WARRANT VALUATION METHODS

Transcription

1 ARRA ALUAIO MEHOD o R. aa Ph.D. FRM Mogom Ivsm cholog Ic. Fal am J 83 Pho: so.saa@fools.com cos h cas of a Euopa sl waa ha ls h coac hol o bu a h sk pc o sha of h ulg sock. BLAK-HOLE-MERO MODEL h valu of h waa s ual o h valu of a call opo wh h sam sk a mul-xpao: BM h call opo valu s sma usg h Black-chols-Mo fomula BM wh s h sock valu s h sk s h m-ul-xpao s h sock volal s h skf s a a s h v l. DILUED BLAK-HOLE-MERO MODEL h valu of h waa s ual o h lu valu of a call opo wh h sam sk a m-ul-xpao: BM h luo faco s h umb of ousag sock shas v b sum of h umbs of ousag sock shas a ssu waas. ouh a Gala 99a o ha pacc waa pcs a of calcula b mulplg h oucom fom h Black-chols-Mo fomula b h luo faco. Gala 989 a ouh a Gala 99a agu ha hs pocu whch s bas upo a mspao of h Gala a chll 978 mol. Accog o ouh a Gala 99a hs mol s hl wog: aa aluao Mhos Mogom Ivsm cholog Ic. / o R. aa Ph.D. FRM Fbua Pag of

2 Pag of aa aluao Mhos Mogom Ivsm cholog Ic. / o R. aa Ph.D. FRM Fbua hs appoach s wog a gal las o a usmao of h waa s pc. A alav appoach s o m h valu of h waas a h fm s oh labls smulaousl wh h valu a h volal of h fm s asss. GALAI-HELLER MODEL IH DIIDED YIELD Dog b h compa s valu a mau h paoff fo a waa s Gala a chll 978; ox a Rubs 985 pags ; Hull 993 pags 8-9: max max 3 max hfo h waa valu s ual o h valu of a call opo aus fo luo o h compa o sock valu Gala a chll 978; ouh a Gala 99a; ouh a Gala 99b: 4 BM I h abov uao s h volal of h compa o sock valu a s L a og 4; L a og 5: 5 Usg h fomula fo h compa valu suls: 6 BM Usg h Black-chols-Mo fomula w g: 7

3 wh: l 8 9 I uao 7 Ф s h cumulav omal sbuo fuco. I shoul b o ha uao 7 s a mplc uao fo h waa valu whch appas o boh h gh a lf ha s of h ual sg. h soluo of hs uao us aos. UKHO-LI-OG MODEL IH DIIDED YIELD Followg Ukhov 3; 4 a L a og 4; 5 h waa pcg uao 7 ca b -w usg h compa valu sa of h sock valu: wh: l Followg Ukhov 3; 4 fom uao 5 w g h followg laoshp Ukhov 3; Ukhov 4: xp 3 Fom uao 4 w g: aa aluao Mhos Mogom Ivsm cholog Ic. / o R. aa Ph.D. FRM Fbua Pag 3 of

4 4 hfo fom uaos 3-4 suls: 5 ock volal s la o fm volal ouh a Gala 994; chulz a auma 994; Ukhov 3; Ukhov 4 as follows s Appx: 6 hfo fom uaos 5-6 suls: 7 Euaos a 7 ps a ssm of wo o-la uaos wh wo ukows a. h soluo of hs ssm s us uao 4 o oba h waa valu. UKHO-DAE-EHRHARD MODEL IH DIIDED YIELD Followg Ukhov 3; 4 a Dav a Ehha 5 fooo h waa pcg uao 7 ca b -w as follows: 8 wh : 9 aa aluao Mhos Mogom Ivsm cholog Ic. / o R. aa Ph.D. FRM Fbua Pag 4 of

5 Pag 5 of aa aluao Mhos Mogom Ivsm cholog Ic. / o R. aa Ph.D. FRM Fbua l Fom uao 9 w g h followg laoshp: 3 h waa valu s ual o h valu of a call opo aus fo luo o h compa o sock valu Gala a chll 978; ouh a Gala 99a; ouh a Gala 99b: 4 BM I shoul b o ha fom uaos 4 a 9- w g:

6 Pag 6 of aa aluao Mhos Mogom Ivsm cholog Ic. / o R. aa Ph.D. FRM Fbua I cocluso w g: 5 hfo fom uaos 3 a 5 suls: 6 Fom uaos 6 a 6 suls: 7 Euaos 8 a 7 ps a ssm of wo o-la uaos wh wo ukows a. h soluo of hs ssm s us uao 4 o oba h waa valu.

7 LAUERBAH-HULZ MODEL IH DIIDED DIRIBUIO Laubach a chulz 99 us a vso of h Gala a chll 978 mol aap fo kow v sbuos. If uao 7 sa of - w us [-PD ] wh PD s h ps valu of h fuu v sbuos h w oba h Laubach a chulz 99 mol. U som mak coos ma b mo alsc o mak assumpos gag h amou of h vs pa a ff as ah ha o assum h valu of h v l. I hs cas h sock pc wh volal ca b s as h sum of wo compos:. O sk-lss compo cospog o h kow vs ug h lf of h coac; a. Aoh couous sk compo * wh volal * wh o vs. A a gv m h sk-lss compo s h ps valu of h fuu vs. h scoug a s h sk-f s a. A h bgg of h smulao assumg ha a m of h v pam s D h valu of * wll b: [ * D ] 8 Hull 993 pag 347 assums ha * s cosa o a mos ha gal * >. Howv Hull 993 os o ca how o compu *. hss 997 pag 57 uao 4.9. suggss h followg fomula fo h cas wh o v sbuo: * D [ D ] 9 I h abov fomula D s h v sbuo a m D a s h ps m. hs fomula s mo also b Hull 3 pag 53. B a os xpa a mpov h hss 997 appoxmao usg a wgh avag of a aus a a uaus vaac wh h wghg ps o h m of h v pam D : aa aluao Mhos Mogom Ivsm cholog Ic. / o R. aa Ph.D. FRM Fbua Pag 7 of

8 Pag 8 of aa aluao Mhos Mogom Ivsm cholog Ic. / o R. aa Ph.D. FRM Fbua * 3 ] [ D I h abov fomula s h umb of kow v sbuos ug h lf of h coac a s h xpao m. Amao Maos al. 6 po ha h B a os appoxmao pfoms sgfcal b ha assumg *. I o o us h Laubach a chulz 99 mol shoul b o ha oc * s kow s cssa o sma.. h volal of h compa o sock valu. UKHO-HE-LI MODEL IH DIIDED DIRIBUIO Fo h cas of kow v sbuos uao 5 bcoms h a L 8: 3 } { ] [ D Followg Ukhov 3; 4 a h a L 8 h waa pcg uao 7 ca b -w as follows: 3 } { ] [ D wh: 33 l 34

9 Pag 9 of aa aluao Mhos Mogom Ivsm cholog Ic. / o R. aa Ph.D. FRM Fbua } { ] [ D 35 Followg Ukhov 3; 4 fom uao 3 w g h followg laoshp: 36 Fom uao 35 w g: 37 hfo fom uaos suls: 38 hfo fom uaos 6 a 38 suls h a L 8: 39 Euaos 3 a 39 ps a ssm of wo o-la uaos wh wo ukows a. h soluo of hs ssm s us uao 35 o oba h waa valu. I shoul b o ha wh hss appoxmao uao 9 ma b us h suls:

10 * { D [ D ] } 4 Euaos 3 a 4 ps a ssm of wo o-la uaos wh wo ukows a. hs ssm s smla o uaos a 7 wh:. Isa of - w us [-PD ] wh PD s h ps valu of h fuu v sbuos; a. Isa of w us *. REFEREE Amao Maos J.; Dlao R.; Fa B. h xac valu fo Euopa opos o a sock pag a sc v. MPRA Pap o. 7. Much: Much Psoal RPEc Achv; Jaua 6. B R.; os. Opo o Dv Pag ocks. I Jogm Yog Eo Rc Dvlopms Mahmacal Fac. Pocgs of h Iaoal ofc o Mahmacal Fac. pags 4-7. hagha: ol cfc;. h J.; L Q. Mofcao of h aas Pcg Mol a alao Aalss. Iaoal Joual of Busss a Maagm 3 : 63-7;Fbua 8. hss. Black-chols a Bo. Opo Pcg Mols. hcago IL: Iw; 997. ox J..; Rubs M. Opos Maks. Eglwoo lffs J: Pc Hall; 985. ouh M; Gala D. ommo os h valuao of waas a opos o fms wh waas. Facal Aalss Joual 47 5: 89-9; pmb-ocob 99a. ouh M.; Gala D. aa aluao a Eu olal Avacs Fuus a Opos Rsach 5: 3 5; 99b. ouh M; Gala D. h aco bw h facal a vsm csos of h fm: h cas of ssug waas a lv fm. Joual of Bakg a Fac 8 5: 86-88; Ocob 994. Davs P. R.; Ehha M.. h Impac of opoa a Psoal axs o h aluao of Emplo ock Opos. Fac Dpam. Koxvll : Uvs of ss; Apl 6 5. hp://copgovc.uk.u/rsach/impodavseh5.pf aa aluao Mhos Mogom Ivsm cholog Ic. / o R. aa Ph.D. FRM Fbua Pag of

11 Gala D. A o o "Eulbum waa pcg mols a accoug fo xcuv sock opos. Joual of Accoug Rsach 7 : 33-35; Auum989. Gala D.; chll M. I. Pcg of waas a h valu of h fm. Joual of Fac 33 5: ; 978. Hull J.. Opos Fuus a Oh Dvav cus. o. Eglwoo lffs J: Pc Hall; 993. Hull J.. Opos Fuus a Oh Dvav cus. 5 h o. Upp al Rv J: Pc Hall; 3. Laubach B.; chulz P. Bass a pof oppous waa maks Avacs Fuus a Opos Rsach 5: 55 66; 99. L F.; og M. H. F. Emplo ock Opos Eu aluao a h aluao of Opo Gas Usg a aa-pcg Mol. Augus 4. Avalabl a R: hp://ss.com/absac54853 L F.; og M. H. F. Emplo ock Opos Eu aluao a h aluao of Opo Gas Usg a aa-pcg Mol. Joual of Accoug Rsach 43 : 97 3; 5. chulz G.U. ; auma. Robusss of opo-lk waa valuao. Joual of Bakg a Fac 8 5: ; Ocob 994. Ukhov A. D. aa pcg usg obsvabl vaabls. Augus 4 3. Avalabl a R: hp://ss.com/absac45 Ukhov A. D. aa pcg usg obsvabl vaabls. h Joual of Facal Rsach 7 3: ; Fall 4. aa aluao Mhos Mogom Ivsm cholog Ic. / o R. aa Ph.D. FRM Fbua Pag of

12 APPEDI L us assum ha h sock follows a gomc Bowa moo pocss: µ w A Accog o Io s Lmma h fuco follows h pocss scb blow: µ w A hs pocss ma b scb as follows: α w A3 hfo fg h coffcs of w fom uaos A-A3 w hav h followg coo: A4 h fuco has o b spcf. om xampls a ls blow: A5 A6 A7 aa aluao Mhos Mogom Ivsm cholog Ic. / o R. aa Ph.D. FRM Fbua Pag of