866 Uni Naions Plaza i 566 Nw Yo NY 7 Phon: + 3 355 Fa: + 4 668 info@gach.com www.gach.com Eoan an Amican oions wih a singl amn of ivins Abo fomla Roll Gs & Whal Ma Ioff Absac Th aicl ovis a ivaion of fomlas fo icing Eoan an Amican all oions wih a singl amn of ivins. Fo a Eoan oion ivaion is bas on h isibion of h soc ic a h iaion im. Fo h Amican oion is iv an nmicall solv an qaion fo h oion ic as a fncion of h soc ic an im nil iaion. Nmical sls a coma o calclaions iv fom calclaions bas on h fomlas cnl s mhos. I ns o ha h Amican oion sls o iff ssniall fom h mho of Roll Gs & Whal. Insingl his aicla cas of oions whn ivins a ai onl onc is analz in ail in h oion ho an in aicla onlin h://www.global-ivaivs.com in h scion on Amican oions. This scion ovis a mho fo calclaing h Amican all oions fo socs wih a singl amn of h ivin vlo b Roll Gs & Whal. Aanl his mho is imlmn in Malab financial ooli fncion osocbgw. L s fis consi how o calcla ic fo h Eoan all wih singl ivin amn. Un sana assmions whn h chang in h soc ic is scib b a iffsion ocss wih consan aams soc's ic a iaion im col b scib b h fomla: D Wh - is h im o a a ivin; - is h im fom amn of a ivin o iaion; - sha ic a h im of slmn; D - ivin amon; - Gass anom vaiabl wih man an sana viaion:.5 Nw Yo 3
866 Uni Naions Plaza i 566 Nw Yo NY 7 Phon: + 3 355 Fa: + 4 668 info@gach.com www.gach.com Nw Yo 3 - Gass anom vaiabl wih man an sana viaion:.5 3 In an 3 - f ins a; σ - volaili. Ranom vaiabls an a innn. Dnsi isibion of hs vaiabls qals: 4 Toal ic of h all oion wih a si obviosl: D D 5 Eqaion 5 can b win as: D D Ln 6 Using h cmlaiv nomal isibion fncion Lalac fncion L : L 7
866 Uni Naions Plaza i 566 Nw Yo NY 7 Phon: + 3 355 Fa: + 4 668 info@gach.com www.gach.com Psn 6 as: D Ln.5 L D L 8 Th ingal in 8 can b calcla fo aml sing Hmi olnomials. W s z 9 bsiing 9 ino 8 w obain.5 D Ln L D L To calcla h ingal in w al Gass nmical ingaion sing Hmi olnomials. Accoing o his fomla: Wh n i w i i i - h oos of n-o Hmi olnomial; w i - wighs. Fomlas an allow on o accal calcla h ic of h Eoan oion wih on ivin. Obviosl in h cas wh h aam D is an shol show h val of h oion ic Nw Yo 3 3
866 Uni Naions Plaza i 566 Nw Yo NY 7 Phon: + 3 355 Fa: + 4 668 info@gach.com www.gach.com calcla accoing o h fomla Blac-chols. Imlmn in Ecl fo an oion wih h aams = = = 7% σ = % D = =.555 9 as =.4978 53 as hs fomlas giv h val of h oion ic.889396. Th Blac-chols fomla givs.889397. In acic Eoan oion icing wih ivins gnall cai o b h Blac-chols oion o b ing h f -coninos ivin a o b cing h. An in boh cass h fnc is o h avag isibion fo h sha ic a h iaion im M obviosl qal o: M D In h fis cas h aam f is h solion of qaion: 3 f D Tha is: f D Ln 4 In h scon cas h aam is h solion of qaion: 5 D Tha is: 6 D Tabl shows h vals of h calcla oion ics wih h abov aams wih 4 vals ivins scifi in colmn D fo h fo mhos of calclaion: in colmns P P ajs coninos ivin a an soc ics scivl in h colmn P3 val obain sing h calclao on h wbsi www.ivolaili.com colmn P4 val obain sing an. Tabl. D P P P3 P4.997577.997.979.33348 4.634375.63438.63.95693 6.68435.684.68.65 8.757.7.3.334 Nw Yo 3 4
866 Uni Naions Plaza i 566 Nw Yo NY 7 Phon: + 3 355 Fa: + 4 668 info@gach.com www.gach.com As shown in Tabl h aiional wa o calcla Eoan oion wih a ivin il vals ha a iffn fom h vals scib in h mol. Howv h iffncs a v small. W n consi h Amican oion. Accoing o h finiion h b of h oion can cis i a an im fom chas o iaion an fo his igh o a in gnal som amon cash. I is gnall acc ha al iaion occs if an onl if h ic of h oion os no c h iffnc bwn h soc ic an h si ic. Pic of Amican all P as a fncion of h soc ic an im o iaion saisfis h following qaion: P Ma 7 P Wh Δ - obabili nsi of ic changs fom h val o h val of in h im Δ. Accoing o h log-nomal mol of soc ic changs w can wi: 8 Wh Δ - Gass anom vaiabl wih man an sana viaion:.5 9 Th anom vaiabls an Δ - innn. Th obabili nsi of ic changs fom h val o h val of in h im Δ can b calcla as follows. Th join cmlaiv isibion fncion of an is: F v P ob v v P ob Ln v Ln Wh P - h nsi of h anom vaiabl ; Nw Yo 3 5
866 Uni Naions Plaza i 566 Nw Yo NY 7 Phon: + 3 355 Fa: + 4 668 info@gach.com www.gach.com PΔ - h nsi of h anom vaiabl Δ. Th nsi isibions of an is: v F v v s v Ln v Fom ha h obabili nsi of ic changs fom h val o h val of in h im Δ which is h coniional obabili nsi is: v v In accoanc wih 9 v Ln s v 3 Fom h obabili nsi of h soc ic changs fom h val o h val of in h im Δ is innn of h isibion of h soc ic ha is h sam bfo an af h amn of a ivin. To calcla h ic of an Amican all oion wih on ivin is ncssa o solv fo h nnown fncion P qaion 7.Th iniial coniion fo h solion is obvios: P ma 4 On h im inval fom h im of amn of a ivin nil iaion his fncion is h Blac-chols fomla fo a Eoan all oion sinc i is nown ha in h absnc of ivins h val of Amican an Eoan all is qal. how i ha is w ov h inqali wih h Blac-chols fomla: Nw Yo 3 6
866 Uni Naions Plaza i 566 Phon: + 3 355 info@gach.com Nw Yo NY 7 Fa: + 4 668 www.gach.com 5 inc h ivaiv of h oion ic b qals L 6 i is alwas osiiv. Thfo h minimm val of h fncion is ach a =. Thfo on = an in his cas w will ov inqali. W s F L L ln ln 7 I is as o show ha Nw Yo 3 7
866 Uni Naions Plaza i 566 Nw Yo NY 7 Phon: + 3 355 Fa: + 4 668 info@gach.com www.gach.com F F L 8 Obviosl w a onl ins in h vals of ln>. Bcas of 8 ha h ivaiv of F wih sc o fo an is non -ngaiv h fncion F has a minimm val a = an sinc L= h minimm val is which ovs h inqali. Ths in h im af h amn of h ivin oion ic is: P L L Ln.5 9 Bcas af h amn of a ivin soc ic is c b h amon of ivins ai io o amn of h oion ic is: P _ P D 3 Fo h calclaion of h oion ic a h iniial im saa fom him fo a im io o h amn of h ivin w ivi h whol inval ino n sgmns an fo h ansiion fom on o h oh w s fomla 7. Tha is P Ma n n n 3 P Nw Yo 3 8
866 Uni Naions Plaza i 566 Nw Yo NY 7 Phon: + 3 355 Fa: + 4 668 info@gach.com www.gach.com Nw Yo 3 9 In accoanc wih fomlas an 3 Ln Ln 3 W s 33 Wih 3 an 33 3 can b win as P P Ma 34 To calcla h ingal in 34 w again s h nmical mho of Gass ha is b fis conving h vaiabl of ingaion. Tha is Ma i P P 35 Ths h P P j i j j n i i i j w Ma 36
866 Uni Naions Plaza i 566 Nw Yo NY 7 Phon: + 3 355 Fa: + 4 668 info@gach.com www.gach.com Tabl shows h vals of h calcla ic of h Amican oion wih h abov aams wih h fo vals of h ivins scifi in colmn D h fo was of coming: h colmn AmM val obain sing 3 an 36 h colmn AmIvol val o bain fom alclao onlin www.ivolaili.com colmn AmBin binomial ajs coninos ivin a h colmn MaLab val obain b MaLab fncion osocbgw. Tabl. D AmM AmIvol AmBin MaLab.38.375.997.37 4.63.7.5645.544 6.46.583.339.68 8.45.577.787.664 As shown in Tabl h calcla b o fomla vals fnamnall iff fom h vals obain b mho Roll Gs & Whal MaLab fncion osocbgw. Nw Yo 3