Pricing Warrants Models: An Empirical Study of the Indonesian Market

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1 Pricig Warras Modls: A Epirical udy of Idosia Mark Zouair BEN HELIFA Waji ABBAI Absrac ai issu durig priods of fiacial crisis is o rsor isor s cofidc ad arac back o fiacial arks. us, warras a b a gra lp o rlauc ay sou Eas Asia fiacial arks jus afr 997 crisis, by couragig isors o fiac rsrucurig of roubld copais. Howr, fw pirical sudis wic wr irsd i is produc, was liid o dlopd arks ad i paricular Arica o. I is papr, w will ry o pasiz warra s spcificiis copard o opio os., w will sk o rlas idal odl rady o proid bs pricig for is produc. copariso will rla o 3 odls: Black ad ols odl B, Diluio Adjusd Black ad cols odl DAB ad fially odifid quar Roo Cosa Elasiciy of Variac RCEV adjusd by diluio ffc. pirical sudy, wic is sprad ou or 3 yars , will rla o a rg ark: Jakara ock Excag JX. y words: Black ad cols, Cosa lasiciy ariac, Diluio, Idosia ark, Moyss, Warra aluaio. JEL Classificaio: G, G3 Profssor of Fiac, Haus Euds Corcials HEC cool of aag, Carag uisia Assisa lcurr of Fiac, Haus Euds Corcials HEC cool of aag, Carag uisia, E-ail : abbassi_waji@yaoo.fr

2 Iroducio Warras ar fiacial isrus issud by fiacial isiuios ad ar grally goiad o opios arks. A warra oldr as rig o buy or o sll a spcific udrlyig ass oc arrid warra xpiry a a fixd pric calld srik pric. A firs sig, o ca cofus warra wi coioal opio. Bo sar sa pricipls of udrlyig ass, srik pric ad xpiry da. Moror, sral paprs usd o alua warra pric as sa as opio s o usig black ad cols forula. Howr, r ar wo ajor diffrcs bw warras ad coioal opios. Firs, warra lifi is qui larg a opio s o. A warra ca a a i o xpiraio goig up o s yars wras a opio cooly as a i o xpiraio o xcdig fw os. cod, opios ar issud by idiiduals so a ual xrcis will oly lad o udrlyig ass rasfr fro o opraor o aor o. Warras ar issud by firs. Cosquly, ir xrcis will caus issu of addiioal sars ad us a fir quiy diluio. I spi of warras succss, a fw paprs wr iold o ir aluaio. Moror, os of s paprs wr focusd o dlopd arks. For is papr, w ar irsd o pirical alidaio of r aluaio odls for warras issud o a rg ark. W slc Idosia ark 3 for rasos. Firs, bcaus of sou Eas Asia fiacial ark dyaic caracr is dos cas aracig all kids of isors ad spculaors. Also, Idosia warra ark rcordd igs grow ra, i r of sock xcag capializaio, copard o or Idosia fiacial arks, paricularly afr 997 crisis. A os circusacs, warras wr issud o courag risig i Idosia socks. W will ry o copar prforacs of r warra aluaio odls usig daa obsrd o Jakara ock Excag for yars 00, 00 ad 003. odls sd ar Black ad cols odl B, Diluio Adjusd Black ad cols odl DAB ad odifid quar Roo Cosa Elasiciy Variac odl odifid RCEV ispird fro Cox Cosa Elasiciy Variac Modl CEV. is papr will b srucurd as follows. firs scio iroducs B ad DAB odls. scod iroducs CEV odl ad cssary odificaios, i ordr o ak Cox Modl or suiabl for warras aluaio. ird scio iroducs saisical suaris of daa usd roug our pirical sudy. four scio copars B ad DAB aluaio prforacs. fif dals abou udrlyig asss olailiy baiour. six scio copars odifid RCEV ad DAB aluaio prforacs. Fially, w coclud by a brif suary of ai rsuls ad so suggsio for fuur rsarc. I- Valuaig warras wi B ad DAB odls. DAB odl: Galai ad cllr approac Lik a coioal Europa call, warra is rig o buy, a srik pric, a udrlyig ass a a xpiraio da. Howr, warras ar issud by copais so a ir xrcis will lad o a issu of addiioal sars ad, rfor, a diluio of issuig copay quiis. W ry o pasiz rlaio bw issuig fir alu ad warra pric. 3 Warras a appard for firs i o Jakara ock Excag dad July 3, 995 ad wr issud by bak IARA AIA. 4 J. Cox 975. Nos o opio pricig: Cosa Elasiciy Variac Diffusios. Workig Papr, aford Uirsiy. roug is papr, Cox as sablisd a w odl for pricig opios assuig a olailiy, is o logr cosa, bu os irsly o udrlyig ass pric. 5 D. Galai ad M. cllr. Pricig of Warras ad Valu of Fir. Joural of Fiac 978.

3 L cosidr a fir A wi 00% of quiis ad sars. uppos a is fir will b liquidad a da wi a socasic fuur aluv. Eac sar alu, a a da, will V b. Now l cosidr a scod fir B, idical o firs, a issus warras a 0, wi srik pric ad xpiraio da. Eac xrcisd warra allows is oldr o possss a w issuig fir sar. o, corsio raio, odλ, is qual o. If warras ar xrcisd a xpiraio da, fir B alu will b igr a fir A o. V * V sar alu of fir B a i, od, will b qual o * V warra will b xrcisd a xpiraio da, oly if udrlyig sock pric xcds warra srik pric. I or ad, V, so > * V * > * > Now, l ry o ifr warra pric usig B Europa call pricig forula. Cosidr a Europa call wi fir A sar as a udrlyig ass. A xpiraio da, call alu will * b qual o Max 0 ; -. A sa da, warra alu will b qual o Max 0 ; -. * If >, bo call ad warra will b xrcisd, so If W C <, call ad warra will b abadod ad will a a ull alu. W coclud a warra ad call prics ar prfcly corrlad. I a ffici ark, ad wi o arbirag assupio, fiacial asss offrig prfcly corrlad oupus will a 0;. o ir prics corrlad wi sa coffici a ay i [ ] W C 7 As w subsiuc by B forula, warra alu ca b wri as W -r [ N d - N ] d 8 5

4 Cosidr a corsio raioλ 6, DAB warra pricig forula as prsd Galai ad cllr is W λ W : warra pric a i : udrlyig ass pric a i : warra srik pric w N rτ d N d r: risklss irs ra obaid by irpolaig ra for wo bods wos auriis sraddl warra xpiraio da. τ : warra i o xpiraio d l d d σ w σ r τ σ τ τ : ubr of issud warras : ubr of udrlyig sars w: warra pric a issu da λ : Corsio raio, i.. for ac warra xrcisd,λ w udrlyig sars ar crad. 9. B odl: Bsoussa, Crouy ad Galai approac is approac assus a udrlyig ass pric rflcs diluio ffc. curr saroldrs aicipa ffc of a ual diluio a issu aouc. A a da, r would b a jup i udrlyig ass pric. a s wy; Bsoussa, Crouy ad Galai cosidr a r is o d o ak odificaios o B odl w i cos o aluaig warras bcaus diluio ffc is dircly icludd io udrlyig ass pric. warra orical pric will b W : warra pric a i : udrlyig ass pric a i : warra srik pric W.Nd. -rτ.nd 0 r: risklss irs ra obaid by irpolaig ra for wo -bills wos auriis sraddl warra xpiraio da. 6 Idd, wr a warra oldr dcids o xrcis, issuig copay will b rquird o craλ w sars for i. 7 Bsoussa, Crouy ad Galai. Black ad cols Approxiaios of Warra Prics. Adacs i Fuurs ad Opios Rsarc

5 τ : warra i o xpiraio d l σ r τ σ τ d d σ τ N.: cuulai oral disribuio II- odifid quar Roo Cosa Elasiciy Variac odl A xrly pracical cosquc of udrlyig ass pric logoraliy assupio is a isorical ariac ca b usd o prdic fuur olailiy ad us o alua opios. Uforualy, pirical ad pro a ir isorical or iplid olailiy ca gi a fuur ariac sia wic is uiqu ad cosa or i. Idd, r ar fw cacs a isorical ariac calculad or fi las yars is qual o isorical ariac calculad or las six os. r is cosidrabl idc i liraur idicaig a sock rurs ar roscdasic, wi a probabiliy disribuio sowig a gai skwss. Black wris I a blid for log i a sock rurs ar rlad o olailiy cags. W socks go up, olailiis s o go dow; ad w socks go dow, olailiis s o go up. Cox 975 proposd a call aluaio odl wic assud a olailiy is rlad, by a gai ad cosa rlaio, o ass pric. diffusio procss caracrizig Cox odl aks for d µ : xpcd ass rur ra δ θ θ µ d δ dz : isaaous sadard diaio of rur ra ass, wiδ adθ big cosas dz : a Wir sadard procss lasiciy ariac s wi rspc o sock pric is θ δ σ s θ θ σ δ I is asily s a odl CEV will b quial o B odl wθ ad a olailiy is a dcrasig fucio of, as rpord by Black 975, ifθ <. Udr diffrial quaio ad s as assupios i B frawork 9, Cox 975 drid quilibriu pric forula of Europa call opio 0 forθ <. W ca clarly s a lasiciy s is cosa 8 Black, F.,976. udis of sock pric olailiy cags. Procdigs of 976 Migs of Arica aisical Associaio, pp Aloug ariac is o logr saioary, accordig o quaio, i s sill a driisic cosa fucio of ass pric. Cox odl is a o sa ariabl odl, wic is oc agai ass pric. Udr s circusacs, i s possibl o sablis a prfcly cord posiio usig call ad is udrlyig ass. W applyig o arbirag assupio, i bcos possibl o obai call pricig forula. Moror, is forula ca b applid risk prfrcs. Idd, i a ural risk cooy, idiiduals rquir o risk priu. All asss a a xpcd rur ra qual o risk fr irs ra. 0 Cox opio pricig forula is dlopd i appdix A. 4

6 ad gai. a s wy Cox odl is of calld Cosa Elasiciy Variac CEV odl. CEV Europa call pric dails ar prsd i appdix. ral pirical sudis, Eaul ad Mac B 98, Bas 995 ad Jos 003, ad sow a CEV odl corrcs a sysaic bias of B odl wic ds o orpric i--oy calls ad udrpric ou-of- oy os. Howr, w olailiy is irsly rlad o ass pric, ig low ll ass prics us a low ig olailiis, so i--oy ou-of- oy calls CEV prics will b lowr igr a B os. As warra lif r is grally igr a o yar, w sould us a aluaio odl a allows ariac o ary roug i. For a raso, usig CEV odl o pricig warras sould b or appropria a us of B odl. o calcula CEV warra pric, w iroducd ai odificaios o gral forula s up by Cox. Firs, w fixθ. is coic will cosidrably siplify Cox forula. Mor, sral sudis, Bckrs 980, Laurbac ad culz 990, Hausr ad Laurbac 997 3, proid couragig sias of warras ad opios prics w kpigθ. CEV odl will b rasford o wa s calld quar Roo Cosa Elasiciy Variac RCEV odl. forula dails i appdix. scod odificaio ad o CEV odl is a w igrad diluio ffc as w did for DAB odl. odifid RCEV odl is W λ q4 ad q0 ar dlopd i appdix B. w N[ q4] - -rτ N[ q0] 3 IV- Daabas daabas usd for is sudy cosiss o: - Warras prics: d-of- day prics for warras goiad o Jakara ock Excag JX durig priod goig fro 0/0/00 uil /3/003. W cou 85 call warras. ss prics wr dircly obaid fro JX couicaio ad public rlaiosip dpar. - Iforaio bullis: also obaid fro JX couicaio ad public rlaiosip dpar. s bullis idica, for ac warra, issu da, xrcis pric, xpiry da. corsio raio is qual o for all warras goiad o JX durig is priod. - Udrlyig sars prics: d-of- day sars prics. oal ubr of obsrd prics is Howr, ir daabas, obaid fro JX couicaio ad. Bckrs 980. Cosa Elasiciy of Variac odl ad is iplicaios for opio pricig. Joural of Fiac, pp B. Laurbac ad P. culz. Pricig Warras: A Epirical udy of Black cols Modl ad Is Alrais. Joural of Fiac 990. is papr, cosidrd as firs o a dal, i a xausi ar, issu of warra pricig usig DAB ad RCEV odls. 3. Hausr ad B. Laurbac 997. Rlai Prforac of Fi Alrai Warra Pricig Modls. Fiacial Aalyss Joural. is papr was irsd i coparig prforac of fi pricig warras odls radd a Arica ark for priod 97 o 980. odls usd wr B odl, DAB odl, Logsaff odl for xdd warras, RCEV odl ad Rick fr a odl. lar ga lows siaio rror 3,6% aad of RCEV odl 3,67% rakd i d posiio. 5

7 public rlaiosip dpar, or a dowloadd fro wb si proids iforaio abou diidd disribuio. - Risklss Irs Ras: irs ras for goral obligaios issud durig 00, 00 ad 003. iforaio wr dowloadd fro Idosia Cral Bak wb si: ral filrs wr usd i ordr o rac a sapl a ca b usd far a rliabl aalysis. W liia warras wi icopl iforaio bulli., w liia warras iappropria for pricig odls usd roug our papr, i.. warras wi a s up xrcis pric followig a sock spli opraio. W also apply a liquidiy filr. W rai warras wi a liquidiy ll igr a 0%. warras prics a do o ac o arbirag assupio ar xcludd fro sapl. Fially, w kp warras a a, a las, 00 usabl obsraios. suarizd saisics for fial sapl ar prsd a abl. disribuio of obsraio accordig o criria of oyss 4 ad i o xpiraio ar prsd a abl. I s od a fial sapl is doiad by dp ou ad ou-of- oy warras, wic accou for 83.6 of oal obsraios. Rgardig i o xpiraio, 73.06% of obsraios sow a i o xpiraio byod a yar rsus 3.7% wi a i o xpiraio lss a a yar. V- Diluio Adjusd Black & cols s. Black & cols W calcula siaio rror as absolu diaio of ark warra pric fro orical pric, rpord o warra pric. is asur illusras xacss wi wic ac odl fis obsrd prics. arag siaio rror AEE, for wol obsraios, ca b prsd as follows AEE j W j od l W W j ark j ark 4 Wr is ubr of obsrd warra prics across all warras ad days of sapl. AEE, calculad roug 5460 obsraios, ar 4.45% ad.76% rspcily for B ad DAB odls. Applyig z-s, iiu sigifica diffrc s, w fid a AEE ar sigificaly diffr fro ac or a ll α%. Now, l calcula AEE by warra. abl 3 proids rsuls. I also proids facors a ay xplai prforac of ac of odls. Excp for warra MPO-W, DAB odl proids br sias a B odl. is suprioriy ca b xplaid by fac a DAB odl succdd o corrcly ra diluio ffc by akig a xplici adjus of B odl, ia iroducio of diluio facor. A iplici adjus for diluio ffc roug udrlyig ass pric, as λ assud by Bsoussa, Crouy ad Galai 995, dos s ry suiabl for Idosia ark. W clarly s a igs diffrc, bw siaio rrors of ac odl, was rτ 4 Moyss is calculad as wr rτ is udrlyig ass pric, warra xrcis pric, r risklss irs ra ad τ i o xpiraio. 6

8 rcordd for warra a as igs diluio facor warra PLA-W wi a diluio facor of 74%. O or sid, salls diffrc was rcordd for warra wi salls diluio facor warra IG-W wi a diluio facor of 9%. W also classify obsraio io 9 subsapls usig warra oyss ad i o xpiraio criria. W ad sa calculus ad sa copariso bw odls for ac of s cagoris. Rsuls ar suarizd i abl 4. W clarly s a DAB arag siaio rror g closr o B o as w o fro ou-of- oy warras subsapls o i--oy os. Moror, w applyig z-s, w fid a DAB siaio rrors ar sigificaly diffr fro B os sigificac a ll α5% oly for dp ou-of- oy subsapls. uc a rsul ca b xplaid by fac a s warras ar prcisly s wo a igs diluio facor s abl. Aor fidig is a, bo DAB ad B odls prfor wors for ou ad dp-ou-of oy warras, xacly as do classic B opio aluaio forula. is is du o fac a s odls sill us udrlyig ass pric logoraliy assupio wi a cosa olailiy. uc a assupio is ardly suiabl for a log lif r ass as warra is. VI- olailiy Baiour As sow i abl 4, AEE aris, draaically, as w o cross subsapls. is ay b rsul of srog rlaiosip bw local olailiy ra ad ass pric. o cofir is assupio, w ca rgrss iplid olailiy o oyss facor. xisc of a sigifica gai rlaio bw ass pric ad is olailiy will, cssarily, riforc posiio of odifid RCEV odl rsus B ad DAB os. o a a or clar ida o is issu, w apply followig rgrssio: ID rτ α α 0 ε rτ 5 Wr ID is warra iplid sadard diaio calculad a da. oyss facor. r rτ τ is Firs of all, w do rgrssio for wol pal daa, across all warras. W apply r diffr pal rgrssio odls: Fixd Effcs Modl, Wigd Fixd Effcs Modl ad Rado Effcs Modl. rgrssio rsuls ar suarizd a abl 5. 7

9 abl - DECRIPIVE AIIC Warra Prsc i sapl Obs. Diluio Expiraio rik Pric Ma «Moyss» Ma ID Ma i o xpiraio Lif r AIM-W 0/08/0 /30/ % 07/9/ yars ANA-W 0/8/0 /30/ % 0/8/ yars BCAP-W 06//0 /30/ % 06/07/ yars CNO-W 04/6/0 /3/ % // yars GEMA-W 08//0 /30/ % 08// yars IDR-W 03//0 /30/ % 03// yars IG-W /6/0 /3/ % /5/ yars AR-W 07/0/0 /30/ % 07/9/ yars REN-W 06/8/0 /30/ % 06/8/ yars MEA-W 07/8/0 07/0/ % 07/7/ yar PLA-W 03/6/0 /30/ % 03/5/ yars MPO-W 0/08/0 /3/ % 0/07/ yars WAPO-W 0/0/0 /30/ % 06// yars ubr of obsraios of ac warra i sapl. iplid sadard diaio. 8

10 abl - Disribuio of Obsraios accordig o oyss ad i o xpiraio criria disribuio of 5460 obsraios will b basd o criria: oyss ad i o xpiraio. Figurs i parss ar prcags of obsraios of ac subsapl copard o wol sapl. Dp i--oy warras ar os wi a oyss facor>0.5. I--oy warras a a oyss facor ] 0.05;0.5]. A--oy warras a a oyss [ 0.05;0.05]. Ou-of--oy warras a a 0.5; Dp ou-of- oy warras a a oyss facor<-0.5. oyss facor [ [ Dp I oy I oy A oy Ou of oy Dp Ou of oy yar > yar %.89% % 7.47% % 3.77% % 7.6% % 35.55% abl 3 - Copariso of Esiaio Errors by Warra A suary abl wic ais o copar prforacs proidd, rspcily, by B odl ad DAB odl. is copariso will b ad by usig arag siaio rror of ac odl, by calculaig diffrc bw arag rrors. Warra Obsraios Ma Moyss Diluio Error B Error DAB Prforac B DAB AIM-W % 4.73% 4.8% 0.46% ANA-W % 3.56% 3.33% 0.3% BCAP-W % 7.5% 5.7%.88% CNO-W %.40% 8.34% 3.06% GEMA-W % 3%.30% 0.70% IDR-W % 5.07% 4.9% 0.6% IG-W %.93%.8% 0.% AR-W % 0.80% 8.8%.98% REN-W % 5.47% 4.94% 0.53% MEA-W % 4.8%.88%.40% 9

11 PLA-W %.49%.7% 8.77% MPO-W % 3.74% 3.89% -0.5% WAPO-W % 5.33% 4.4% 0.9% abl 4 - Esiaio Errors classifid by Moyss ad i o xpiraio A suary abl wic ais o copar prforacs proidd, rspcily, by B odl ad DAB odl. is copariso will b ad by usig arag siaio rror of ac odl, by calculaig diffrc bw arag rrors. W apply z-s for ac diffrc. Dp i-oy warras ar os wi a oyss facor>0.5. I--oy warras a a oyss facor ] 0.05;0.5]. A--oy warras a a oyss [ 0.05;0.05]. Ou-of--oy warras a a 0.5; Dp ou-of- oy warras a a oyss facor<-0.5. oyss facor [ [ B DAB Diffrc Probabiliy 7 Dp I yar 4.44% 4.87% % 0.56 I yar 5.3% 5.6% 0.05 % I - yar 5.58% 35.76% -0.8 % A - yar 6.86% 5.7%.5 % 0.76 A yar 0.07% 9.9% 0.5 % Ou yar.7% 0.7%.0 % 0.3 Ou yar 4.55%.99%.56 % 0.93 Dp ou - yar 6.% 4.3%.07 % Dp ou yar.% 8.3% 3.89 % ull yposis probabiliis for z-s. 0

12 abl 5 - Coparai abl of Rgrssio Modls Coparai abl aid o idify siaio rsuls gi by ac rgrssio odl. criria usd i is copariso ar: calculad sud, rgrssio liariy coffici ad su of rsidual squars. W copar Fixd Effcs Modl FEM, wigd FEM ad Rado Effcs Modl REM. FEM wigd FEM REM α REUL α R² 6.69% 78.% 70.8% R² adjusd 6.6% 78.5% 70.8% CR All rgrssio odls sow a srog irs rlaiosip bw iplid olailiy ad oyss. rα siaios ar sigificaly gai a ll α%. bs rgrssio odl ss o b Wigd Fixd Effcs Modl proidig os sigificaα siaio wi igs sud, bs liariy ad adjusd liariy cofficis ad lows su of squard rsiduals. uc a rsul suggss a odifid RCEV odl would b or appropria o fi warras prics a DAB ad B odls. Accordig o socasic olailiy odls, Wiggis 987 8, Hull ad Wi 987 9, Hso 993 0, olailiy aris roug i wiou big likd o udrlyig ass pric by a driisic rlaio. I or rs, s odls allow arbirary corrlaio bw olailiy ad ass pric. uc olailiy baiour, as sow by Hso 993, producs a probabiliy disribuio of ass rurs wi ig kurosis. us, rlaio bw iplid olailiy ad udrlyig ass pric ca ary as warra is i or ou-of- oy. o rify wr olailiy is purly socasic or o, w rpa rgrssio, usig wigd Fixd Effc Modl, oly for i--oy warras. i--oy obsraios rprs % of wol pal 665 obsraios aog a oal of 5460 warras prics. rgrssio rsuls ar suarizd i abl 6. 8 Wiggis, J.B., 987. Opio alus udr socasic olailiy: ory ad pirical sias. Joural of Fiacial Ecooics 9, Hull, J. ad A. Wi pricig of opios o asss wi socasic olailiis. Joural of Fiac, pp Hso,.L., 993. A closd-for soluio for opios wi socasic olailiy wi applicaios o bod ad currcy opios. Riw of Fiacial udis 6,

13 abl 6 - Rgrssio rsuls for i--oy obsraios Rgrssio wi wigd FEM 665 Obsraios Warra α 0i α AIM-W 0.44 ANA-W BCAP-W IDR-W IG-W 0.69 AR-W MEA-W PLA-W 0.45 MPO-W α R 93.8% R adjusd 93.7% Prob0.000 Aloug irs rlaio is or prooucd for ou-of- oy warras, rgrssio as sow a suc rlaio is aiaid for i--oy os wi aα siaio sigificaly gai a llα %., w ca dduc a olailiy rais irsly rlad o udrlyig ass pric, as assud by RCEV odl, ad o purly socasic. VII- Modifid RCEV s. DAB AEE, calculad roug 5460 obsraios, ar 8.95% ad.76% rspcily for odifid RCEV ad DAB odls. uc a rsul was prdicabl as odifid RCEV is adapd o warra spcificiis, i.. diluio ffc ad warra log lif r, by icorporaig diluio facor i origial pricig forula ad usig a ariabl olailiy. z-s sows a AEE ar sigificaly diffr fro ac ors a llα %. abl 7 copars AEE calculad for ac warra usig odls. abl 7 - Copariso of Esiaio Errors by Warra A suary abl wic ais o copar prforacs proidd, rspcily, by DAB odl ad RCEV odl. is copariso will b ad by usig arag siaio rror of ac odl, by calculaig diffrc bw arag rrors. Warra Arag Moyss Error DAB Error RCEV Diffrc AIM-W % 8.0% 6.075% ANA-W % 0.47%.86% BCAP-W % 5.7% 0.0%

14 CNO-W % 4.3% 4.% GEMA-W % 8.85% 3.45% IDR-W % 3.49%.4% IG-W %.9% 0.53% AR-W % 9.8% 9.0% REN-W % 0.7% 4.77% MEA-W % 5.3% 7.566% PLA-W %.4% 0.58% MPO-W % 9.7% 4.7% WAPO-W % 8.59% 5.83% Now l calcula AEE across subsapls ad upo oyss ad i o xpiraio criria. Rsuls ar suarizd a abl 8. W fid a odifid RCEV siaio rrors ar sigificaly diffr, a ll α 5%, fro DAB rrors, for ou-of- oy subsapls. O or ad, w ca s a odls siaio rrors ar o sigificaly diffr fro ac ors, for a ad i--oy warras. odls proid a alos siilar prforac for s cagoris of warras. uc a rsul ca b xplaid by odifid RCEV assupio for fixd θ across all oyss cagoris ad for all warras. is assupio ay b suiabl for ou-of- oy warras, wr odifid RCEV odl ad prford is bs prforac, bu i ss a i dos work so good for a ad i-- oy subsapls. For s cagoris of warras, r is sill a irs rlaio bw olailiy ad ass pric as sow i prious scio bu corrlaio bw s ariabls sould b or odra a for ou-of- oy warras. Isors d o b ry frigd fac ay dowrd i ark ad o dosra a ry odra opiis w o bcos bullis. uc baiour is qui rarkabl rspos o priods of fiacial criss. olailiy dcrass oooically as ass pric gos upward bu ra of dcras sould b lss a ra of olailiy icras w ass pric gos dowward. usθ us ary as w o fro ou-of- oy o i-oy subsapls. 3

15 abl 8 - Esiaio Errors classifid by Moyss ad i o xpiraio A suary abl wic ais o copar prforacs proidd, rspcily, by DAB odl ad RCEV odl. is copariso will b ad by usig arag siaio rror of ac odl, by calculaig diffrc bw arag rrors. W apply z-s for ac diffrc. Dp i-oy warras ar os wi a oyss facor>0.5. I--oy warras a a oyss facor ] 0.05;0.5]. A--oy warras a a oyss [ 0.05;0.05]. Ou-of--oy warras a a 0.5; Dp ou-of- oy warras a a oyss facor<-0.5. oyss facor [ [ DAB RCEV Diffrc Probabiliy Dp I yar 4.87 % 3.98 % 0.89 % 0.38 I yar 5.6 % 4.36 % 0.9 % 0.4 I - yar % 4.76 % % 0.6 A - yar 9.9 % 9.0 % 0.9 % 0.40 A yar 5.7 % 5.54 % 0.7 % Ou yar 0.7 % 8.6 %.45 % Ou yar.99 %.03 %.96 % Dp ou - yar 8.3 % 0.89 % 7.4 % 0 Dp ou yar 4.3 % 9.7 % 4.4 % 0 Coclusio As xpcd, odifid RCEV odl ld o lows AEE for wol pal, for ac warra ad for diffr oyss subsapls. is rsul ca b xplaid by fac a odifid RCEV is oly o of 3 odls usd i our pirical sudy, wic ris o xploi ajor warra spcificiis. Firs, odifid RCEV is xplicily adjusd for diluio ffc wic sould allow offsig gai ipac a diluio could a o qualiy of siaio. Also, ulik B ad DAB odls, odifid RCEV is or adapd o warra log lif r, proposig a diffusio procss wi ariabl udrlyig ass pric olailiy. is assupio ss o b or cosis a a of cosa olailiy usd by B ad DAB odls. Howr, odifid RCEV odl ay b subjc o so criicis. W ca rfr, i paricular, is us of a cosa lasiciy ariac across all udrlyig asss. is lasiciy ca ary for sa udrlyig ass dpdig if spo pric is goig upward or dowward. o rdy is dficicy, o ca calcula iplid lasiciy usig obsrd warras prics ad calibra odifid RCEV odl so i ca b adapd for udrlyig ass spcificiis. ull yposis probabiliis for z-s. 4

16 Bibliograpy Bas, D sig Opio Pricig Modls. Workig papr. Naioal Burau of Ecoois Rsarc. Bas, D Jups ad socasic olailiy: Excag ra procsss iplici i Dusc Mark opios. Riw of Fiacial udis, pp Bckrs, cosa lasiciy of ariac odl ad is iplicaios for opio pricig. Joural of Fiac, pp Black, F ad M. cols pricig of opios ad corpora liabiliis. Joural of Poliical Ecooy, pp Bsoussa, A., M. Grouy ad D. Galai Black-cols approxiaios of warra prics. Adacs i Fuurs ad Opio Rsarc, pp. -4. Crisoffrs, P. Jacobs,. ad Mioui, Modls for &P500 Dyaics: Eidc fro Ralizd Volailiy, Daily Rurs, ad Opio Prics. Workig papr. Faculy of aag, McGill Uirsiy. Crouy, M. ad D. Galai. 99. Warra aluaio ad quiy olailiy. Adacs i Fuurs ad Opios Rsarc, pp Cox, J Nos o opio pricig : Cosa lasiciy of ariac diffusios. Workig Papr. aford Uirsiy. Cox, J cosa lasiciy of ariac opio pricig odl. Joural of Porfolio Moag, pp Galai, D. ad M. cllr 978. Pricig warras ad alu of fir. Joural of Fiac, pp Hausr,. ad B. Laurbac ss of warra pricig odls: radig profis prspci. Joural of Driais, pp Hausr,. ad B. Laurbac rlai prforac of fi alrai warra pricig odls. Fiacial Aalyss Joural, pp Hsu, L. L, C. ad Li, Cosa lasiciy of ariac CEV opio pricig odl: igraio ad daild driaio. Maaics ad copurs siulaio. Hull, J. ad A. Wi pricig of opios o asss wi socasic olailiis. Joural of Fiac, pp Jacquir, E., Polso, N.G ad Rossi, P.E. 00. Baysia aalysis of socasic olailiy wi lrag ffc ad fa ails. Joural of Ecoorics, forcoig. Jos, C dyaics of socasic olailiy: idc fro udrlyig ad opios arks. Joural of coorics, pp Laa, H. ad R. Rdla 976. adard diaios of sock pric raios iplid i opio prics. Joural of Fiac, pp

17 Laurba, B. ad P. culz 990. Pricig warras: a pirical sudy of Black-cols odl ad is alais. Joural of Fiac, pp L, C. ad. Wu. 00. cosa lasiciy of ariac odls: Nw idc fro &P 500 idx opios. Workig papr. Rugrs Busiss cool. Logsaff, F Pricig opios wi xdibl auriis: Aalysis ad applicaios. Joural of Fiac, pp Macb, J. ad L. Mrill 979. A pirical xaiaio of Black-cols call opio pricig odl. Joural of fiac, pp Macb, J. ad L. Mrill 980. ss of Black-cols ad Cox call opio aluaio odls. Joural of Fiac, pp Pa, J., 00. jup-risk pria iplici i opios: idc fro a igrad i-sris sudy. Joural of Fiacial Ecoorics 63, pp Rick, P Opio: ory, ragy ad Applicaios. Gliw, IL: co, Forsa ad Copay. culz, G. ad. raua 994. Robusss of opio-lik warra aluaio. Joural of Bakig ad Fiac, pp co, L Opio pricig w ariac cags radoly: ory, Esiaio ad Applicaio. Joural of Fiacial ad Quaiai Aalysis, pp kiadopoulos, G., 000. Volailiy sil cosis opio odls: a sury. Iraioal Joural of orical ad Applid Fiac 4, urs M Coiuous-i Mods i Fiac: A Riw ad a Assss. Joural of Fiac, Vol. 55, No. 4,, pp uis, J. ad Corl, A 007. Log ory opios: LM idc ad siulaio. Rsarc i Iraioal Busiss ad Fiac, pp opkis, R.G., 00. Iplid olailiy surfacs: ucorig rgulariis for opios o fiacial fuurs. Europa Joural of Fiac 7, Waly, R. 98. Valuaio of Arica call opios o diidd payig socks: Epirical ss. Joural of Fiacial Ecooics, pp Yag, H Esiaio i cosa lasiciy of ariac odl.workig papr. Uirsiy of Hog og. Yu, C. ad ig, C 00. Warra pricig: ocasic olailiy s. Black-cols. Pacific Basi Fiac Joural, pp

18 7 Appdix A - Cox Opio Pricig Forula Cox opio pricig forula udr assupio of a cosa lasiciy ariac is dfid i followig ar:,,,, 0 0 G g G g C r r r φ τ α φ τ φ τ α φ λ φ λ φ λ λ C is a Europa call pric wi a srik pric ad a i o xpiraio τ ; θ φ ; φτ φ δ λ r r ; Γ α 0 d : gaa fucio; x x g x Γ, : dsiy gaa fucio; α 0,, dx x g a G : coplary gaa disribuio fucio. Appdix B - RCEV opio pricig forula as proposd by Bckrs 980 For 0 or 4, { } 5 0, ] 3 [ ] [ 3 p p y z p p, 3 3 y y y, y y p, 4 r W N M r y σ W N M y z.

19 Pricig Warras Modls: A Epirical udy of Idosia Mark Auors: Zouair BEN HELIFA ; Waji ABBAI Lcurr: Waji ABBAI Absrac ai issu durig priods of fiacial crisis is o rsor isor s cofidc ad arac back o fiacial arks. us, warras a b a gra lp o rlauc ay sou Eas Asia fiacial arks jus afr 997 crisis, by couragig isors o fiac rsrucurig of roubld copais. Howr, fw pirical sudis wic wr irsd i is produc, was liid o dlopd arks ad i paricular Arica o. I is papr, w will ry o pasiz warra s spcificiis copard o opio os., w will sk o rlas idal odl rady o proid bs pricig for is produc. copariso will rla o 3 odls: Black ad ols odl B, Diluio Adjusd Black ad cols odl DAB ad fially odifid quar Roo Cosa Elasiciy of Variac RCEV adjusd by diluio ffc. pirical sudy, wic is sprad ou or 3 yars , will rla o a rg ark: Jakara ock Excag JX. y words: Black ad cols, Cosa lasiciy ariac, Diluio, Idosia ark, Moyss, Warra aluaio. JEL Classificaio: G, G3 : Fiacial crisis ad ass pricig Profssor of Fiac, Haus Euds Corcials HEC cool of aag, Carag uisia Assisa lcurr of Fiac, Haus Euds Corcials HEC cool of aag, Carag uisia, él , -ail : abbassi_waji@yaoo.fr

20 Iroducio Warras ar fiacial isrus issud by fiacial isiuios ad ar grally goiad o opios arks. A warra oldr as rig o buy or o sll a spcific udrlyig ass oc racd warra xpiry a a fixd pric calld srik pric. A firs sig, o ca cofus warra wi coioal opio. Bo sar sa pricipls of udrlyig ass, srik pric ad xpiry da. Moror, sral paprs usd o alua warra pric as sa as opio s o usig black ad cols forula. Howr, r ar wo ajor diffrcs bw warras ad coioal opios. Firs, warra lifi is qui larg a opio s o. A warra ca a a i o xpiraio goig up o s yars wras a opio cooly as a i o xpiraio o xcdig fw os. cod, opios ar issud by idiiduals so a ual xrcis will oly lad o udrlyig ass rasfr fro o opraor o aor o. Warras ar issud by firs. Cosquly, ir xrcis will caus issu of addiioal sars ad us a fir quiy diluio. I spi of warras succss, a fw paprs wr iold o ir aluaio. Moror, os of s paprs wr focusd o dlopd arks. For is papr, w ar irsd o pirical alidaio of r aluaio odls for warras issud o a rg ark. W slc Idosia ark 3 for rasos. Firs, bcaus of sou Eas Asia fiacial ark dyaic caracr is dos cas aracig all kids of isors ad spculaors. Also, Idosia warra ark rcordd igs grow ra, i r of sock xcag capializaio, copard o or Idosia fiacial arks, paricularly afr 997 crisis. A os circusacs, warras wr issud o courag risig i Idosia socks. W will ry o copar prforacs of r warra aluaio odls usig daa obsrd o Jakara ock Excag for yars 00, 00 ad 003. odls sd ar Black ad cols odl B, Diluio Adjusd Black ad cols odl DAB ad odifid quar Roo Cosa Elasiciy Variac odl odifid RCEV ispird fro Cox Cosa Elasiciy Variac Modl CEV. is papr will b srucurd as follows. firs scio iroducs B ad DAB odls. scod iroducs CEV odl ad cssary odificaios, i ordr o ak Cox Modl or suiabl for warras aluaio. ird scio iroducs saisical suaris of daa usd roug our pirical sudy. four scio copars B ad DAB aluaio prforacs. fif dals abou udrlyig asss olailiy baiour. six scio copars odifid RCEV ad DAB aluaio prforacs. Fially, w coclud by a brif suary of ai rsuls ad so suggsio for fuur rsarc. I- Valuaig warras wi B ad DAB odls. DAB odl: Galai ad cllr approac Lik a coioal Europa call, warra is rig o buy, a srik pric, a udrlyig ass a a xpiraio da. Howr, warras ar issud by copais so a ir xrcis will lad o a issu of addiioal sars ad, rfor, a diluio of issuig copay quiis. W ry o pasiz rlaio bw issuig fir alu ad warra pric. 3 Warras a appard for firs i o Jakara ock Excag dad July 3, 995 ad wr issud by bak IARA AIA. 4 J. Cox 975. Nos o opio pricig: Cosa Elasiciy Variac Diffusios. Workig Papr, aford Uirsiy. roug is papr, Cox as sablisd a w odl for pricig opios assuig a olailiy, is o logr cosa, bu os irsly o udrlyig ass pric. 5 D. Galai ad M. cllr. Pricig of Warras ad Valu of Fir. Joural of Fiac 978.

21 L cosidr a fir A wi 00% of quiis ad sars. uppos a is fir will b liquidad a da wi a socasic fuur aluv. Eac sar alu, a a da, will V b. Now l cosidr a scod fir B, idical o firs, a issus warras a 0, wi srik pric ad xpiraio da. Eac xrcisd warra allows is oldr o possss a w issuig fir sar. o, corsio raio, odλ, is qual o. If warras ar xrcisd a xpiraio da, fir B alu will b igr a fir A o. V * V sar alu of fir B a i, od, will b qual o * V warra will b xrcisd a xpiraio da, oly if udrlyig sock pric xcds warra srik pric. I or ad, V, so > * V * > * > Now, l ry o ifr warra pric usig B Europa call pricig forula. Cosidr a Europa call wi fir A sar as a udrlyig ass. A xpiraio da, call alu will * b qual o Max 0 ; -. A sa da, warra alu will b qual o Max 0 ; -. * If >, bo call ad warra will b xrcisd, so If W C <, call ad warra will b abadod ad will a a ull alu. W coclud a warra ad call prics ar prfcly corrlad. I a ffici ark, ad wi o arbirag assupio, fiacial asss offrig prfcly corrlad oupus will a 0;. o ir prics corrlad wi sa coffici a ay i [ ] W C 7 As w subsiuc by B forula, warra alu ca b wri as W -r [ N d - N ] d 8 5

22 Cosidr a corsio raioλ 6, DAB warra pricig forula as prsd Galai ad cllr is W λ W : warra pric a i : udrlyig ass pric a i : warra srik pric w N rτ d N d r: risklss irs ra obaid by irpolaig ra for wo bods wos auriis sraddl warra xpiraio da. τ : warra i o xpiraio d l d d σ w σ r τ σ τ τ : ubr of issud warras : ubr of udrlyig sars w: warra pric a issu da λ : Corsio raio, i.. for ac warra xrcisd,λ w udrlyig sars ar crad. 9. B odl: Bsoussa, Crouy ad Galai approac is approac assus a udrlyig ass pric rflcs diluio ffc. curr saroldrs aicipa ffc of a ual diluio a issu aouc. A a da, r would b a jup i udrlyig ass pric. a s wy; Bsoussa, Crouy ad Galai cosidr a r is o d o ak odificaios o B odl w i cos o aluaig warras bcaus diluio ffc is dircly icludd io udrlyig ass pric. warra orical pric will b W : warra pric a i : udrlyig ass pric a i : warra srik pric W.Nd. -rτ.nd 0 r: risklss irs ra obaid by irpolaig ra for wo -bills wos auriis sraddl warra xpiraio da. 6 Idd, wr a warra oldr dcids o xrcis, issuig copay will b rquird o craλ w sars for i. 7 Bsoussa, Crouy ad Galai. Black ad cols Approxiaios of Warra Prics. Adacs i Fuurs ad Opios Rsarc

23 τ : warra i o xpiraio d l σ r τ σ τ d d σ τ N.: cuulai oral disribuio II- odifid quar Roo Cosa Elasiciy Variac odl A xrly pracical cosquc of udrlyig ass pric logoraliy assupio is a isorical ariac ca b usd o prdic fuur olailiy ad us o alua opios. Uforualy, pirical ad pro a ir isorical or iplid olailiy ca gi a fuur ariac sia wic is uiqu ad cosa or i. Idd, r ar fw cacs a isorical ariac calculad or fi las yars is qual o isorical ariac calculad or las six os. r is cosidrabl idc i liraur idicaig a sock rurs ar roscdasic, wi a probabiliy disribuio sowig a gai skwss. Black wris I a blid for log i a sock rurs ar rlad o olailiy cags. W socks go up, olailiis s o go dow; ad w socks go dow, olailiis s o go up. Cox 975 proposd a call aluaio odl wic assud a olailiy is rlad, by a gai ad cosa rlaio, o ass pric. diffusio procss caracrizig Cox odl aks for d µ : xpcd ass rur ra δ θ θ µ d δ dz : isaaous sadard diaio of rur ra ass, wiδ adθ big cosas dz : a Wir sadard procss lasiciy ariac s wi rspc o sock pric is θ δ σ s θ θ σ δ I is asily s a odl CEV will b quial o B odl wθ ad a olailiy is a dcrasig fucio of, as rpord by Black 975, ifθ <. Udr diffrial quaio ad s as assupios i B frawork 9, Cox 975 drid quilibriu pric forula of Europa call opio 0 forθ <. W ca clarly s a lasiciy s is cosa 8 Black, F.,976. udis of sock pric olailiy cags. Procdigs of 976 Migs of Arica aisical Associaio, pp Aloug ariac is o logr saioary, accordig o quaio, i s sill a driisic cosa fucio of ass pric. Cox odl is a o sa ariabl odl, wic is oc agai ass pric. Udr s circusacs, i s possibl o sablis a prfcly cord posiio usig call ad is udrlyig ass. W applyig o arbirag assupio, i bcos possibl o obai call pricig forula. Moror, is forula ca b applid risk prfrcs. Idd, i a ural risk cooy, idiiduals rquir o risk priu. All asss a a xpcd rur ra qual o risk fr irs ra. 0 Cox opio pricig forula is dlopd i appdix A. 4

24 ad gai. a s wy Cox odl is of calld Cosa Elasiciy Variac CEV odl. CEV Europa call pric dails ar prsd i appdix. ral pirical sudis, Eaul ad Mac B 98, Bas 995 ad Jos 003, ad sow a CEV odl corrcs a sysaic bias of B odl wic ds o orpric i--oy calls ad udrpric ou-of- oy os. Howr, w olailiy is irsly rlad o ass pric, ig low ll ass prics us a low ig olailiis, so i--oy ou-of- oy calls CEV prics will b lowr igr a B os. As warra lif r is grally igr a o yar, w sould us a aluaio odl a allows ariac o ary roug i. For a raso, usig CEV odl o pricig warras sould b or appropria a us of B odl. o calcula CEV warra pric, w iroducd ai odificaios o gral forula s up by Cox. Firs, w fixθ. is coic will cosidrably siplify Cox forula. Mor, sral sudis, Bckrs 980, Laurbac ad culz 990, Hausr ad Laurbac 997 3, proid couragig sias of warras ad opios prics w kpigθ. CEV odl will b rasford o wa s calld quar Roo Cosa Elasiciy Variac RCEV odl. forula dails i appdix. scod odificaio ad o CEV odl is a w igrad diluio ffc as w did for DAB odl. odifid RCEV odl is W λ q4 ad q0 ar dlopd i appdix B. w N[ q4] - -rτ N[ q0] 3 IV- Daabas daabas usd for is sudy cosiss o: - Warras prics: d-of- day prics for warras goiad o Jakara ock Excag JX durig priod goig fro 0/0/00 uil /3/003. W cou 85 call warras. ss prics wr dircly obaid fro JX couicaio ad public rlaiosip dpar. - Iforaio bullis: also obaid fro JX couicaio ad public rlaiosip dpar. s bullis idica, for ac warra, issu da, xrcis pric, xpiry da. corsio raio is qual o for all warras goiad o JX durig is priod. - Udrlyig sars prics: d-of- day sars prics. oal ubr of obsrd prics is Howr, ir daabas, obaid fro JX couicaio ad. Bckrs 980. Cosa Elasiciy of Variac odl ad is iplicaios for opio pricig. Joural of Fiac, pp B. Laurbac ad P. culz. Pricig Warras: A Epirical udy of Black cols Modl ad Is Alrais. Joural of Fiac 990. is papr, cosidrd as firs o a dal, i a xausi ar, issu of warra pricig usig DAB ad RCEV odls. 3. Hausr ad B. Laurbac 997. Rlai Prforac of Fi Alrai Warra Pricig Modls. Fiacial Aalyss Joural. is papr was irsd i coparig prforac of fi pricig warras odls radd a Arica ark for priod 97 o 980. odls usd wr B odl, DAB odl, Logsaff odl for xdd warras, RCEV odl ad Rick fr a odl. lar ga lows siaio rror 3,6% aad of RCEV odl 3,67% rakd i d posiio. 5

25 public rlaiosip dpar, or a dowloadd fro wb si proids iforaio abou diidd disribuio. - Risklss Irs Ras: irs ras for goral obligaios issud durig 00, 00 ad 003. iforaio wr dowloadd fro Idosia Cral Bak wb si: ral filrs wr usd i ordr o rac a sapl a ca b usd far a rliabl aalysis. W liia warras wi icopl iforaio bulli., w liia warras iappropria for pricig odls usd roug our papr, i.. warras wi a s up xrcis pric followig a sock spli opraio. W also apply a liquidiy filr. W rai warras wi a liquidiy ll igr a 0%. warras prics a do o ac o arbirag assupio ar xcludd fro sapl. Fially, w kp warras a a, a las, 00 usabl obsraios. suarizd saisics for fial sapl ar prsd a abl. disribuio of obsraio accordig o criria of oyss 4 ad i o xpiraio ar prsd a abl. I s od a fial sapl is doiad by dp ou ad ou-of- oy warras, wic accou for 83.6 of oal obsraios. Rgardig i o xpiraio, 73.06% of obsraios sow a i o xpiraio byod a yar rsus 3.7% wi a i o xpiraio lss a a yar. V- Diluio Adjusd Black & cols s. Black & cols W calcula siaio rror as absolu diaio of ark warra pric fro orical pric, rpord o warra pric. is asur illusras xacss wi wic ac odl fis obsrd prics. arag siaio rror AEE, for wol obsraios, ca b prsd as follows AEE j W j od l W W j ark j ark 4 Wr is ubr of obsrd warra prics across all warras ad days of sapl. AEE, calculad roug 5460 obsraios, ar 4.45% ad.76% rspcily for B ad DAB odls. Applyig z-s, iiu sigifica diffrc s, w fid a AEE ar sigificaly diffr fro ac or a ll α%. Now, l calcula AEE by warra. abl 3 proids rsuls. I also proids facors a ay xplai prforac of ac of odls. Excp for warra MPO-W, DAB odl proids br sias a B odl. is suprioriy ca b xplaid by fac a DAB odl succdd o corrcly ra diluio ffc by akig a xplici adjus of B odl, ia iroducio of diluio facor. A iplici adjus for diluio ffc roug udrlyig ass pric, as λ assud by Bsoussa, Crouy ad Galai 995, dos s ry suiabl for Idosia ark. W clarly s a igs diffrc, bw siaio rrors of ac odl, was rτ 4 Moyss is calculad as wr rτ is udrlyig ass pric, warra xrcis pric, r risklss irs ra ad τ i o xpiraio. 6

26 rcordd for warra a as igs diluio facor warra PLA-W wi a diluio facor of 74%. O or sid, salls diffrc was rcordd for warra wi salls diluio facor warra IG-W wi a diluio facor of 9%. W also classify obsraio io 9 subsapls usig warra oyss ad i o xpiraio criria. W ad sa calculus ad sa copariso bw odls for ac of s cagoris. Rsuls ar suarizd i abl 4. W clarly s a DAB arag siaio rror g closr o B o as w o fro ou-of- oy warras subsapls o i--oy os. Moror, w applyig z-s, w fid a DAB siaio rrors ar sigificaly diffr fro B os sigificac a ll α5% oly for dp ou-of- oy subsapls. uc a rsul ca b xplaid by fac a s warras ar prcisly s wo a igs diluio facor s abl. Aor fidig is a, bo DAB ad B odls prfor wors for ou ad dp-ou-of oy warras, xacly as do classic B opio aluaio forula. is is du o fac a s odls sill us udrlyig ass pric logoraliy assupio wi a cosa olailiy. uc a assupio is ardly suiabl for a log lif r ass as warra is. VI- olailiy Baiour As sow i abl 4, AEE aris, draaically, as w o cross subsapls. is ay b rsul of srog rlaiosip bw local olailiy ra ad ass pric. o cofir is assupio, w ca rgrss iplid olailiy o oyss facor. xisc of a sigifica gai rlaio bw ass pric ad is olailiy will, cssarily, riforc posiio of odifid RCEV odl rsus B ad DAB os. o a a or clar ida o is issu, w apply followig rgrssio: ID rτ α α 0 ε rτ 5 Wr ID is warra iplid sadard diaio calculad a da. oyss facor. r rτ τ is Firs of all, w do rgrssio for wol pal daa, across all warras. W apply r diffr pal rgrssio odls: Fixd Effcs Modl, Wigd Fixd Effcs Modl ad Rado Effcs Modl. rgrssio rsuls ar suarizd a abl 5. 7

27 abl - DECRIPIVE AIIC Warra Prsc i sapl Obs. Diluio Expiraio rik Pric Ma «Moyss» Ma ID Ma i o xpiraio Lif r AIM-W 0/08/0 /30/ % 07/9/ yars ANA-W 0/8/0 /30/ % 0/8/ yars BCAP-W 06//0 /30/ % 06/07/ yars CNO-W 04/6/0 /3/ % // yars GEMA-W 08//0 /30/ % 08// yars IDR-W 03//0 /30/ % 03// yars IG-W /6/0 /3/ % /5/ yars AR-W 07/0/0 /30/ % 07/9/ yars REN-W 06/8/0 /30/ % 06/8/ yars MEA-W 07/8/0 07/0/ % 07/7/ yar PLA-W 03/6/0 /30/ % 03/5/ yars MPO-W 0/08/0 /3/ % 0/07/ yars WAPO-W 0/0/0 /30/ % 06// yars ubr of obsraios of ac warra i sapl. iplid sadard diaio. 8

28 abl - Disribuio of Obsraios accordig o oyss ad i o xpiraio criria disribuio of 5460 obsraios will b basd o criria: oyss ad i o xpiraio. Figurs i parss ar prcags of obsraios of ac subsapl copard o wol sapl. Dp i--oy warras ar os wi a oyss facor>0.5. I--oy warras a a oyss facor ] 0.05;0.5]. A--oy warras a a oyss [ 0.05;0.05]. Ou-of--oy warras a a 0.5; Dp ou-of- oy warras a a oyss facor<-0.5. oyss facor [ [ Dp I oy I oy A oy Ou of oy Dp Ou of oy yar > yar %.89% % 7.47% % 3.77% % 7.6% % 35.55% abl 3 - Copariso of Esiaio Errors by Warra A suary abl wic ais o copar prforacs proidd, rspcily, by B odl ad DAB odl. is copariso will b ad by usig arag siaio rror of ac odl, by calculaig diffrc bw arag rrors. Warra Obsraios Ma Moyss Diluio Error B Error DAB Prforac B DAB AIM-W % 4.73% 4.8% 0.46% ANA-W % 3.56% 3.33% 0.3% BCAP-W % 7.5% 5.7%.88% CNO-W %.40% 8.34% 3.06% GEMA-W % 3%.30% 0.70% IDR-W % 5.07% 4.9% 0.6% IG-W %.93%.8% 0.% AR-W % 0.80% 8.8%.98% REN-W % 5.47% 4.94% 0.53% MEA-W % 4.8%.88%.40% 9

29 PLA-W %.49%.7% 8.77% MPO-W % 3.74% 3.89% -0.5% WAPO-W % 5.33% 4.4% 0.9% abl 4 - Esiaio Errors classifid by Moyss ad i o xpiraio A suary abl wic ais o copar prforacs proidd, rspcily, by B odl ad DAB odl. is copariso will b ad by usig arag siaio rror of ac odl, by calculaig diffrc bw arag rrors. W apply z-s for ac diffrc. Dp i-oy warras ar os wi a oyss facor>0.5. I--oy warras a a oyss facor ] 0.05;0.5]. A--oy warras a a oyss [ 0.05;0.05]. Ou-of--oy warras a a 0.5; Dp ou-of- oy warras a a oyss facor<-0.5. oyss facor [ [ B DAB Diffrc Probabiliy 7 Dp I yar 4.44% 4.87% % 0.56 I yar 5.3% 5.6% 0.05 % I - yar 5.58% 35.76% -0.8 % A - yar 6.86% 5.7%.5 % 0.76 A yar 0.07% 9.9% 0.5 % Ou yar.7% 0.7%.0 % 0.3 Ou yar 4.55%.99%.56 % 0.93 Dp ou - yar 6.% 4.3%.07 % Dp ou yar.% 8.3% 3.89 % ull yposis probabiliis for z-s. 0

30 abl 5 - Coparai abl of Rgrssio Modls Coparai abl aid o idify siaio rsuls gi by ac rgrssio odl. criria usd i is copariso ar: calculad sud, rgrssio liariy coffici ad su of rsidual squars. W copar Fixd Effcs Modl FEM, wigd FEM ad Rado Effcs Modl REM. FEM wigd FEM REM α REUL α R² 6.69% 78.% 70.8% R² adjusd 6.6% 78.5% 70.8% CR All rgrssio odls sow a srog irs rlaiosip bw iplid olailiy ad oyss. rα siaios ar sigificaly gai a ll α%. bs rgrssio odl ss o b Wigd Fixd Effcs Modl proidig os sigificaα siaio wi igs sud, bs liariy ad adjusd liariy cofficis ad lows su of squard rsiduals. uc a rsul suggss a odifid RCEV odl would b or appropria o fi warras prics a DAB ad B odls. Accordig o socasic olailiy odls, Wiggis 987 8, Hull ad Wi 987 9, Hso 993 0, olailiy aris roug i wiou big likd o udrlyig ass pric by a driisic rlaio. I or rs, s odls allow arbirary corrlaio bw olailiy ad ass pric. uc olailiy baiour, as sow by Hso 993, producs a probabiliy disribuio of ass rurs wi ig kurosis. us, rlaio bw iplid olailiy ad udrlyig ass pric ca ary as warra is i or ou-of- oy. o rify wr olailiy is purly socasic or o, w rpa rgrssio, usig wigd Fixd Effc Modl, oly for i--oy warras. i--oy obsraios rprs % of wol pal 665 obsraios aog a oal of 5460 warras prics. rgrssio rsuls ar suarizd i abl 6. 8 Wiggis, J.B., 987. Opio alus udr socasic olailiy: ory ad pirical sias. Joural of Fiacial Ecooics 9, Hull, J. ad A. Wi pricig of opios o asss wi socasic olailiis. Joural of Fiac, pp Hso,.L., 993. A closd-for soluio for opios wi socasic olailiy wi applicaios o bod ad currcy opios. Riw of Fiacial udis 6,

31 abl 6 - Rgrssio rsuls for i--oy obsraios Rgrssio wi wigd FEM 665 Obsraios Warra α 0i α AIM-W 0.44 ANA-W BCAP-W IDR-W IG-W 0.69 AR-W MEA-W PLA-W 0.45 MPO-W α R 93.8% R adjusd 93.7% Prob0.000 Aloug irs rlaio is or prooucd for ou-of- oy warras, rgrssio as sow a suc rlaio is aiaid for i--oy os wi aα siaio sigificaly gai a llα %., w ca dduc a olailiy rais irsly rlad o udrlyig ass pric, as assud by RCEV odl, ad o purly socasic. VII- Modifid RCEV s. DAB AEE, calculad roug 5460 obsraios, ar 8.95% ad.76% rspcily for odifid RCEV ad DAB odls. uc a rsul was prdicabl as odifid RCEV is adapd o warra spcificiis, i.. diluio ffc ad warra log lif r, by icorporaig diluio facor i origial pricig forula ad usig a ariabl olailiy. z-s sows a AEE ar sigificaly diffr fro ac ors a llα %. abl 7 copars AEE calculad for ac warra usig odls. abl 7 - Copariso of Esiaio Errors by Warra A suary abl wic ais o copar prforacs proidd, rspcily, by DAB odl ad RCEV odl. is copariso will b ad by usig arag siaio rror of ac odl, by calculaig diffrc bw arag rrors. Warra Arag Moyss Error DAB Error RCEV Diffrc AIM-W % 8.0% 6.075% ANA-W % 0.47%.86% BCAP-W % 5.7% 0.0%

32 CNO-W % 4.3% 4.% GEMA-W % 8.85% 3.45% IDR-W % 3.49%.4% IG-W %.9% 0.53% AR-W % 9.8% 9.0% REN-W % 0.7% 4.77% MEA-W % 5.3% 7.566% PLA-W %.4% 0.58% MPO-W % 9.7% 4.7% WAPO-W % 8.59% 5.83% Now l calcula AEE across subsapls ad upo oyss ad i o xpiraio criria. Rsuls ar suarizd a abl 8. W fid a odifid RCEV siaio rrors ar sigificaly diffr, a ll α 5%, fro DAB rrors, for ou-of- oy subsapls. O or ad, w ca s a odls siaio rrors ar o sigificaly diffr fro ac ors, for a ad i--oy warras. odls proid a alos siilar prforac for s cagoris of warras. uc a rsul ca b xplaid by odifid RCEV assupio for fixd θ across all oyss cagoris ad for all warras. is assupio ay b suiabl for ou-of- oy warras, wr odifid RCEV odl ad prford is bs prforac, bu i ss a i dos work so good for a ad i-- oy subsapls. For s cagoris of warras, r is sill a irs rlaio bw olailiy ad ass pric as sow i prious scio bu corrlaio bw s ariabls sould b or odra a for ou-of- oy warras. Isors d o b ry frigd fac ay dowrd i ark ad o dosra a ry odra opiis w o bcos bullis. uc baiour is qui rarkabl rspos o priods of fiacial criss. olailiy dcrass oooically as ass pric gos upward bu ra of dcras sould b lss a ra of olailiy icras w ass pric gos dowward. usθ us ary as w o fro ou-of- oy o i-oy subsapls. 3

33 abl 8 - Esiaio Errors classifid by Moyss ad i o xpiraio A suary abl wic ais o copar prforacs proidd, rspcily, by DAB odl ad RCEV odl. is copariso will b ad by usig arag siaio rror of ac odl, by calculaig diffrc bw arag rrors. W apply z-s for ac diffrc. Dp i-oy warras ar os wi a oyss facor>0.5. I--oy warras a a oyss facor ] 0.05;0.5]. A--oy warras a a oyss [ 0.05;0.05]. Ou-of--oy warras a a 0.5; Dp ou-of- oy warras a a oyss facor<-0.5. oyss facor [ [ DAB RCEV Diffrc Probabiliy Dp I yar 4.87 % 3.98 % 0.89 % 0.38 I yar 5.6 % 4.36 % 0.9 % 0.4 I - yar % 4.76 % % 0.6 A - yar 9.9 % 9.0 % 0.9 % 0.40 A yar 5.7 % 5.54 % 0.7 % Ou yar 0.7 % 8.6 %.45 % Ou yar.99 %.03 %.96 % Dp ou - yar 8.3 % 0.89 % 7.4 % 0 Dp ou yar 4.3 % 9.7 % 4.4 % 0 Coclusio As xpcd, odifid RCEV odl ld o lows AEE for wol pal, for ac warra ad for diffr oyss subsapls. is rsul ca b xplaid by fac a odifid RCEV is oly o of 3 odls usd i our pirical sudy, wic ris o xploi ajor warra spcificiis. Firs, odifid RCEV is xplicily adjusd for diluio ffc wic sould allow offsig gai ipac a diluio could a o qualiy of siaio. Also, ulik B ad DAB odls, odifid RCEV is or adapd o warra log lif r, proposig a diffusio procss wi ariabl udrlyig ass pric olailiy. is assupio ss o b or cosis a a of cosa olailiy usd by B ad DAB odls. Howr, odifid RCEV odl ay b subjc o so criicis. W ca rfr, i paricular, is us of a cosa lasiciy ariac across all udrlyig asss. is lasiciy ca ary for sa udrlyig ass dpdig if spo pric is goig upward or dowward. o rdy is dficicy, o ca calcula iplid lasiciy usig obsrd warras prics ad calibra odifid RCEV odl so i ca b adapd for udrlyig ass spcificiis. ull yposis probabiliis for z-s. 4

34 Bibliograpy Bas, D sig Opio Pricig Modls. Workig papr. Naioal Burau of Ecoois Rsarc. Bas, D Jups ad socasic olailiy: Excag ra procsss iplici i Dusc Mark opios. Riw of Fiacial udis, pp Bckrs, cosa lasiciy of ariac odl ad is iplicaios for opio pricig. Joural of Fiac, pp Black, F ad M. cols pricig of opios ad corpora liabiliis. Joural of Poliical Ecooy, pp Bsoussa, A., M. Grouy ad D. Galai Black-cols approxiaios of warra prics. Adacs i Fuurs ad Opio Rsarc, pp. -4. Crisoffrs, P. Jacobs,. ad Mioui, Modls for &P500 Dyaics: Eidc fro Ralizd Volailiy, Daily Rurs, ad Opio Prics. Workig papr. Faculy of aag, McGill Uirsiy. Crouy, M. ad D. Galai. 99. Warra aluaio ad quiy olailiy. Adacs i Fuurs ad Opios Rsarc, pp Cox, J Nos o opio pricig : Cosa lasiciy of ariac diffusios. Workig Papr. aford Uirsiy. Cox, J cosa lasiciy of ariac opio pricig odl. Joural of Porfolio Moag, pp Galai, D. ad M. cllr 978. Pricig warras ad alu of fir. Joural of Fiac, pp Hausr,. ad B. Laurbac ss of warra pricig odls: radig profis prspci. Joural of Driais, pp Hausr,. ad B. Laurbac rlai prforac of fi alrai warra pricig odls. Fiacial Aalyss Joural, pp Hsu, L. L, C. ad Li, Cosa lasiciy of ariac CEV opio pricig odl: igraio ad daild driaio. Maaics ad copurs siulaio. Hull, J. ad A. Wi pricig of opios o asss wi socasic olailiis. Joural of Fiac, pp Jacquir, E., Polso, N.G ad Rossi, P.E. 00. Baysia aalysis of socasic olailiy wi lrag ffc ad fa ails. Joural of Ecoorics, forcoig. Jos, C dyaics of socasic olailiy: idc fro udrlyig ad opios arks. Joural of coorics, pp Laa, H. ad R. Rdla 976. adard diaios of sock pric raios iplid i opio prics. Joural of Fiac, pp