On the Accuracy of Binomial Model for the Valuation of Standard Options with Dividend Yield in the Context of Black-Scholes Model

Transcription

1 IG Innaional Jonal o li Mahmais 44: IJM_44 5 On h a o Binomial Mol o h Valaion o ana Oions ih Divin Yil in h onx o Blak-hols Mol hma Rahal oo na mmanl Fagba bsa his a sns h aa o binomial mol o h valaion o sana oions ih ivin il in h onx o Blak-hols mol. I is obsv ha h binomial mol givs a b aa in iing h mian oion han h Blak-hols mol. his is o a ha h binomial mol onsis h ossibiliis o al xis an oh as lik ivin. I is also obsv ha h binomial mol is boh omaionall iin an aa b no aqa o i ah nn oions. Inx ms mian Oion Binomial Mol Blak- hols Mol Divin Yil oan Oion ana Oion Mahmais b lassiiaion : B 9B4 9B5 I. IRODUIO Finanial ivaiv is a ona hos val ns on on o mo siis o asss all nling asss. n oion is a oningn laim ha givs h hol h igh b no h obligaion o b o sll an nling ass o a min i all h sik o xis i ing a ain io o im. Oions om in a vai o "lavos". sana oion os h igh o b o sll an nling si b a ain a a a s sik i. In omaison o oh oion ss sana oions a no omlia. h oions ma b ll-knon in h maks an as o a. Inasingl hov h m sana oion is a laiv mas o omlxi siall hn invsos a onsiing vaios oions an ss. xamls o sana oions a mian oions hih allo xis a an oin ing h li o h oion an oan oions ha allo xis o o onl a xiaion o mai a. Blak an hols blish hi smina ok on oion iing [] in hih h sib a mahmaial am ok o ining h ai i o a oan oion. h s a no-abiag agmn o sib a aial inial qaion hih govns h volion o h oion i ih s o h mai im an h i Mansi iv mb 3 3; vis ovmb 9 3. hma Rahal oo is ih h Damn o Mahmais Univsi o Ibaan Oo a igia hon: ; -mail: noo@ahoo.om noo@gmail.om. na mmanl Fagba is ih h Damn o Mahmaial ins kii a Univsi o kii igia mail: mmasa6@ahoo.om mmasa@gmail.om. o h nling ass. Moov in h sam a [9] xn h Blak-hols mol in sval imoan as. h sb o nmial mhos in h aa o oion iing an hging is v boa ing mo mans on omaional s an iin. i ang o in s o onas a availabl an in man ass h a sval ania mols o h sohasi volion o h nling sa vaiabls []. W sn an ovvi o binomial mol in h onx o Blak-hols-Mon [ 9] o iing sana oions bas on a isk-nal valaion hih as is sggs an iv b [4] an assms ha sok i movmns a omos o a lag nmb o small binomial movmns. Oh os a ini in mhos o iing ivaiv b [3] Mon alo mho o iing oan oion an ah nn oions ino b [] an a lass o onol vaias o iing sian oions n sohasi volaili mols onsi b [5]. h omaaiv s o ini in mho an Mon alo mho o iing oan oion as onsi b [6]. om nmial mhos o oions valaion onsi b []. [] onsi Mon alo mho o iing som ah nn oions. On h aa o binomial mol an Mon alo mho o iing oan oions as onsi b [7]. hs os ovi mh o h inass in hih man onibions o h il ov h as h as hav bn n. In his a shall onsi onl h aa o binomial mol o h valaion o sana oions naml; mian an oan oions ih ivin il in h onx o Blak-hols mol. II. BIOMIL MODL FOR H VLUIO OF DRD OIO WIH DIVIDD YILD his sion sns binomial mol o h valaion o sana oions ih ivin il.. Binomial Mol his mol is a siml b ol hniq ha an b s o solv h Blak-hols an oh omlx oioniing mols ha qi solions o sohasi inial qaions. h binomial oion-iing mol o-sa oion-iing mol is mahmaiall siml an i is bas on h assmion o no abiag. h assmion o no abiag imlis ha all isk- invsmns an h isk- a o n o ollas o van onlin bliaion: 3 Fba 4

2 IG Innaional Jonal o li Mahmais 44: IJM_44 5 invsmn b il osiiv ns. I is h aivi o man inivials oaing ihin h onx o inanial maks ha in a hols hs oniions. h aiviis o abiags o slaos a on malign in h mia b hi aiviis ins ha o inanial mak ok. h ins ha inanial asss sh as oions a i ihin a nao olan o hi hoial vals []. Binomial Oion Mol his is in as an iaiv solion ha mols h i volion ov h hol oion valii io. Fo som s o oions sh as h mian oions sing an iaiv mol is h onl hoi sin h is no knon los om solion ha is i ov im. h a o s o binomial mol naml i. Rombining ii. on-ombining h in bn ombining an nonombining s is omaional onl. h ombining ih n aing ios has n inal nos in n ios h an b 3... n s an a non- n ombining ih n aing ios has inal nos. In binomial h nmb o inal nos is h nmb o os o a oksh hn imlmning h binomial mol. h Blak-hols mol sms o hav omina oion iing b i is no h onl ola mol h ox- Ross-Rbinsin RR Binomial mol is also ola. h binomial mols is sggs b [4] in hi a il Oion iing: imlii oah in 979 hih assms ha sok i movmns a omos o a lag nmb o small binomial movmns. Binomial mol oms in han ailal hn h hol has al xis isions o mak io o mai o hn xa omla a no availabl. hs mols an aommoa omlx oion iing oblms []. RR on a b sok movmn mol oh han h gomi Bonian moion mol ali b Blak- hols h binomial mols. Fis ivi h li im [ ] o h oion ino im sbinval o gh h os ha is h sok i a h bginning o a givn io. hn h binomial mol o i movmns assms ha a h n o ah im io h i ill ih go o ih obabili o on o ih obabili h an a h an. on aos ih W all ha b h inil o isk nal valaion h x n om all h a oions is h isk ins a. W an val ash los b isoning hi x vals a h isk- ins a. h aams an sais h oniions o h isk-nal valaion an lognomal isibion o h sok i an hav h x sok a im as. n xlii xssion o is obain as ollos: ons a oolio omising a long osiion in nis o h nling ass i an a sho osiion hn. W alla h val o ha maks h oolio isklss. I h is an movmn in h sok i h val o h oolio a h n o h li o oion is an i h is a on movmn in h sok i h val boms las o xssions a qal hn hav. in h In h abov as h oolio is isklss an ms an h isk- ins a. hos ha is h aio o h hang in h oion i o h hang in h sok i as mov bn h nos. I no h isk- ins a b h sn val o h oolio is. h os o sing h oolio is i ollos ha 3 bsiing ino 3 an simliing hn 3 boms 4 Fo 5 hn hav a on-s binomial mol. qaions 4 an 5 bom sivl an 7 nabl an oion o b i sing a on-s binomial mol. lhogh o no n o mak an assmions abo h obabiliis o h an on movmns in o o obain 4. h xssion is h x ao om h oion. Wih his inaion o 4 hn sas ha h val o h oion oa is is x val ison a h isk- a. Fo h x n om h sok hn h obabili o an movmn is assm o b h x sok i a im is givn b ho van onlin bliaion: 3 Fba 4

3 IG Innaional Jonal o li Mahmais 44: IJM_ bsiing 7 ino 8 ils o ho 9 Whn onsing a binomial o sn h movmns in a sok i hoos h aams an o mah h volaili o a sok i. h n on h ass i in a small inval o im is W h = Man n ni im = Volaili o h ass i W = ana Bonian moion an W W. gling os o W o o o an abov i ollos om ha h vaian o h n is W W W W Fo h on io binomial mol hav ha h vaian o h n o h ass i in h inval as o mah h sok i volaili ih h 's aams ms ho hav ha bsiing 7 ino hav ha Whn ms in on solion o his qaion is an high os o a igno 3 h obabili obain in 7 is all h isk nal obabili. I is h obabili o an a movmn o h sok i ha nss ha all bs a ai ha is i nss ha h is no abiag. Hn ollos om h assmion o h isk-nal valaion. ox-ross-rbinsin Mol h ox-ross-rbinsin mol [8] onains h Blak- hols analial omla as h limiing as as h nmb o ss ns o inini. on im io h sok i an mov o ih obabili o on o obabili as shon in h Fig. blo. Fig. : ok an Oion is in a Gnal On- ih ho h osoning val o h all oion a h is im movmn is givn b [] max max h an 4 a h vals o h all oion a a an ona movmns sivl. W n o iv a omla o alla h ai i o h oion. h isk nal all oion i a h sn im is 5 o xn h binomial mol o o ios l o o onsiv no h all val a im a sok movmns o on a an on ona movmn an o o onsiv ona movmn o h sok i as shon in h Fig. blo. Fig. : Binomial o h siv ss an all i in a o-io Mol van onlin bliaion: 3 Fba 4

4 IG Innaional Jonal o li Mahmais 44: IJM_44 5 hn hav max max max h vals o h all oions a im a bsiing 7 ino 5 hav: is all h n all val sing im h h nmbs an a h isk nal obabiliis ha h nling ass is sivl a aain. an W gna h sl in 8 o val an oion a as h max max!!! an 9 is h binomial oiin. W assm ha m is h smalls ing o hih h oion's ininsi val in 9 is ga han o. his imlis ha m m m m. hn 9 an b xss as givs s h sn val o h all oion. h m is h isoning ao ha s o is sn val. h is m o is h binomial obabili o a movmns o o a h is aing ios an is h osoning val o h ass a a mov o h sok i. h son m is h sn val o h oion`s sik i. ing R in h is m in obain R ho m m m R R m m; o l nos h binomial isibion nion givn b m ; m is h obabili o a las m sss in innn ials ah sling in a sss ih obabili an in a ail ih obabili. hn l R an R onsqnl i ollos om ha m; m;. 3 h mol in 3 as vlo b ox-ross- Rbinsin [4] an ill o i as RR mol o h valaion o oan all oion. h osoning val o h oan oion an b obain sing all- ai o h om. W sa a lmma o RR binomial mol o h valaion o oan all oion. Lmma [7]: h obabili o a las m sss in innn ials ah sling in a sss ih obabili an in a ail ih obabili q is givn b m ; L m R an q R hn i ollos ha m; m; B. os o h Imlmnaion o h Mli- io Binomial Mol Whn sok i movmns a govn b a mlis binomial an a ah binomial s saal. h mli-s binomial an b s o h mian an oan sl oions. Lik h Blak-hols mol h RR omla in 3 an onl b s in h iing o oan sl oions an is asil imlmnabl in Malab. o ovom his oblm s a in mli-io binomial mol o h mian sl oions on boh h ivin an nonivin aing soks. h no-abiag agmns a s an no assmions a qi abo h obabiliis o h an on movmns in h sok i a ah no. van onlin bliaion: 3 Fba 4

5 IG Innaional Jonal o li Mahmais 44: IJM_44 5 Fo h mli io binomial mol h sok i $$ is knon a im o. im h a o ossibl sok is an sivl. im h a h ossibl sok is an an so on. In gnal a im i h i sok i a onsi givn b i o... 4 h is h oal nmb o movmns an is h oal nmb o movmns. h mli-io binomial mol an l nmos sok i ooms i h a nmos ios. h binomial oion iing mol is bas on ombining s ohis h omaional bn qikl bom ovhlming as h nmb o movs in h inass. Oions a vala b saing a h n o h a im an oking baka. W kno h oh o a all an a im is max 5 max sivl. Bas a assming h isk nal ol h val a ah no a im an b alla as h x val a im ison a a o a im io. imilal h val a ah no a im an b alla as h x val a im ison o a im io a a an so on. B oking bak hogh all h nos obain h val o h oion a im o. os ha h li o a oan oion on a nonivin aing sok is ivi ino sbinvals o h gh. Dno h no a im i as h. Din i no h i an i as h val o h oion a h i i i a h i no is oan all an an b xss as max no. h sok. hn h siv 6 max o... 7 h is a obabili o moving om h i no a im i o h i no a im i an a obabili o moving om h i no a h i o h i no a im i. hn h nal valaion is i [ i i i i ] 8 Fo an mian oion hk a ah no o s hh al xis is abl o holing h oion o a h im io. Whn al xis is akn ino aon his val o ms b oma ih h i oion's ininsi val [7]. Fo h mian oion hav ha i i i [ i i i i 9 givs o ossibiliis: I I i i hn al xis is avisabl. i i hn al xis is no avisabl. ] 9. Vaiaions o Binomial Mols h vaiaions o binomial mols is o o oms naml nling sok aing a ivin o knon ivin il an nling sok ih oninos ivin il. Unling sok aing a ivin o knon ivin il h val o a sha ls h val o h oman. a ivin is ai h val o h oman is so h val o h sha. o i o... i i 3 3 Unling sok ih oninos ivin il sok inx is omos o sval hn in shas. ah sha givs ivin aa a in im so h sok inx an b assm o ovi a ivin oninosl. W xlo Mon's mol h asmn o h Blak-hols mol o a o oan oions on soks ha as ivin. H h isk- ins a is h is h oninos moii om o ivin il. W al h sam inil in o binomial mol o h valaion o h oions. h isk nal obabili in 5 is moii b h oh aams mains h sam. 3 hs aams al hn gnaing h binomial o sok is o boh h mian an oan oions on soks aing a oninos ivin an h ill b inial in boh ass. h obabili o a sok i inas vais invsl ih h lvl o h oninos ivin a. III. BL HOL QUIO Blak an hols iv h amos Blak-hols aial inial qaion ha ms b saisi b h van onlin bliaion: 3 Fba 4

6 i o an ivaiv nn on a non-ivin aing sok. h Blak-hols mol an b xn o al ih oan all an oions on ivin-aing soks his ill b shon la. In h sql shall sn h ivaion o Blak-hols mol sing a noabiag aoah.. Blak-hols aial Dinial qaion W onsi h qaion o a sok i W 33 h is h a o n is h volaili an W ollos a Win oss on a il obabili sa B F B in hih ilaion { B B F h B is h sigma-algba gna b }. : { o sos ha is h ai i o a all oion o oh ivaiv oningn laim o h nling ass i a im. ssming ha ] [ R hn b h Io s lmma givn blo; s s s W x U x U U W s U W U 34 W obain h Blak-hols aial inial qaion o h om 35 olving h aial inial qaion abov givs an analial omla o iing h oan sl oions. hs oions an onl b xis a h xiaion a. h mian sl oions a xis anim o h mai a. hs h analial omla ill iv is no aoia o iing hm o his al xis ivilg [8]. In h as o a oan all oion hn h k bona oniion is max 36 In h as o a oan oion hn h k bona oniion is max 37 B. olion o h Blak-hols qaion W shall al h bona oniions o h oan all oion o solv h Blak-hols aial inial qaion. h ao oniion is max 38 h lo an bona oniions a givn b hs a h oniions ha ms b saisi b h aial inial qaion. L h is h xiaion a an is h sn im. in 39 bsiing 39 ino 35 ils 4 aking hn ho 4 W no ino a n solion an hn hav ha 4 bsiing 4 ino 4 hav 43 lso sbsiing 4 ino 43 ils IG Innaional Jonal o li Mahmais 44: IJM_44 5 van onlin bliaion: 3 Fba 4

7 ho is all a ision qaion hih has a namnal solion as a nomal nion. x 45 o x 46 in hn 47 h ao o all oion boms max max max max max 48 h solion o 44 is 49 W s h ao oniion in 48 an h namnal solion o 46 o obain x max 5 W no h isibion nion o a nomal vaiabl b x x x 5 hn 5 boms x x 5 L o 5 boms x x 53 W onsi h son m in h igh-han si o 53 ha is x ing 54 hn an h limis o 53 sing 54 a givn blo hn hn IG Innaional Jonal o li Mahmais 44: IJM_44 5 van onlin bliaion: 3 Fba 4

8 55 hanging h vaiabl om o h son m in h igh-han si o 54 boms 56 h is ingan o h is m in 53 is xss as x x x x 57 W s h iniion o o hav ho hn 57 boms x x 58 bsiing 58 ino h is m o 53 hav x B hanging h vaiabls as i in h vios as g 59 Wh an 6 Whn 53 boms 6 Rall ha hn 6 his is h Blak-hols omla o h i a im o o a oan all oion on a non-ivin aing sok []. W an iv h osoning oan oion omla o a non-ivin aing sok b sing h all- ai givn b. h oan analial omla is 63 h h oan all an anali omla hav gain olai in h ol o inan o h as ih hih on an s h omla o oions valaion h oh aams aa om h volaili an asil b obsv om h mak. hs i boms nssa o in aoia mhos o simaing h volaili.. Divin aing ok W lax h assmion ha no ivin a ai ing h li o h oion an xamin h o ivin on h val o oan oions b moii h Blak- hols aial inial qaion o a o hs ivins amns. o shall onsi h oninos ivin il mol l no h onsan oninos ivin il hih is knon. his mans ha h hol ivs a ivin ih h im inval. h sha val is lo a h ao o h ivin an so h x a o n o a sha boms. h gomi Bonian moion mol in 33 boms W 64 an h moii Blak-hols aial inial qaion in 35 is givn b 65 L solving 65 b aling h sam mho h oan all oion o a ivin aing sok is givn b 66 an h oan oion is IG Innaional Jonal o li Mahmais 44: IJM_44 5 van onlin bliaion: 3 Fba 4

9 IG Innaional Jonal o li Mahmais 44: IJM_44 5 h h sls in 66 an 67 an similal b ahiv b onsiing h non-ivin aing sok omla in 6 an 63. h ivin amn los h sok i om o an h isk- ins a hih is h a o n om o [7]. D. Bona oniion o Blak-hols Mol h bona oniions o Blak-hols mol o iing a sana oion a givn b lim max 69 W shall sa blo som homs iho oo as ollos: hom [7]: Un h binomial mol o sok iing h i o a oan sl oion ih xiaion a is givn b * 7 oolla 3 [7]: Un h binomial mol o sok iing h i o a oan sl oion ih xiaion a is givn b * * max h obabili * * 7 * nos x val n h isk nal * o sok i. h abov hom an b in in os as h i o h oion is qal o h sn val o h x ao o h oion n h isk nal mas. hom 4: oninos Blak hols Fomla [7] ssm ha an in sh ha an. hn h hols lim x 7 x h. is h mlaiv nion o h sana nomal isibions. hom 5: L b h n n RR 73 io RR binomial la o a sana oan oion ih xn an B n RR h is h la. ho n B n n n O n n n an m is h lags ing hih saisis m nm. IV. UMRIL XML D RUL his sion sns som nmial xamls an sls gna as ollos;. mial xamls xaml : onsi a sana oion ha xis in h monhs ih an xis i o$ 65. ssm ha h nling sok as no ivin as a $ 6 an has a volaili o 3% annm. h isk- a is 8% annm. W om h vals o boh oan an mian sl oions sing Binomial mol agains Blak-hols mol as inas h nmb o ss ih h olloing aams: van onlin bliaion: 3 Fba 4

10 IG Innaional Jonal o li Mahmais 44: IJM_ h Blak-hols i o all an oions a.334 an sivl. h sls obain a shon in abl I blo. xaml : Binomial iing sls in a all i o $ 3.87 an a i o$ h ins a is 5 % h nling i o h ass is $ an h xis i o h all an h is $ 85. h xiaion a is in h as. Wha aions an an abiag ak o mak a isklss oi i h all is aall slling o $ 35? olion: in h all is ovval an abiag ill no an o i h all b h b h sok an boo h sn val o h xis i sling in h olloing ash lo oa as shon blo Wi all $ 35 B $5. 3 B sha $ Boo $85.53 $ $ 3.3 h val o h oolio in h as ill b ohlss galss o h ah h sok aks ov h h-a io. xaml 3: onsi iing a sana oion on a sok aing a knon ivin il. 5 ih h olloing aams: h sls obain a shon in abl II blo. xaml 4: W onsi h oman o Binomial mol agains h Blak-hols i o mian an oan oions ih h sls obain a shon in h abl III blo. h onvgn o h binomial mol o h Blak- hols val o h oion is also ma mo iniiv b h gah in Fig. 3 blo. Fig. 3: onvgn o h oan all i o a on- Divin aing ok Using Binomial Mol o h Blak- hols Val O 7. 6 B. abl o Rsls W sn h sls gna in h abls I II an III blo. BL I: H OMRIV RUL LYI OF H BIOMIL MODL D BL HOL VLU B.334 B OF H DRD OIO oan all mian all oan mian BL II: OU-OF-H MOY -H-MOY D I-H-MOY DRD OIO O O YIG OW DIVIDD YILD van onlin bliaion: 3 Fba 4

11 IG Innaional Jonal o li Mahmais 44: IJM_44 5 BL III: OMRIO OF H RR BIOMIL MODL O BL-HOL VLU W IR B 7.6 B Disssion o Rsls s an s om abls I an III ha Blak-hols omla o h oan all oion an b s o i mian all oion o i is nv oimal o xis an mian all oion bo xiaion. s inas h val o h val o h mian oion is high han h osoning oan oion bas o h al xis mim... omims h al xis o h mian oion an b oimal. abl II abov shos ha mian oion on ivin aing sok is alas oh mo han is oan ona. v in h mon mian oion has a high al xis mim. h mim o boh an all oions ass as h oion gos o o h mon. h mian an oan all oions a no oh h sam as i is oimal o xis h mian all al on a ivin aing sok. o o h mon mian an oan all oions a oh h sam. his is o h a ha h migh no b xis al as h a ohlss. h abov sls a obain sing Malab os. V. OLUIO Oions om in man in lavos sh as ah nn o non-ah nn ix xis im o al xis oions an so on. Divin is a amn ma o h on o a sok. On an isingish h kins o ivin. ash ivin is a amn in ash. ok ivin is a amn in soks. h hi is a mix o h o vios ivins. Rla o all- ai on an a a snhi sok o o a long all an a sho o h sam sik. gains his snhi sok on an sll h al sok. I on bs h nomal sok an sll h snhi sok Long Long ho on as a onvsion. I on os h vs ansaion Long ho ho his is all a vsal. Wih s o hangs in h nling val h is almos no isk involv in hs aing sagis. Hov on o h biggs isks assoia o hs aing sagis is a hang in ivin. Whn a ivin is lo his ill hav a osiiv on h val o h all oion an a ngaiv on h val o h oion. his ol b asil nsoo i on alis ha a lo ivin ill sl in a high val o h i o h sok. s on is long on oion an sho h oh oion h hangs in vals o a hang in ivin ok in h sam iion. lo ivin ill sl in a loss o hos as ha hav s a onvsion. hs as a sai o b long h ivins. I ill sl in a oi o hos ho hav s a vsal i.. o hos as ho a sai o b sho h ivins. I ill b no sis ha mian oions a mo omlia an insing. ivin an igg h al xis o an mian oion. In h absn o ivins oan an mian oions oh h sam val an i is nv oimal o xis an mian all oion bo is xiaion. On shol alas hol h mian all oion ill xiaion. On h oh han i migh b avanagos o xis an mian oion bo xiaion i h oion is siinl in h mon. Whn ivins a ai ing h li o h oion i migh b avanagos o xis an mian all oion bo xiaion. Fo mian all oion al xis is ossibl hnv h bnis o bing long h nli a si o ommoi hih is sb o liv on xis o an oion ona o onvibl si oigh h os o sning h oion al. Fo insan on h a bo an x-ivin a i ma mak sns o xis an qi all oion al in o o oll h ivin. Fo an mian oion i is nv oimal o xis h oion immial bo a ivin amn. hangs in x ivin an mak h in bn al xis an no al xis o h oion Binomial mol is si o aling ih som o hs oion lavos. In gnal binomial mol has is avanags an isavanags o s. his mol is goo o oions valaion ih al xis ooniis aa onvgs as an i is laivl as o imlmn b an b qi ha o aa o mo omlx siaions. W onl ha binomial mol is goo o h valaion o sana oions b ah nn oion mains oblmai. RFR [] F. Blak an M. hols h iing o oions an ooa liabiliis Jonal o oliial onom []. Bol Oions: Mon alo aoah Jonal o Finanial onomis [3] M. Bnnan an. ha Fini in mhos an m osss aising in h iing o oningn laims Jonal o Finanial an Qaniaiv nalsis [4] J. ox. Ross an M. Rbinsin Oion iing: simlii aoah Jonal o Finanial onomis [5]. D G. Li an G. G lass o onol vaias o iing sian oions n h sohasi volaili mols IG Innaional Jonal o li Mahmais [6]. Fagba. oo an. Babalola h omaaiv s o ini in mho an Mon alo mho o iing oan oion Mahmaial ho an Moling [7].. Fagba. R. oo J.. Oknlola O.. mo an. O. ai On h aa o binomial mol an Mon alo van onlin bliaion: 3 Fba 4

12 IG Innaional Jonal o li Mahmais 44: IJM_44 5 mho o iing oan oions Innaional Jonal o Mahmais an aisis is [8] J. Hll Oions Fs an oh Divaivs ason aion In. Fih iion ni Hall Js 3. [9] R.. Mon ho o aional oion iing Bll Jonal o onomis an Managmn in []. R. oo an.. Fagba om nmial mhos o oions valaion ommniaions in Mahmaial Finan []. R. oo an.. Fagba Mon alo mho o iing som ah nn oions Innaional Jonal o li Mahmais [] J. Wson. olan an. hasi Finanial ho an ooa oli Foh iion Yok ason ison Wsl 5.. van onlin bliaion: 3 Fba 4