WORKING PAPER No. 5/2009

Transcription

1 WORKING AER No. 5/9 Risk-ural Opio ricig a Iformaio Emb i CEZ Warra rics: Impli Volailiy Smil Impli Dsiy Fucio Mari Cícha 9

2 Řaa suií Workig aprs Cra výzkumu kokurčí schoposi čské koomiky j vyáváa s poporou projku MŠM výzkumá cra M54. ISSN Ig. Mari Cícha Novákových 3 8 raha 8 mari.cicha@mfcr.cz -mail:

3 RISK-NEURAL OION RICING AND INFORMAION EMBEDDED IN CEZ WARRAN RICES: IMLIED VOLAILIY SMILE IMLIED DENSIY FUNCION Absrac: Risk-ural approach is o of wo mai approachs o rivaivs pricig. his papr iroucs a rivaivs pricig ool riv from a biomial r of urlyig ass pric ur h assumpio of cosa volailiy. his aricl focuss o socks bu h propos approach ca b graliz a appli o ohr urlyig asss as wll. h assumpio of cosa volailiy las o a logormal isribuio of urlyig ass pric. Howvr h volailiy obsrv i h mark is of a fucio of opio srik a im o mauriy rahr ha a cosa. Hc usig logormal isribuio i pricig formulas las o icorrc prics. I his papr provi vic of h ologormaliy of CEZ sock usig Amrica-syl call warras. As warras ar Amrica-syl w hav o ak io accou h possibiliy of arly xrcis. I cosruc a volailiy smil im sris for CEZ warras. Furhr I riv impli risk-ural siy fucio of CEZ sock pric from obsrv warras mark prics. h rsulig impli siy fucio ca b us for corrc rivaivs pricig. Impli siy fucio also provis us wih a usful way of aalyzig mark racios a xpcaios of fuur sock pric o compay arigs rlass chags i official irs ra govrm bo aucios c. I provi som xpcaio of fuur isribuio of CEZ sock pric usig highr moms of impli siy fucio. Rczovala: Ig. Ira Jiřichovská CSc.

4 . INRODUCION Opioal coracs ar a larg group of fiacial isrums a uiqu approach o hir corrc pricig has o b irouc y. h rplicaio sragy a risk-ural approach ar wo mai approachs ha la o iical rsuls. Ur h assumpio of cosa volailiy w riv logormal isribuio of a urlyig ass. Usig logormal isribuio i pricig formulas w g h horical pric of h opio. I is of h cas ha rars us volailiy as a fucio of opio srik pric a opio im o mauriy rahr ha as a cosa valu which las o volailiy surfac. Logormal isribuio of urlyig ass ca o b us for pricig ay mor. I is possibl o us h iformaio ha is mb i opios prics o riv impli isribuio of urlyig ass. hs siy fucios sima from cross-scio of obsrv opio prics ar gaiig icrasig aio. hy ar us for pricig complx rivaivs a rivaivs o ra i h mark. Usig impli isribuio of urlyig ass w ca also fi h mispric rivaivs a ar xcss rurs abov h risk-fr ra of irs by buyig opios ha ar urvalu by h mark a sllig opios ha ar ovrvalu by h mark. A umbr of auhors hav us impli siy fucios as iicaor of mark sim o xami whhr opios marks aicipa major coomic vs. Cral baks i paricular hav b irs i usig impli siy fucios o assss mark paricipas xpcaios of fuur chag i irs ras sock prics a xchag ras. May rsarch paprs o his may b fou issu i rsarch bullis of cral baks. S for xampl FED a Bak of Egla (Bliss a aigirzoglou Bak of Egla (Bahra 997 ECB (Schir a Glazr 3. h mho for simaig impli siy fucio fall io fiv groups: sochasic procss mhos impli biomial rs fii-iffrc mhos impli volailiy smoohig mhos a siy fucio approximaig mhos. Sochasic procss mhos bgi by assumig a mol for sochasic procss rivig h prics of h urlyig ass. his approach ca b us i h absc of opios prics. For isac Horahl appli h Logsaff-Swcharz mol o Swish irs ras i (Horahl. h impli biomial r mho was vlop by (Rubisi 994. h smooh impli volailiy mho was origially vlop i (Shimko 993. his is a approximaig fucio mho o h volailiy smil rahr ha o h siy fucio. Approximaig fucio mhos ar bas o miimizig h iffrc bw obsrv opios prics a fi prics prouc by paricular fucioal form chos o allow for a variy of possibl shaps. For isac Mlick a homas us mixurs of logormals i (Mlick a homas 994 Maa a Mil us Hrmi

5 polyomials i (Maa a Mil 994 a alraivly Ai-Sahalia us o-paramric krl simaor i (Ai-Sahalia a Lo 998. h papr is orgaiz as follows. Scio iroucs biomial r mol i h ral worl a shows h rasiio from h ral o h risk-ural worl. Furhr Scio iscusss ral a risk-ural isribuio of h sock pric. Scio iroucs h risk-ural opio pricig approach a applis i o h ral mark opios. Scio 3 iscusss violc i cosa volailiy assumpio. Scio 3 also shows ha warras o CEZ sock o o hav cosa volailiy hc CEZ sock pric is o logormally isribu a ay fuur im. Furhrmor Scio 3 iroucs rivig volailiy smil a volailiy rm srucur from Amrica-syl call warras o CEZ. Scio 4 iscusss wo-logormal mixur impli probabiliy isribuio of h CEZ sock pric shows a simaio of paramrs a als wih h compuig of highr moms. Usig hs saisics w ca mak a assumpio abou fuur CEZ sock pric. Scio 5 is h coclusio.. HE BINOMIAL REE MODEL AND EXECAION AROACH A rivaiv is h fucio of urlyig ass say sock a a spcific im i h fuur. L us assum ha h pric of h sock follows h procss of biomial r. Du o his oly wo higs ca happ o h sock a ach im : a 'up' mov or 'ow' mov. W sar a h im a coiu owars o h im usig im sp. W will assig probabiliis o hs movs: probabiliy p o mov up a hus p o mov ow. W also a cash bo B o rprs h im-valu of moy. hr will b som coiuously compou irs ra r ha will hol for h prio a will r caus ha $ oay will b worh $ a ick lar. W ca us wo approachs o h rivaiv pricig a his mom. h firs o is h so- call rplicaio sragy. h rplicaio sragy is bas o h fac ha ay rivaiv of a sock rivaiv pay-off paricularly ca b cosruc i avac from a appropria porfolio of bo a sock. h valu of h rivaiv woul xacly cacl h valu of h porfolio whavr h sock pric was a h of h ick-prio s (Joshi 5 for ails. hrfor if o kows h valu of his porfolio o also kows h valu of h rivaiv. h ohr rivaivs pricig approach is bas o xpcaio a chag of h masur. W ca imagi h valu of h rivaiv as a xpcaio of iscou claim ur marigal masur. I will al oly wih h xpcaio approach i his papr.

6 L us assum ha h sock pric procss { } S has a cosa growh ra μ a cosa ois σ. h sock jumps from h valu S o h valus a im. h valu S afr up jump rmis (. h valu S afr ow jump rmis (. up S S xp( μ σ. ( ow S S xp( μ σ. ( All jumps ar qually likly i h ral worl i.. ur a Π masur. h ral worl masur Π ur which S occurs i h ral worl is irrlva for corrc pricig (Baxr 999. W hav o fi h masur Θ ur which h procss of iscou sock { B S } is a marigal. L us fi h variabl ow S xp( r S q up ow. S S (3 If h coiio of raioal mark is fulfill followig rlaio hols ow r up S < S < S which forc q io (. W ma h sam cosrai for probabiliy. I ca b show ha q is h marigal masur probabiliy (Baxr 999. W ca calcula q approximaly qual o q μ σ r. σ (4 Cosquly q ps o logr o im bu oly o im sp. Hc h whol r has oly o uiqu probabiliy q. h biomial r mol is paramriz by h im sp. As ha quaiy gs smallr h mol shoul vr mor closly approxima h ral worl. For a fix im w g sock valu ur Π.

7 S S X μ σ (5 whr is h umbr of im sps ill im a umbr of h spara jumps which wr up-jumps. X is h oal X is biomially isribu raom variabl wih ma a variac 4 X so ha has zro ma a variac quals. Applyig h cral limi horm his isribuio covrgs o h saar ormal isribuio. As gs smallr h isribuio of S bcoms log- σ / μ ormal wih ma S a variac μ σ σ S μ. Disribuio of S is uslss for corrc pricig ur ral worl Π masur a hus w apply h sam procur ur marigal masur Θ. Usig up-mov probabiliy X is sill biomially X isribu wih ma q a variac q( q. Hc has μ σ r ma a variac asympoically approachig o. σ As gs smallr h cral limi horm lls us ha his formula covrgs o a ormal variabl wih h sam ma a variac xacly o. h rasformig his ormal raom variabl w g h isribuio of S which is log-ormal jus lik ur Π masur bu r r S a variac ( σ S wih ma. hus h isribuio of S ur Θ marigal masur is fully rmi by im a ois σ (so-call volailiy a os o p o growh ra μ. Sic w hav fou h margial isribuio of S ur Θ w ca sar alig wih h rivaivs pricig. W o kow h sock isribuio claim im of xrcisig h claim a h risklss irs ra r. h pric of h claim X (i.. rivaiv pric is h h xpc valu of h iscou claim ur Θ : V EΘ ( B X (6

8 whr is im of xrcisig h claim a r coupo bo which is fi as B. 3. RISK NEURAL DERIVAIVES RICING B is iscou zro I orr o compu h rivaiv pric (6 w hav o fi h isribuio of iscou claim ur Θ usig h rasformaio of raom variabl chiqu. o b abl o o ha i is asir o us ormal isribuio prfrably N ( isa of log-ormal isribuio of S. Disribuio of S is fi by: r σ σ S S (7 whr Z is N( ur Θ (Joshi 5. Z 3.. ricig a call opio Call opio (Europa givs h holr h righ bu o h obligaio o buy sock a a spcifi pric (srik a im. A raioal ivsor will xrcis h opio oly if h sock valu is grar ha h srik pric k a im. hus h claim X is h fucio of sock X ( S k a im. rimarily w hav o rmi h moms of h claim or of h iscou claim ur Θ. Usig (7 w g siy (8 a isribuio fucio (9 of iscou claim Y B X. f ( y σ π r σ y k log S σ (8 r ( y k

9 F( y rf σ (9 log y k S σ r Havig appropria siy fucio w compu h xpc valu (6. W hav o kp i mi ha h claim X a iscou claim Y may rach oly posiiv valus. Hc h pric V of h call opio is V ( S y f ( y y ( Suppos a sock wih cosa volailiy σ of % a cosa rif μ of 5% wih coiuously compou irs ra r cosa a 3%. Wha is h pric of a opio o buy h sock for $8 i 5 yars im giv h curr sock pric of $5? h siy (8 a h isribuio fucio (9 ar isplay i Figur. W solv h quaio ( umrically wih h oucom of hus h pric of h call opio is $.36 a im. All ohr prics woul la o arbirary risk-fr profis hrough cosrucig appropria rplicaio porfolio of a bo a a sock (Joshi 5. L us ak a look a h Black-Schols formula (Hull 6. Fischr Black a Myro Schols cosruc h slf fiacig rplicaio porfolio of bo a sock i coiuous worl. hus hy ha o solv a DE (arial Diffrial Equaio o g h corrc opio pric. h DE has a xplici soluio hrfor hir formula ( bcam so popular. Applyig formula ( h call opio pric quals $.36 which is xacly h sam pric as usig risk ural approach. h xplici soluio o B-S for call opios is giv by S log Φ k V ( S S ( k r S log k Φ r σ σ r σ σ

10 whr h oaio Φ (x os h isribuio fucio of N (. Figur : Dsiy a isribuio fucio of iscou call opio claim wih σ. k 8 S 5 5 r ricig a pu opio u opio (Europa givs h holr h righ bu o h obligaio o sll sock a a spcifi pric (srik a im. A raioal ivsor will xrcis h opio oly if h sock valu is lss ha h srik pric a im. hus h claim X is h fucio of sock X ( k S a im. Applyig h rasformaio of raom variabl chiqu w g h siy ( a h isribuio fucio (3 of iscou claim Y. f ( y σ k log r r y S σ ( π σ r r ( k y.

11 . σ σ log ( r S y k rf y F r (3 Havig appropria siy fucio w compu h xpc valu ( whr for < y a y for > y. Dsiy a isribuio fucios of iscou claim ur Θ marigal masur for. σ 8 k 5 S 5.3 r ar show i Figur. Agai w solv h igral ( umrically. h pric of h pu opio is $.457. W g xacly h sam pric applyig h Black-Schols formula for pu opio. h Black-Schols formula for pu opio is giv by xplici DE soluio (4 s (Hull 6 for ails. log ( Φ Φ r k S S r k S log k S V r σ σ σ σ (4 whr h oaio (x Φ os h isribuio fucio of ( N.

12 Figur : Dsiy a isribuio fucio of iscou pu opio claim wih σ. k 8 S 5 5 r ricig a arbirary rivaiv L us suppos a corac payig off som arbirary fucio of sock pric a im. h corac also has guara miimum payou a maximum payou accorig o gais of h sock. Mor prcisly i is a fiv-yar corac which pays ou 9% of h raio of h rmial a iiial valus of h sock. Or i pays ou 3% of iiial sock pric if ohrwis i woul b lss or 8% of iiial sock pric if ohrwis i woul b mor. How much is his corac worh a im? hus h claim X is S X mi max.3s.9.8s S (5 o pric his corac w hav o compu h xpc valu of h iscou claim ur Θ marigal masur (6 i h sam way as pricig call a pu opios. h bs way o o ha is sparaig h claim io hr parial claims. h w rmi h valu of ach parial claim usig a corrspoig irval of sock pric S a im. A h w sum all valus of parial claims a w g h oal valu of h rivaiv (5. h parial claims a corrspoig irvals ar show i (6-(8.

13 S S S.3 S X.3S..9 (6.3 S.9 S S X.9 S (7 ( S X.8. 3 S (8. W assum sock rif μ 7% sock volailiy σ 5% iiial sock valu S a irs ra r 6.5%. W compu h valu of h parial claim (6 a firs. o o ha w hav o fi h probabiliy.3 S S ur Θ which is h probabiliy of xrcisig h.9 firs claim. h isribuio of S is giv by (7. h siy a isribuio fucios of S ar show i Figur 3. h rsulig probabiliy is Θ S.3.9 S.66. Hc usig h rivaiv pricig formula (6 mulipli by h corrspoig probabiliy w g h valu of firs parial claim..3 r.3 ( B X S S X S S. V EΘ Θ Θ.9.9 arial claim X has h valu of Applyig h sam procur w fi valu of X 3 (8. Havig 3 3 r V E ( B X ( S ( S X ( S ( S. whr Θ 3 Θ 3 Θ Θ ( S ( S.9. arial claim X 3 has h valu of.338. Now w ar o pric h rmaiig parial claim X. Agai w apply (6

14 r ( B X.9 f ( y EΘ V y.3 whr y for y ( S S a ohrwis f (y os.9 siy fucio of S ur Θ marigal masur. h valu of parial claim X is h valu of h rivaiv is which is sum of valus of h parial claims. Cosrucig h rplicaio porfolio o fully hg h claim (5 a solvig corrspoig DE w g h valu of which is xacly h sam pric as applyig h riskural approach (Baxr 999. Figur 3: Dsiy a isribuio fucio of S wih σ.5 S 5 r MARKE RACICE L m ak a look a ral mark opios prics. How clos ar h mark prics of opios o hos pric by Black-Schols a riskural approach usig logormal isribuio? Ar h probabiliy isribuios of h ass prics rally logormal a ay fuur im? rars us h Black-Schols mol bu o xacly i h way ha Black a Schols origially i. h risk-ural approach is gral ough o prouc corrc prics ur chag coiios i h ral mark. A lo of iformaio abou urlyig icluig probabiliy isribuio ca b riv from opios prics. I will aalyz h isribuio of CEZ sock pric usig CEZ warras prics.

15 4.. Daa scripio I chos h opios o CEZ sock u o hir rlaivly high mark aciviy comparig o h opios o h ohr Czch socks. h oly opioal isrums o CEZ sock ar warras omia i EUR a ra o Bors Sugar. I us all pric call warras o CEZ ra o Bors Sugar as of 3 April 9. S Appix B for h compl lis of us warras. Mark mi prics wr ak from Bors Sugar via Bloombrg. Warras ar vry similar o call opios a will of cofr h sam righs as a quiy opio a ca v b ra o scoary marks. Howvr hy hav svral ky iffrcs. Firsly warras ar issu by priva paris ypically h corporaio o which a warra is bas a by ivsm baks rahr ha a public opios xchag. Scoly warras ar o saariz lik xchag-lis opios. Whil ivsors ca wri sock opios o h CEZ shar hy ar o prmi o o so wih CEZ-lis warras. As all CEZ warras ra o Bors Sugar ar wri by baks hy ar o iluiv hc w ca hal h warras as opios rspcig h raio aurally. h raio rmis how may warras ar i orr o acquir h righ o buy or sll o ui of h urlyig isrum. Bcaus all cosir warras ar Amrica-syl calls o ivi payig sock w ar forc o moify our pricig formula for Europa call (. Ulik h Europa xrcisig syl Amrica syl mas ha h righ o xrcis h warra ca b ivok o ay raig ay urig h lif of h warra. I ca b show ha i is vr opimal o xrcis a Amrica call opio o o-ivi payig sock bfor h xpiraio a. Wh urlyig sock pays off ivis i is opimal o xrcis h opio ihr a a im immialy bfor h sock gos x-ivi or a a im of xpiraio (Hull 6. CEZ pays off h ivi oc a yar mor imporaly oc urig h lif of all cosir opios. Usig Black's approximaio w pric wo Europas opios wih xrcis as a mauriy a a im immialy bfor ivi. h valu of Amrica-syl opio is h h grar of h valus (Hull Volailiy smil Volailiy is a uobsrvabl paramr a mus b sima. If hr ar o raabl opios o h mark w ca sima h volailiy from a hisory of h sock pric. W fac h problm of h lgh of im sris i his cas. For raabl opios w rmi h volailiy from h opio's mark pric a h pricig formula. Hc his ki of volailiy is impli. Our pricig formulas assum ha h probabiliy siy fucio (DF of h urlyig ass is logormal Warras issu by h compay islf ar iluiv. Wh h warra issu by h compay is xrcis h compay issus w shars of sock hc h umbr of ousaig shars icrass.

16 wih cosa volailiy a ay giv fuur im. his assumpio is o h o ma by rars. Accorig o mpirical suis h mark assums h probabiliy isribuio of a quiy pric wih havir lf ail a lss havy righ ail ha h logormal isribuio. h raso is ha rars hik ha h probabiliy of a largr owwar movm i h sock pric is highr ha ha pric by h logormal probabiliy isribuio. For saar opios rars us volailiy smils o allow for ologormaliy. hy ajus h volailiy paramr accorig o h volailiy smil a h hy us logormal isribuio. h volailiy smil fis h rlaioship bw h impli volailiy of a opio a is srik pric. For quiy opios h volailiy smil s o b owwar slopig. his fac mas ha call opios wih srik pric blow h mark pric of h sock (i-hmoy o hav highr impli volailiy ha call opios wih srik pric abov h mark pric of h sock (ou-of-h-moy. Hc i-h-moy call opios ar rlaivly mor xpsiv ha ou-of-h-moy calls. W ca also fi h impli volailiy as a fucio of srik a opio's im o mauriy which is saar pracic i h mark. W call his rlaioship h volailiy surfac. Havig a opio mark pric w h pricig formula o riv h impli volailiy of paricular opio. Carr has furhr vlop a graliz h Gsk-Johso approach a has irouc gral approach o h valuaio of h Amrica-syl opio s appix a (Carr 995 for ails. Sic CEZ pays off ivi oc urig h lif of all cosir opios i is opimal o xrcis h opio ihr immialy bfor h sock gos x-ivi or a h opio's mauriy. For ha raso w cosir for ω ( k a for ω ( i (8 a i (9 rspcivly which las o usig h bivaria ormal isribuio i h pric formula. Usig h Carr riskural approach h valu of Amrica call is V kω ( S ω ( k whr ω ( k a ( (9 ω ar fi i appix. Solvig h quaio (9 for σ w obai h opio's impli volailiy. I solv h quaio for all opios wih ay mark aciviy maurig o 7 Ju 9 for all raig ays sic Jauary 9 ill 3 April 9. h rsulig impli volailiy smil im sris is isplay i Figur 4. W ca obsrv ha h volailiy s o icras wih h opio's icrasig im o mauriy a s o cras wih icrasig srik pric. his oucom is i accorac wih h hory. rars of call his owwar slopig shap as a volailiy skw. h warra wih srik valu of CZK 9 sms o hav cosaly lowr impli volailiy hc i sms o b cosaly urvalu. o xplai his iscrpacy bw h hory a h

17 mark w hav o ak a look a mark aciviy. h raig volum is far highr ha by h ohr warras a hc h bi-offr spra is much smallr for all aalyz ays. Du o highr liquiiy o his warra is pric rprss h mark opiio br ha h prics of h ohr warras. I will us wighs o h warras subjc o hir liquiiy o xprss mark pric sigificac lar i h papr. Figur 4: Volailiy skw for Amrica call opios o CEZ sock 4.3. Impli isribuio h variac ha is impli by a opio's pric is h mark x a sima of h urlyig ass's rur volailiy ovr h rmaiig lif of a opio. Mor irsigly i is possibl o riv h highr moms of fuur ass valus from h mark opios prics. hs ca b xrac i h form of a x a risk-ural probabiliy isribuio of h urlyig ass a paricular fuur a. Rahr h spcifyig h urlyig ass pric yamics o ifr h riskural siy fucio i is possibl o mak a assumpio abou h fucioal form of siy fucio islf a o rcovr is paramrs by miimizig h isac bw h obsrv prics a hos ha ar gra by h assum probabiliy isribuio. Sic all cosir warras o CEZ sock ar Amrica-syl calls wih o ivi payou urig warras lif i is impora o kow whhr i is opimal o xrcis h warra immialy bfor xivi a or a mauriy. o iify his I us Black's approximaio approach. Black's approximaio aks accou of arly xrcis i call opios. his ivolvs calculaig h pric of Europa opios ha maur a opio's mauriy a a a im immialy bfor h sock gos x-ivi ( a h sig h Amrica pric qual o h grar of h wo. W sar

18 cosirig h possibiliy of arly xrcis jus prior o h x-ivi a (i.. a im. If h warra is xrcis a im h ivsor rcivs S K. If h warra is o xrcis h sock pric rops o S D whr D is h ivi. h ivi o CEZ is rmi i amou of CZK 5 pr sock for 9. h x-ivi a is 3 May 9. Bcaus h lowr bou for call opios is S 6 i follows ha if r( D k (Hull ha is S D k r( S D r( D k( i cao b opimal o xrcis h opio a im. O h ohr ha if r( D > k( ( for ay rasoabl assumpio abou h sochasic procss follow by h sock pric i ca b show ha i is always opimal o xrcis h opio a im. h iqualiy i ( will o b saisfi wh x-ivi a is fairly clos o h mauriy of h opio a h ivi is larg. I us forwar mi ra bw RIBOR a RIBID as h risk-fr irs ra. Applyig his mho o all cosir warras I go h iformaio whhr i is opimal o xrcis h warra a im or a im. h oucoms ar isplay i Figur (6. Usig his iformaio w ca riv h impli isribuio of CEZ quiy pric. h pric of h Europa call opio is giv by (6 a im whr h claim is X ( S k. Ulik h siy fucio us i chapr ''ricig a call opio'' I assum ha h risk-ural isribuio of sock is h mixur of wo logormals rahr ha logormal a ay fuur im. W ca us h arbirary siy fucio which fulfills h criria of probabiliy siy fucio a has fii variac. h problm wih usig ohr ha h Gaussia siy fucios is ha h urlyig pric isribuio chags as h holig prio chags. Ur h assumpio of logormally isribu aily rurs h arbirary lgh holig prio pric isribuios mus also

19 b logormal. No ohr fii variac isribuio is similarly sabl (Bahra 997. Morovr h fucioal form assum for h siy fucio shoul b rlaivly flxibl. I paricular i shoul b abl o capur h mai coribuios o h volailiy smil amly h skwss a h kurosis of h urlyig isribuios. A wigh sum of ip logormal siy fucios fis hs criria. h probabiliy isribuio of ay fuur sock pric is giv by ( S μ σ S ( θ L( S μ σ f ( S θl S ( whr L ( S is h logormal siy fucio giv by ( a θ is θ. h probabiliy wigh saisfyig h coiio ha [ ] ( logs (.5 logs μ σ τ ( L S xp S σ πτ σ τ ( whr τ os im o fuur a o which w riv h siy fucio a { μ σ μ σ θ} ar h ukow paramrs. rasformig h sock pric siy fucio (( for X ( S k w g h siy fucio of call warra pric giv by f ( x x k log S xp σ πτ σ τ (3 ( x k ( μ.5σ hus h fi valu for a call pric giv paramrs μ σ μ σ } is giv by { θ τ. Cˆ rτ ( k τ μ σ μ σ θ θ xf ( x μ σ x ( θ xf ( x μ σ x. (4 Havig h fi valus of calls warras w ca miimiz h sum of squar rrors wih rspc o h fiv isribuioal paramrs a risk-fr ra r bw h warra prics gra by h mixur isribuio mol a hos acually obsrv i h mark. I pracic I us h corrspoig mi ra bw RIBID a RIBOR o fix r a hrby ruc h imsioaliy of h problm. W shoul iclu pu warras i our miimizaio problm as wll sic hy ar pric off h sam urlyig isribuio.

20 hr wr oly wo pu warras ra o Bors Sugar a h im of wriig. Boh of hm ar ra clos o a-h-moy sa. I my opiio icluig hs warras i h opimizaio problm will o provi sigificaly br iformaio abou impli isribuio. Morovr boh pus ar Amrica-syl warras o ivi payig sock urig hir lif. his mas ha i ca b opimal o xrcis hm a ay im urig hir lif. his fac woul la o compuig problms (Mlick al o avoi h arbirag opporuiis h ma of h impli siy fucio shoul qual h forwar pric of urlyig ass. h forwar pric of h urlyig asss is qual o rτ S i risk-ural worl. h xpc valu of impli isribuio ( is giv by ( S μ σ S S ( θ L S L( S μ σ S S. E( S θl S L (5 Som warras ar fairly illiqui paricularly a p i-h-moy a ou-of-h-moy srik prics. Warra prics a hs our sriks ar lss iformaiv abou mark xpcaios or ar o availabl. Hc I ci o us wighs o ach warra which qual raig volums o ach warra o 4 April 9. I xclu o-pric warras from h simaio sampl. Giv obsrvaios of call prics c i I sima h paramrs { μ σ μ σ θ} of h impli wologormal siy fucio miimizig h objciv fucio mi N c { μ σ μ σ θ} i subjc o w i ( ˆ r c C ( k τ μ σ μ σ θ ( E( S S i (6 i N c i w i a w i i whr N c is umbr of call warras i h simaio sampl. Miimizig objciv fucio (6 w hav o sur ha h igral of h rsulig impli siy fucio ( from o is qual o σ > o i.. f ( S S. Furhrmor h cosrais o paramrs ar σ a θ (. h paramrs simas ar isplay i abl. Figur 5 shows h wo-logormal mixur impli siy h mark pric of h urlyig ass is ar h srik pric

21 fucio of CEZ sock wih is wigh compo logormal siy fucios o 7 Ju 9. Aiioally Figur 5 shows h logormal siy fucio wih h sam ma a saar viaio as h impli isribuio. Figur 5 isplays h xpc valu of impli isribuio a forwar pric as wll. W ca obsrv ha h xpc valu of CEZ sock riv from impli isribuio a forwar pric of CEZ sock ar pracically iical which was o of h criria us by miimizig h objciv fucio. W ca obsrv ha h rsulig impli isribuio of CEZ sock pric has grar kurosis ha logormal isribuio a givs highr pric o i-hmoy warras a lowr pric o ou-of-h-moy warras ha h iiially cosir logormal isribuio. Figur 6 shows h iffrc bw warras mark prics i.. obsrv prics a hir prics impli by wo-logormal mixur isribuio. h avrag iffrc is.8 EUR. W ca obsrv lss accura fi by lss liqui warras a vic vrsa. Each warra is labl by is opimal xrcis im. os xrcisig h warra a im immialy bfor h sock gos x-ivi a os h im of warra mauriy. W ca obsrv ha arly xrcis os o affc h qualiy of h fi. Uforualy all warras o CEZ ar oly ra a iscrly spac srik pric lvls a for vry limi rag ihr si of h a-h-moy srik (AM which is from CZK 7 o CZK 55 i our cas. AM pric is CZK 835. Hc all impli siy fucios irpola bw obsrv srik prics a xrapola ousi of hir rag o mol h ail probabiliis. h ails spcially h lf o i h call ou-of-h moy ara ar giv by h fucio formula rahr h by obsrvaios. his issu cocrs o almos all warras a opio o h mark v hos ra o vlop highly liqui marks. abl : aramrs simas ˆμ ˆμ ˆ σ ˆ σ Impli isribuio is us primarily for pricig o ra rivaivs o urlyig ass bu impli isribuio coais much mor iformaio. I provis us wih iformaio abou fuur bhaviour of CEZ quiy i rms of probabiliis. For xampl h ma of 787 CZK is h xpc fuur valu of CEZ sock o 7 Ju 9. h mo of CZK 77 o h ohr ha is h mos likly fuur oucom o his a. h saar viaio of CZK is h masur of h ucraiy arou h ma. h posiiv skwss of. lls us ha hr is lss probabiliy aach o oucoms highr ha h ma ha o oucoms blow h ma. h kurosis of 3.48 is a masur of how pak h isribuio is a h liklihoo of xrm oucoms: h grar h kurosis h far h ails of h isribuio. hs θˆ

22 saisics provi a usful way of rackig h bhaviour of impli siy fucios ovr h lif of a sigl corac a makig comparisos across coracs. Impli isribuio may also b usful for aalyzig mark racios o compay arigs rlass moy mark opraios govrm bo aucios c. Impli summary saisics for CEZ sock o 7 Ju 9 ar isplay i abl. h highr moms ca b riv ircly from siy fucio usig ' h moms of impli siy fucio. h h mom μ is giv by ' μ ( S S f ( S S whr f ( S is wo-logormal mixur siy fucio (. For w g xpc valu of f ( S. Furhr w o compu h cral moms μ usig formula: μ ( S E{ ( S E( S }. For w g h variac of S. Skwss μ 3( U is ormaliz hir cral mom giv by μ3( S μ 3( U μ ( S μ ( S a kurosis μ ( is h ormaliz fourh cral mom giv by 4 U μ4( S μ 4( U ( μ ( S. abl : Impli risk-ural summary saisics for CEZ sock o 7 Ju ma 787 saar viaio skwss. kurosis 3.48 I is impora o rmmbr ha h riv impli siy fucio is risk-ural ha is i is quival o h ru mark siy fucio oly wh h ivsors ar risk ural. Hc h impli summary saisics ar risk-ural as wll. I raliy h ivsors ar likly o b risk avrs a warra prics will icorpora hs prfrcs owars risk as wll as blifs abou fuur oucoms. Disiguishig bw hs wo facors woul rquir spcificaio of h

23 uobsrvabl aggrga mark uiliy fucio a simaio of h corrspoig coffici of risk avrsio. Rubisi assums a mark risk prmia of bw 3.3 a 5 pr c for a quiy ix a blu chips quiis. Mor imporaly h fis ha h subjciv isribuio is oly slighly shif o h righ rlaiv o h risk-ural isribuio a h qualiaiv shaps of h wo isribuios ar qui similar (Rubisi 994. Figur 5: wo-logormal mixur impli siy fucio of CEZ sock o 7 Ju 9 wih is wigh compo logormal siy fucios logormal siy fucio wih h sam ma a saar viaio as h impli isribuio xpc valu of impli isribuio forwar pric of CEZ sock

24 Figur 6: Absolu iffrcs bw warras mark prics i.. obsrv prics a hir prics impli by wo-logormal mixur isribuio. Opimal xrcis im whr os h im immialy bfor h sock gos x-ivi a mauriy. 5. CONCLUSIONS I his papr I hav prs h risk-ural opio pricig approach a hav show is applicaio i pricig ral mark rivaivs. I hav riv h probabiliy isribuio of a urlyig ass pric sock i paricular from h biomial r procss of h sock pric i ral worl. Sic ral worl is irrlva for pricig I chag h masur a hav riv risk-ural isribuio of h sock pric. h iscr rs ar oly a approximaio of h way ha prics acually mov. I pracic a pric ca chag a ay isa rahr ha jus a som fix ick-ims. Usig cral limi horm I cam ovr o coiuous im. Ur assumpio of cosa volailiy is h isribuio of sock pric logormal. I hav pric hr iffr rivaivs a show o bias bw prics prouc by risk-ural approach a hos prouc by rplicaio sragy approach. h rplicaio sragy approach las o solvig parial iffrial quaios DE (havily us Black-Schols formula for isac. h prs risk-ural approach is suiabl for pricig ay Europasyl claim fucio of urlyig ass. I s is mai avaag i is rlaiv simpliciy compar o h coiuous sochasic procss approach which las o solvig h sochasic iffrial quaios SDE. h rsuls i his papr provi srog vic of ologormaliy of CEZ sock pric. I us Amrica-syl call warras o CEZ wih

25 ivi pay off urig h lif of warra o cosruc h volailiy smil im sris. As CEZ pays off h ivi w hav o ak io accou h possibiliy of arly xrcis. W ca obsrv ha volailiy is h fucio of srik pric rahr ha a cosa valu. Furhrmor I hav riv h impli isribuio of CEZ sock pric from warras prics obsrv o h mark. I us wo-logormal mixur siy fucio which is i my opiio flxibl ough o capur h mai coribuios o h volailiy smil amly h skwss a h kurosis of h urlyig isribuio. Miimizig h objciv fucio subjc o highly oliar cosrai I hav fou paramrs simas of h impli siy fucio. No all warras ar qually liqui a hus hir prics ar o qually rliabl. Hc I us ra volum o ach warra as wighs. h rsulig impli isribuio of CEZ sock pric has grar kurosis ha h logormal isribuio a givs highr pric o i-hmoy warras a lowr pric o ou-of-h-moy warras ha iiially cosir logormal isribuio. Usig impli siy fucio w ca pric o-ra rivaivs or sarch for mispric os. Impli isribuio provis us wih iformaio abou h fuur i bhaviour of urlyig ass i rms of probabiliis as wll. I hav riv highr moms of CEZ sock pric lik skwss a kurosis. hs saisics provi a usful way of rackig h bhaviour of impli siy fucios ovr h lif of a sigl corac a makig comparisos across coracs. Obsrvig ay-o-ay chags i hs saisics w may also aalyz mark racios a fuur xpcaios o compay arigs rlass moy mark opraios govrm bo aucios c.

26 AENDIX Appix A Gral valuaio formula o h imig opio Gral valuaio formula o h imig opio 3 is fi by ( ( S k S k V ω ω (7 whr ( N k k ω (8 K N k ; ( ( ; Ω k N τ τ K a ( N ω (9 K N ; ( ( ; Ω N τ τ K whr k S / τ (whr τ is opio's im o mauriy τ k a ar h corrspoig ivi yils (for Amrica call o sock r for Amrica pu o sock r k. Furhr for Amrica call k r for Amrica pu r. 3 imig opio givs h buyr h righ o a choic of iffr xrcis as.

27 Furhr i N is h saar i-varia isribuio fucio wih corrlaio marix Ω whr ( ( τ σ τ σ G l G (3 a ( ( τ σ G G (3 whr σ is h volailiy. Furhr k is h uiqu valu of h pric raio. ric raio is h criical valu of iqualiy which lls us ha h opio will o b xrcis a mi-lif if h opporuiy cos of xrcis xcs h cash procs of xrcis (Carr 995. For isac is h soluio o ( N N k k ; ( N N ; (3 whr is h soluio o ( ( ( (. N N k (33 S (Carr 995 for ails

28 Appix B abl 3: Call warras o CEZ sock us o riv wo-logormal mixur impli siy fucio of CEZ. hs warras ar ra o Bors Sugar as a 3 April 9. Warra cos wr ak from Bloombrg. warra co CM5LQM GR Equiy CM5LQN GR Equiy CM6AV GR Equiy CM6AW GR Equiy CM6AX GR Equiy CM6AY GR Equiy CM6AZ GR Equiy CMNXJ GR Equiy CMNXK GR Equiy CM5LQQ GR Equiy CM5LQR GR Equiy CM5LQS GR Equiy CM6AUA GR Equiy CM6AUB GR Equiy mauriy srik i a CZK 6/7/9 9 6/7/9 6/7/9 7 6/7/9 75 6/7/9 8 6/7/9 85 6/7/9 95 9/6/9 9/6/9 3 9/6/9 9 9/6/9 9/6/9 9/6/9 7 9/6/9 75 CM6AUC GR 9/6/9 8

29 Equiy CM6AUD GR Equiy CM6AUE GR Equiy EB5EKE Equiy EB5D93 Equiy EB5D94 Equiy EB5EKF Equiy DB4L87 Equiy DB4L88 Equiy GR GR GR GR GR GR CM6AUF GR Equiy CM6AUG GR Equiy CM6AUH GR Equiy CM6AUJ GR Equiy CM6AUK GR Equiy CM6AUL GR Equiy CM6AUM GR Equiy EB5E Equiy EB5E3 Equiy EB5E4 Equiy GR GR GR 9/6/9 85 9/6/9 95 9/3/9 8 9/3/9 9/3/9 3 9/3/9 9 //9 //9 55 /6/9 7 /6/9 75 /6/9 8 /6/9 85 /6/9 9 /6/9 95 /6/9 3/3/ 8 3/3/ 9 3/3/

30 6. REFERENCES [] Ai-Sahalia Y a Lo A W (998: Noparamric simaio of sa-pric siis impli i fiacial ass prics.joural of Fiac 53(: [] Bahra B (997: Impli risk-ural probabiliy siy fucios from opios prics.bak of Egla workig apr(66 [3] Baxr M a Ri A (999Fiacial Calculus. Cambrig Uivrsiy rss Cambrig. [4] Bliss R R a aigirzoglou N (: sig h sabiliy of impli probabiliy siy fucios.joural of Bakig a Fiac6(3 [5] Carr (995 h Valuaio of Amrica Exchag Opios wih Applicaio o Ral Opios. I rigorgis L:Ral Opios i Capial Ivsm: Nw Coribuios.ragr ublishrs Wspor C 9-. [6] Glazr E a Schichr (3:Mollig h Impli robabiliy of Sock Mark Movms.ECB Workig apr( [7] Horahl (999: Esimaig h impli isribuio of h fuur shor-rm irs ra usig h Logsaff-Schwarz mol.ecb Workig apr [8] Hull J C (6 Opios Fuurs a Ohr Drivaivs. 6 ric Hall [9] Joshi M (5 h Cocps a racic of Mahmaical Fiac. Cambrig Uivrsiy rss Cambrig [] Maa D B a Mil F (994: Coig claims valu a hg by pricig a ivsig i a basis. Mahmaical Fiac [] Mlick W R a homas C (994: Rcovrig a Ass's Impli DF from Opios rics:a Applicaio o Cru Oil urig h Gulf Crisis.Fral Rsrv Boar workig papr [] Rubisi M (994: Impli Biomial rs. Joural of Fiac 69 ( [3] Shimko D C (993: Bous of probabiliy.risk 6(