform mhod s gv h followg rsul. Thorm 3 L S b h rc of h udrlyg ass, b h srk rc, r b h rsk rs ra, q b h dvdd yld ad T b h m o maury. Assum S ylds dvdd, h h gral rrsao A S, s gv by for h rc of h Amrca owr u oo * T c σ αα c A S, S d π r π S c T c q c T S π c S y * σ α α y S y * σ α α y dd y dd y Proof. Cosdr h o-homogous Black-Schols aral dffral quao for h Amrca owr u oo wh dvdd yld gv by, S A S, A S, σ rq S S A S S ra S f S σ,,

6 o, [,T, r qs, f < S S, f S > S f S ad S h fr boudary of h Amrca owr u oo wh dvdd yld. Th hgh coac codo s gv by, max, A S T φ S S S o [,. Th ohr codos ar gv by 7 ad 8. Wh h smooh asg codos gv by ad Th Mll rasform of 4 gvs Pug, 5 A S S 6, A S S,, σ A r q r A, f, 8 σ σ r q α σ *, r α, 8 ylds σ ad A σ * α α A, f, 9 q S ˆ S r S f, f S, d d S S r q S S 3 Followg h sam rocdurs for h cas of o-dvdd yld, h gral soluo o 3 s obad as * * σ α α T r S σ α α y T y A, dy T * y σ α α y q S Th Mll vrso of 3 s gv by dy * T c σ αα c A S, S d π r π S c T y c q c T y S π c S * σ α α y * S σ α α y dd y dd y Equao 3 s h gral rrsao for h rc of Amrca owr u oo wh dvdd yld, S, {, [, T] }, c, ad { C < R < }. Rmark 3 Th frs rm 3 s h gral rrsao for h rc of h Euroa owr u oo sms from h mmum guarad ayoff of h Amrca owr u wh dvdd yld ad h las wo rms do h arly xrcs rmum h valu arbuabl o h rgh of xrcsg h oo arly for h S,. Thrfor 3 bcoms Amrca owr u oo wh dvdd yld dod by

7 ,,, A S E S S 33 * T c σ αα, d c E S S π S * σ α α y r c T y S, S dd y π c q c T y S π c * S σ α α y dd y Sg S S 33 ad usg h smooh asg codos gv by 6 ad 7, w hav h gral rrsao for h fr boudary of h Amrca owr u oo wh dvdd yld as S * σ α α y r c T y S E S, S dd y π c q c T y S π c c π c * S σ α α y dd y * σ α α T ˆ, d E S S From Rmark 3, w hav h followg rsuls. A S, whch ays dvdd yld sasfs h dcomoso Thorm 4 Th Amrca owr u oo * c T y y r S σ αα A S, E S, S dd y π c q c T y S π c r q α σ C < R <. * { } α ad * S σ α α y r σ dd y, S, {, [, T }, c, Thorm 5 Usg h smooh asg codos gv by, ad A S S ad, A S h fr boudary formulao of h Amrca owr u oo wh dvdd yld s gv by r π S c T y c, S E S q c T y S π c S * σ α α y * S σ α α y dd y dd y S 34. Th Th followg rsuls rs som scal cass of ad 3. Thorm 6 If T ad, h Th gral rrsao for h Amrca owr u oo whch ays o dvdd yld rducs o h gral quao drvd by m [6] for h rc of h la Amrca u oo gv by 55

8 rη η A S, E S, r N d dη 35 d η S σ l ˆ r η S η 36 σ η Th fr boudary for h Amrca owr u oo whch ays o dvdd yld rducs o h gral quao drvd by m [6] for h rc of h la Amrca u oo gv by rη S ˆ E S ˆ, r N d d η 37 η Proof. Sg ad T ylds d η ˆ S σ l ˆ r η S η 38 σ η c σ α α c A S, S d π r π S c c Sˆ y σ α α y r α σ ad r α. Equao 39 ca b b wr as σ E S, ad,,,, dd y 39 A S E S S 4 S do h rc of h Euroa u oo wh o dvdd yld ad fr boudary for h Amrca u oo wh o dvdd yld rscvly. L ad Th arly xrcs fuco s gv by ˆ Ω y S, S, S,, y dy 4 c ˆ ˆ ˆ y Ω S, S,, y f, y ξ, y S d π c 4 f ˆ r, f S, S S, y, f S > S σ α α y Usg h covoluo rory of h Mll rasform, 4 bcoms y y 43 ˆ ξ, 44 y ˆ y S Ω S, S,, y f vy, ξ, y dv v v 45 Subsug h valu of h arly xrcs fuco f vy, from 43 ad 56

9 o 45, w hav ha ξ S α σ α l S y S σ y σ π y 46 ˆ y Usg h rasformao gv by 47 bcoms σ α y α S α σ π l S σ y Sˆ y Ω S, S,, y r dv 47 y v S λ l σ σ y v y α λ ˆ r y,,, r y S S y y r dλ r N d y Ω 49 d π y Subsug 49 o 4 w hav h arly xrcs rmum for h Amrca u oo wh o-dvdd yld as y y 48 r S, r N d dy 5 d y S σ y l r S ˆ y σ y 5 Sg η y, h 5 bcoms rη η S, r N d dη 5 d η S σ l ˆ r η S η σ η Subsug 5 o 4 w g h gral quao 35 obad by m [6] as rη η A S, E S, r N d dη Hc s sablshd. For h scod rduco, sg usg h smooh asg codos gv by ˆ, S A S Sˆ ad Sˆ h las gral quao abov ad A S ˆ, dary Ŝ of h Amrca u oo whch ay o dvdd yld 37 drvd by m [6] as rη ˆ Sˆ E Sˆ, r N d d η η Sˆ, w oba h fr bou- 57

10 dˆ η ˆ S σ l ˆ r η S η σ η Thorm 7 If T, h h omal xrcs boudary S of h Amrca owr u oo wh wh dvdd yld s gv by r lm S m, 53 T q Proof. L T ad, 34 bcoms ad α σ S π c S * y σ α α y r c S E S, S dd y π c r q * q c S * y σ α α y dd y r ad α. Facorzg ad rarragg, 54 bcoms σ S r N d S,, ri q N d S,, qj,, d S,, d S S l r q σ 56 σ S l rq σ 57 σ * c S y σ α α y dd 58 I S y π c * c S y σ α α y dd 59 J S y π c Noc frs ha crcal sock rc s boudd from abov.. S, >. Takg h lms of 56 ad 57 as, w hav ha ad lm d S,, lm d S,,, for S, for S <, for S, for S <

11 rscvly. If W hav lm S Usg 63, h lm of 55 s obad as Sc ad Th 64 bcoms If W hav ha 6 lm N d S,, lm N d S,, 63 lm S r N d S,, ri lm q N d S,, qj lm S Th frs gral I ca also b wr as lm ri lm qj lm I lm J 64 S lm 65 lm S < S r I lm lm q J * c S y σ α α y dd 67 I S y π c Alyg h rsdu horm of comlx umbr gv by Th h r gral 67 bcoms Subsug 69 o 67 ylds k w w j 66 f w dw π Rs f w, w 68 j S * σ α α y c y r y S dd y π c 69 I r 7 r 59

12 Smlarly, J q Subsug 7 ad 7 o 66 for q r, w hav ha Usg h l Hosal rul, for q Combg 7 ad 73 q 7 q r r S r lm lm r lm q q q > r, 64 bcoms lm S 7 r 73 q lm S m, r q Hc 53 s sablshd. Rmark 4 Th abov rsuls cofrm h formula of m ad Yu [6] Thorm 8 If h udrlyg ass rc follows a logormal dffuso rocss ad h rs ra s a osv cosa, h h omal xrcs boudary of h Amrca owr u oo wh a maury s gv by lm S r, for q > r q, for q r Proof. L T. I ordr o vsga h bhavour of h omal xrcs boudary S of h Amrca owr u oo wh ar maury, w cosdr 55 whch s of h form S r N d S,, ri q N d S,, qj If q > r, h lm of h rgh had sd of 55 as ca b valuad usg h l Hosal s rul w hav ha If q lm S 74 r 75 q r, h lm of h rgh had sd of 55 as s obad drcly as lm S 76 Combg 75 ad 76, w hav h omal xrcs boudary of h Amrca owr u oo wh a maury gv by Hc 74 s sablshd. Rmark 5 lm S r, for q > r q, for q r 6

13 From 75, w oc ha wh q > r h Amrca u ca hav a osv valu a xrao gv ha has o b xrcsd arlr. Ths dcas ha larg dvdd ayous rduc h cvs of arly xrcs. From 76, w dduc ha wh q r h Amrca u wll hav a zro ayoff a xrao v f has o b xrcsd arlr. Ths s bcaus s o ossbl for h udrlyg ass rc a xrao o fall blow whou crossg h xrcs boudary a a arlr m. Thorm 9 Th gral rrsao for h rc of h Amrca owr u oo whch ays dvdd yld gv by 3 ca b rducd o gral rrsao drvd by m [6]. r η η η η A S, E S, r N d S, S, η dη η r η qs N d S, S, d, η, η d S S, η, η d S S Proof. Sg T, h 3 bcoms S σ η l r q S η σ η S σ η l rq S η σ η S T S S π c S * y σ α α y r c A S, E S, S dd y π c r q α σ * { C < R < }. q c α r σ ad Usg h rocdurs of [3], 78 ca b wr as * S y σ α α y dd y, S, {, [, T] }, c, ad c A S, E S, f, y ξ, y S d dy π c 79 wh h Mll rasforms of f S, y ad S, y r S y ξ gv by q f, y S y y σ αα * ξ, y 8 rscvly. Usg h covoluo horm of h Mll rasform w hav ha S A S, E S, f vy, ξ, y dd vy v v Th rc of h Amrca owr u oo whch ays dvdd yld ca b xrssd as 6

14 Th gral I S, y y A S, E S, I S, y dy 8 s valuad as follows S I S, y f vy, ξ, y dv v v S l * v ρ * σ y ρ ρ α S S y ρ y I S, y r dv σ π v σ α ρ ρ rasformao gv by ad S l * v * ρ σ y ρ ρ α S S y * ρ y q dv σ π v r q * *, σ S y v λ l ρ σ ad 83 r α. Usg h followg varabls σ y S λ l ρ σ y v y For h frs ad scod grals 83 rscvly, w hav ha Subsug 84 o 8 ylds By chagg y r y,,, y, I S y r N d S S y r q σ y, y q N d S, S, y r y, A S, E S, r N d S, S, y dy y η, 85 bcoms r q σ y, y q N d S, S, y dy r η, η A S, E S, r N d S, S, η dη r q σ η, η q N d S, S, η dη Hc by sg, hs rovs Alcao of h Rsuls o Prual Amrca Powr Pu Oo Valuao Now, w aly h rsuls grad for h gral quaos ad 3 o owr oos whch hav o xry da. Th followg rsuls shows h drvao of h xrsso for h fr boudary of rual h Amrca owr u oo ad s closd form soluo for boh o-dvdd ad dvdd ylds, usg h Mll rasform mhod

15 Thorm No-Dvdd Yld If T ad R α rual Amrca owr u oo s gv by ˆ ˆ S S α ad h rc of h rual Amrca owr oo bcoms α α, ˆ S for ˆ ˆ < S A S S S S < <, h h fr boudary of h r α 89 σ Proof. Th gral rrsao for h rc of h Amrca owr u oo whch ays o dvdd yld gv by ca b xrssd as, wh ad E S ad, P S ar gv by,,, A S E S P S 9 r σ T rt,,, E S N d S N d 9 d, S l σ σ T d, r T S σ l r T σ T ˆ S y σ α α y r c T P S, S dd y π c 9 rscvly. For 9 o hold as T, s cssary ha R α α <.. R α < <, α s gv by 89. Th scod smooh asg codo for a rual owr u ca b wr as Dffrag 9 a S ˆ ˆ,, E S P S as T ˆ ˆ S S ˆ w hav ha S ˆ, r σ T E S Sˆ N dˆ, dˆ, ˆ S l σ σ T r T 95 63

16 T dˆ As,, ad hrfor Also dffrag 9 ylds, ˆ, E S Sˆ P S r c T y S σ αα S dy d π c S ˆ S y Takg h lm of 97 as T, w hav Thrfor,, P S r c S S π c ˆ S S σ αα y dyd 98 P S r α α, c d S π σ 99 Sˆ αα c wh α < <. Th lmg cass α ad ar h roos of α α. Hc 99 bcoms ˆ, d σ α P S r c ˆ π c ˆ S S Sc < R < α, alcao of h rsdu horm gv by 68 lads o ˆ, Subsug 96 ad o 93 ylds P S α α S S S α ˆ ˆ ˆ α ˆ S α ˆ S α α Equao s h xrsso for h fr boudary of a rual Amrca owr u oo. Nx, w us o drv a xrsso for h rc of rual Amrca owr u oo A S,. No ha h rc of a rual Euroa owr u oo s zro, sc ca vr b xrcsd. Thrfor, akg h lm as T 9, h rc of rual Amrca u oo for ˆ S > s gv by R α α σ α α y r c S A S, dy d π c ˆ S S 3 <. Igra h r gral, bcoms r c S π c σ Sˆ A S, d α 4 Oc aga w aly h rsdu horm 68 o g 64

17 α, ˆ S for ˆ ˆ < S A S S S S Equao 5 s h rc of a rual Amrca owr u oo obad as a lm of h rc of a f-lvd Amrca owr u oo. * Thorm Dvdd Yld If T ad < R < α, h h fr boudary of h rual Amrca owr u oo s gv by S S ad h rc of rual Amrca owr u oo quals ad S, for < S A S S S S * * α α 4α 8 r q r α, α σ σ * Proof. Th gral rrsao for h rc of h Amrca owr u oo whch ays dvdd yld gv by 3 ca b xrssd as,,, A, S E S P S P S E S,, P S, ad P S, ar gv by wh rt 9,, r q σ T, E S N d S N d d, S l σ σ T d, r q T S σ l rq T σ T S * σ α α y r c T y P S, S dd y π c * S σ α α y q c T y P S, S dd y π c * rscvly. Th roos of α α ar 3 * * α α 4α 65

18 ad Thus w wr ha wh α R α * * α α 4α α α * < <. For o hold as T, s cssary ha R α * α <.. * < <, α ad α ar gv by 9. Th scod smooh asg codo for a r- ual owr u whch ays dvdd yld ca b wr as,,, E S P S P S as T S S S Dffrag a S w hav ha T d As,, S, r q σ T E S d, S ad hrfor Now dffrag w.r., N d, S l σ σ T r q T, E S S S ad akg h lm T w hav ha P S r c S S π c S S Thrfor, by sg S w hav ha S * σ α α y dyd P S r, c d S π σ 9 S αα r L α σ c * ad α * α P S, c α d π c S S I h sam mar, sg T ad dffrag 3 w.r. Thrfor,, q c P S S π c S S S, w hav ha * σ α α y dy d 66

19 ad σ P S, rq r π S σ c S c S Sg S ad solvg furhr w hav ha S σ P S, r q r * αα d c d π c S σ Oc aga by h alcao of rsdu horm 68, h ad yld P S, α α S S S α P S, * α α S rscvly. Subsug 7, 3 ad 4 o 4, w oba S Equao 5 s calld h fr boudary of h rual Amrca owr u oo whch ays dvdd yld. Th rc for h rual Amrca owr u oo s gv by α c S, d c A S π S σ r q r π σ c S S c S Usg h rsdu horm 68, h 6 bcoms d S α σ rq r, S σ A S S S 5. Numrcal Exrms S S S S I hs sco w rs som umrcal xrms ad dscusso of rsuls. Exrm W cosdr h valuao of h Amrca owr u oo for {.9,.95,.,.5,.} whch ays dvdd yld q.6 wh h followg aramrs:

20 { } S $, $, σ.,.5,.,.5,.3, r.8, T.5 Th rsul grad s show Tabl blow. Exrm W cosdr h valuao of h Amrca owr u oo for whch ays o-dvdd yld wh h followg aramrs: { } { } S $4, $35, $4, $45, T.83, r.5, σ.,.3,.4 Th rsuls grad for h rc of h Amrca owr u oo va Black-Schols modl BSM, bomal modl BM ad h Mll rasform mhod MTM ar show Tabls -4 blow. Also h rsuls grad for h fr boudary of h Amrca owr u oo ar show Tabls 5-7 blow. Exrm 3 W cosdr h valuao of h Amrca Powr u oo wh h followg aramrs: { } S $4, $35, $4, $45, T.583, r.5, σ.4,, c Th comarav rsuls aalyss of h Mll rasform mhod MTM h cox of Black-Schols modl BSM, bomal modl BM, rcursv mhod RM ad F dffrc mhod FDM ar show Tabl 8. Tabl. Amrca owr u valus. σ \ Tabl. Th rc of Amrca owr u oo usg, c, σ., r.5, S $4, T.83. Black-Schols Modl BSM Bomal Modl BM Mll Trasform Mhod MTM Tabl 3. Th rc of Amrca owr u oo usg, c, σ.3, r.5, S $4, T.83. Black-Schols Modl BSM Bomal Modl BM Mll Trasform Mhod MTM Tabl 4. Th rc of Amrca owr u oo usg, c, σ.4, r.5, S $4, T.83. Black-Schols Modl BSM Bomal Modl BM Mll Trasform Mhod MTM

21 Tabl 5. Th fr boudary of Amrca owr u oo usg, c, σ., r.5, T.83. Srk Prc, Udrlyg Ass Prc, S Fr Boudary S ˆ Tabl 6. Th fr boudary of Amrca owr u oo usg, c, σ.3, r.5, T.83. Srk Prc, Udrlyg Ass Prc, S Fr Boudary S ˆ Tabl 7. Th fr boudary of Amrca owr u oo usg, c, σ.4, r.5, T.83. Srk Prc, Udrlyg Ass Prc, S Fr Boudary S ˆ Tabl 8. Th comarav rsuls aalyss of som umrcal mhods for h valuao of Amrca owr u oo. S σ c r T BSM BM MTM RM FDM Dscusso of Rsuls From Fgur blow, w obsrv ha h hghr h volaly, h hghr h valus of h Amrca owr u oo. Also h hghr h owr of h Amrca u oo, h lowr h valus of h oo. Fgurs -4 blow show ha h Mll rasform mhod s muually coss, rforms vry wll, accura ad agrs wh h valus of Black-Schols modl BSM. I Fgur 5 blow, w lo h fr boudary S ˆ as a fuco of h srk rc for dffr valus of volaly σ. W obsrv ha h hghr h volaly, h lowr h omal xrcs boudary of h Amrca owr u oo. Fgur 6 blow dmosras ha h Mll rasform mhod s a br alrav chqu comard o h Black-Schols modl BSM, bomal modl BM, rcursv mhod RM ad f dffrc mhod BM for h valuao of h Amrca owr u oo. Hc h Mll rasform mhod s a good chqu for h valuao of h Amrca owr u oo. 6. Cocluso I hs ar, w hav drvd h gral rrsaos for h rc ad h fr boudary of h Amrca owr u oo for o-dvdd ad dvdd ylds usg h Mll rasform mhod. W also xdd h gral quao for h rc of h Amrca owr u oo o drv h xrsso for h fr boudary ad h rc of h rual Amrca owr u oo whch ays boh o-dvdd ad dvdd ylds as h lm of a f-lvd oo by mas of smooh asg codo. I gral, umrcal xrms hav show ha h Mll rasform mhod s accura, flxbl, ffc ad roducs accura rcs for h omal xrcs boudary for a wd rag of aramrs. 69

22 Fgur. Amrca owr u oo valus usg Tabl. Fgur. Th comarav rsuls aalyss usg Tabl. Fgur 3. Th comarav rsuls aalyss usg Tabl 3. 7

23 Fgur 4. Th comarav rsuls aalyss usg Tabl 4. Fgur 5. Th fr boudars of Amrca owr u oo wh. Fgur 6. Th comarav rsuls aalyss usg Tabl 8. 7

24 Rfrcs [] Samulso, P.A. 965 Raoal Thory of Warra Prcg. Idusral Maagm Rvw, 6, 3-3. [] Pa, R. ad Srvasav, R.P. 5 Prcg Prual Oos Usg Mll Trasforms. Ald Mahmacs Lrs, 8, h://dx.do.org/.6/j.aml.4.3. [3] Froczak, R. ad Schöbl, R. 8 Prcg Amrca Oos wh Mll Trasforms. Workg Par. [4] Nwozo, C.R. ad Fadugba, S.E. 4 Mll Trasform Mhod for h Valuao of Som Valla Powr Oos wh No-Dvdd Yld. Iraoal Joural of Pur ad Ald Mahmacs, 96, h://dx.do.org/.73/jam.v96.7 [5] m, I. 99 Th Aalyc Valuao of Amrca Oos. Th Rvw of Facal Suds, 3, h://dx.do.org/.93/rfs/ [6] m, I. ad Yu, G.G. 996 A Alrav Aroach o h Valuao of Amrca Oos ad Alcaos. Rvw of Drvavs Rsarch,, h://dx.do.org/.7/bf [7] Erdly, A., Magus, W., Obrhgr, F. ad Trcom, F. 954 Tabls of Igral Trasforms, Vol. -. McGraw- Hll, Nw York. [8] Fadugba, S.E. ad Nwozo, C.R. 5 Igral Rrsaos for h Prc of Valla Pu Oos o a Bask of Two-Dvdd Payg Socks. Ald Mahmacs, 6, h://dx.do.org/.436/am [9] Froczak, R. ad Schöbl, R. 9 O Modfd Mll Trasforms, Gauss-Lagurr Quadraur ad h Valuao of Amrca Call Oos. Tübgr Dskussosbräg, No. 3. [] Gradshy, I. ad Ryshk, I. 7 Tabl of Igrals Srs ad Producs. 7h Edo, Acadmc Prss, Walham. [] Pa, R. ad Srvasav, R.P. 4 Oo Prcg wh Mll Trasforms. Mahmacal ad Comur Modllg, 4, h://dx.do.org/.6/j.mcm [] Vaslva, O. 9 A Nw Mhod of Prcg Mul-Oos Usg Mll Trasforms ad Igral Equaos. Masr s Thss Facal Mahmacs, School of Iformao Scc, Comur ad Elcrcal Egrg, Halmsad Uvrsy, Halmsad. [3] Nguy, T.H. ad Yakubovch, S.B. 99 Th Doubl Mll-Bars Ty Igrals ad Thr Alcaos o Covoluo Thory. World Scfc, Srs o Sov Mahmacs, Vol. 6, Sgaor, 95. [4] Zmaa, A.H. 987 Gralzd Igral Trasformaos. Dovr Publcaos, Nw York. [5] Zb, A.E. ad Rokah, R.A. Aalycal Soluo for a Arhmc Asa Oo Usg Mll Trasforms. Iraoal Joural of Mahmacal Aalyss, 5,