MATURITY AND VOLATILITY EFFECTS ON SMILES

Transcription

1 5// MATURITY AND VOLATILITY EFFECTS ON SMILES Or Dyng Smlng? João L. C. Dqe Unversdade Técnca de Lsboa - Inso Speror de Economa e Gesão Ra Mgel Lp,, LISBOA, PORTUGAL Paríca Texera Lopes Unversdade do Poro Facldade de Economa do Poro BDP Bolsa de Dervados do Poro Av. da Boavsa 4, 4 PORTO, PORTUGAL

2 MATURITY AND VOLATILITY EFFECTS ON SMILES Or Dyng Smlng? Absrac The smle effec s a resl of an emprcal observaon of he opons mpled volaly wh he same expraon dae, across dfferen exercse prces. I descrbes a U-shape form showng hgh mpled volales for n and o-of-he-money opons and low volaly fgres for a-he-money opons. We can fnd emprcal evdence of hs phenomenon. The reasons sggesed n he lerare were sochasc volaly, raders' behavor, ransacon coss, and he effec of dvdends on prcng Amercan opons. B he mos recen lerare seems o conclde ha he sophscaon of fnancal modellng for opon prcng s no enogh for removng he smle. In hs paper we sed lqd eqy opons on 9 socks raded on he London Inernaonal Fnancal Fres and Opons Exchange (LIFFE) beween Ags 99 and December 99. We esed wo dfferen hypohess ryng o verfy he exsence of wo dfferen phenomena: () he ncrease of he smle as mary approaches; () and he assocaon beween he smle and he volaly of he nderlyng sock. In order o esmae mpled volales for navalable exercse prces, we modelled he smle sng cbc B-splne crves. We fond emprcal sppor for he smle nensfcaon (he U-shape s more prononced) as mary approaches as well as when volaly rses. However, hs ncrease n he crvare s asymmerc. As mary approaches he mpled volaly of o-of-he-money opons ends o be hgher han he mpled volaly of n-hemoney opons and, as he volaly of he nderlyng ncreases, he mpled volaly of n-he-money opons end o be hgher han mpled volaly of o-of-he-money opons. We clam o have deeced new emprcal reasons for prevos emprcal fndngs where he smle was, for some ahors, a symmerc smle, whle for ohers convers no a wry grn or a reverse grn. Key words: opons, volaly, mpled volaly, smle effec. JEL classfcaon: G

3 MATURITY AND VOLATILITY EFFECTS ON SMILES Or Dyng Smlng?. Inrodcon The so called smle effec s a resl of an emprcal observaon of he opons mpled volaly wh he same expraon dae, across dfferen exercse prces. I ypcally descrbes a U-shape form showng hgh mpled volaly paerns for n and o-of-he-money opons and low volaly fgres for a-he-money opons. We can fnd emprcal evdences of hs phenomena n Heynen [994], Taylor and X [994], Dqe and Paxson [994], Gemmll [99], Dmas, Flemmng and Whaley [998] and Peña, Serna and Rbo [997]. Some ahors presen heorecal reasons for sch evdence as n Hll and Whe [987], Taylor and X [994] or Heynen [994]. Among ohers, he sggesed reasons were sochasc volaly, raders behavor, ransacon coss, and he effec of dvdends on prcng Amercan opons. B he mos recen lerare seems o conclde ha he sophscaon of fnancal modellng for opon prcng s no enogh for removng he smle. Ths paper refnes he emprcal analyss of he smle, ryng o esablsh sascally sgnfcan lnks beween some varables whch mpac on opon valaon and he shape of he smle. We sed cbc B-splne crves n order o nfer he mpled volaly smle and observe how he smle changes wh he approach of expraon as well as changes n volaly. We checked for he smle asymmery and lnked me o mary, volaly and he smle U-shape paern. We beleve ha descrbng he smle and s assocaon wh oher varables whch mpac on opon prcng, may modesly conrbe o he refnemen of fre opon prcng models. The paper s organsed as follows. We sar by revewng he relaed lerare and some heorecal relaons beween volaly, me o mary and prcng bas. Nex we descrbe he daabase and presen he mehodology n se. Fnally we presen he emprcal resls and a shor conclson. We hank Professor Dean A. Paxson, Anóno Sosa Câmara, Ser-Hang Poon and he aendees of he h EFA Annal Conference for helpfl commens on prevos versons of hs paper.

4 . Lerare revew When marke prces are sed o esmae mpled volales sng he Black and Scholes [97] model, hese mpled vales end o dffer across opon seres. Typcally, when mpled volales are ploed agans a moneyness rao, a-he-money opons end o have lower vales han n or o-of-he-money opons. Ths s he well known smle effec, sally descrbed as showng a U-shape form and emprcally deeced by dfferen ahors for opons on dfferen nderlyng asses. Alhogh he smle effec as a research opc s a relavely recen one, early sdes on opon prcng heory had already noced possble opon model bas. In her semnal paper, Black and Scholes [97] presened evdence for bas on opon vales when comparng marke warran prces o heorecal prces reslng from her model. Laer, Macbeh and Mervlle [979], sdyng sock opons lsed on he CBOE, fond evdence ha he Black and Scholes [97] model was nderprcng n-he-money opons and overprcng o-of-he-money opons. However, Rbnsen [985], also sdyng sock opons lsed on he CBOE, fond some confsng paerns. I seemed ha he Black and Scholes [97] model was overprcng o-of-he-money opons and nderprcng n-he-money opons for a me perod beween Ags 97 and Ocober 977. However, he same model was overprcng n-he-money opons and nderprcng o-of-he-money opons for a me perod beween Ocober 977 and Ocober 978. More recenly, oher sdes show emprcal evdence for hs exercse prce bas. Clewlow and X [99], sdyng opons on sock ndex fres lsed on he CME, fond asymmerc paerns for he smle. Heynen [994] also fond emprcal evdence for he exercse prce bas when observng sock ndex opons drng 989 on he EOE. Taylor and X [994], sdyng crrency opons raded n Phladelpha beween 984 and 99, fond emprcally ha opon bas was wce he sze hey were expecng, ncreasng s magnde as mary approaches. Dqe and Paxson [994] also fond he smle effec for opons raded on LIFFE drng March 99, and speclaed ha here s a possble emprcal relaon beween me o mary and he U-shape forma of he smle. Gemmll [99] also fond he same effec for opons on he FTSE drng a 5 year perod (from 985 o 99), alhogh he smle showed dfferen paerns B-splne crves n her cbc verson. 4

5 for dfferen days exraced from he sample. Dmas, Flemng and Whaley [998] also fond emprcal smle paerns for opons on he S&P 5 sock ndex, b s shape seemed o be asymmerc and changng along me o mary. Peña, Serna and Rbo [997] also fond emprcal smles for sock ndex opons wren on he Ibex 5 lsed on MEFF, for a me perod beween 994 and 99, deecng a day of he week effec for he smle. Oher ahors red o exrac speror nformaon from he smle as n Baes [99], concldng ha perssence on he smle paern cold mply ha he marke expeced he 987 crash. In order o accommodae hs emprcal bas, researchers have followed wo dfferen approaches: developng more sophscaed models ryng o compensae for ha effec; and modellng he bas self. Dfferen reasons have been sggesed for opon prcng bas sch as he sochasc volaly of he nderlyng sock, he observed changes n neres raes or he dfference beween he observed and he assmed sock prce pah (lke he jmp processes). Meron [97], Hll and Whe [987], Sco [987], Wggns [987], Johnson and Shanno [987], Sen and Sen [99], Baes [99], Baksh and Chen [997], Cox [99], Corrado and S [99], Madan, Carr and Chan [998], are examples of models where he basc Black-Scholes assmpons are dropped. Gkamas and Paxson [999], comparng several opons prcng models assmng sochasc volaly, fond ha hey remove a sgnfcan par of he bas, b hey dd no remove compleely. Some oher ahors mpe he smle o he behavor of raders or o her rsk averson, sch as Booksaber [99], Grossman and Zho [99], Gemmll [99], Dmas, Flemng and Whaley [998] and Gemmll and Kamyama [997]. Some ohers arbe he smle o ransacon coss, lke Clewlow and X [99] and [99], Consanndes [997] and Peña, Serna and Rbo [997]. Fnally, Geske and Roll [984] clam ha smles are a resl of errors n valng Amercan opons on socks payng dvdends pror o mary. We conclde ha he lerare s no nanmos n fndng cases for he smle effec and he models developed n order o cover hs bas have only parally solved he problem. When dscssng he model for prcng opons on socks, assmng boh sock prce rerns and sock rern volaly follow a geomerc Brownan moon process, Hll and Whe [987] fond ha he heorecal prces wold generally dffer from he 5

6 Black and Scholes solon. These dfferences cold be based on several varables. However, some were dscssed n deal: he volaly, he me o mary and he correlaon beween he nderlyng prce rern and he nsananeos sock prce rern volaly. The same dea was laer prsed n Hll and Whe [988] when a dfferen sochasc process was proposed o govern he nsananeos sock prce rern volaly. In hs paper Hll and Whe [988] derved an approxmae eqaon for he prcng bas when he Black and Scholes [97] eqaon s sed. Le s assme ha boh, he sock prce rerns and s volaly follow sochasc processes, sch as ds S d d dz Eq. a b d dw Eq. where S sands for he nderlyng sock prce, represens he nsananeos sock prce rern volaly and represens he volaly of he volaly. The parameers a, b and are consans and dz and dw are Wener processes wh an nsananeos correlaon. The varance n he mean reverson model revers o he long-erm a varance a a reverson rae of -b. The model sggesed by eqaon s b qe general, snce adms several redced forms for he varance: Parameer Model Conseqence = d a b d he varance s no consan b deermnsc a = d b d dw he varance s sochasc wh a consan proporonal drf b = d a d dw he varance s sochasc wh a consan drf a = and b = d dw he varance s sochasc wh no drf a > and b < d a b d dw he varance s sochasc followng, a mean reverng process In he las model, as volaly changes, he expeced varance a me s condonal on he acal nsananeos varance observed a me (s) s: E b s, e s Eq.

7 where represens he long erm mean o whch he process revers and represens he acal varance observed a me. For sock prces, a range of [-, -] s reasonable for b and a range of [.5,.] s reasonable for. Assmng he sochasc processes gven by eqaons and, Hll and Whe [998] showed ha he Black and Scholes model shold bas he call opon heorecal prce. The bas was defned as he dfference of he observed opon premm (calclaed accordng o he sochasc volaly model based on eqaons and ) and he Black and Scholes heorecal prce: B c C BS Eq. 4 The bas can be expressed n erms of an expanded eqaon as a fncon of whch represens he volaly of he volaly: B f f f... Eq. 5 If we assme s small, we may approach he bas vale by droppng he hrd and followng erms of eqaon 5 who losng oo mch precson. Therefore, he bas eqaon comes: B f f Eq. where: f C C a b e e a e S C Eq. 7 f Eq. 8 C b N' d S d S d ln S X r T T d d bt T 7

8 8 C s he Black and Scholes opon prce for a gven nsananeos varance, and s he average expeced varance rae n he nerval (T-) a me. Ths can be expressed as:, E T b T e T b ds s T Eq. 9 Eqaon 9 ells s ha when he acal nsananeos volaly ( ) eqals he long-erm average volaly ( ), s expeced average condonal on he presen level eqals he long-erm average volaly. In sch a case and C C. Then, eqaon redces o: f B Eq. Anoher parclar saon may be explored assmng ha. In sch a case f and, n order o mprove he approxmaon, we may need an exra erm n eqaon. f f B Eq. where 4 4 C e e a e e b a T b f Eq. ' 4 d d d N T S C Eq. and he res defned as prevosly. B, n general, for all oher saons, eqaon s appropraed: ' d d N S e a e e b a b C C B Eq. 4 From eqaon 4, we observe ha here are several varables and parameers wh a drec nflence on he bas. As noed by Hll and Whe [987] and [988], he

9 bas s affeced by he correlaon beween he sochasc processes, he me o mary T, he nsananeos volaly and he exercse prce X. We shold now be n place o remove he bas when prcng opons sng he Black and Scholes eqaon and, herefore, he so-called "smle effec" shold no be observed. Assmng ha he model sggesed by Hll and Whe [988] capres he marke premm of eqy opons, he bas obaned by eqaon 4 shold dffer accordng o he moneyness n order o accommodae he emprcal fndng known as he smle effec. Table shows he resls of he comped bas when prcng opons wh a me o mary rangng from monh o year, an nderlyng sock prce wh an nsananeos volaly rangng from.4% o 44.7% (. 5,. and. ), for fve dfferen sock prce levels ( S 7., S 94., S., S. and S. ). For he remanng assmpons we assmed a. 5, b 5,. 5, r f 4% and X.. We assmed a negave as a resl of prevos emprcal research fond n Chrse [98] and Kon [984]. Table - Bas ncrred when prcng opons accordng o he Black and Scholes eqaon f call premms are comped wh a mean reverng sochasc volaly process BIAS S/X T m e o M a r y =.5,,,48,88,4 / =. -, -,97 -,,87,7 =. -,49 -,77 -,4 -,9 -,5 =.5,99,585,77,58,59 / =. -,54 -,54 -,8,87,74 =. -,95 -,79 -,49 -,9 -,4547 =.5,854,884,98,877,87 =. -,99 -,4 -,49,,89 =. -,4 -,778 -,8597 -,85 -,9 9

10 Then we comped he nsananeos sock prce rern volaly ha wold resl n an nsgnfcan bas B (see Table ). Table - Impled volales comped n order o remove he bas when prcng opons accordng o he Black and Scholes eqaon f call premms are comped wh a mean reverng sochasc volaly process S/X IMPLIED VOLATILITY T m e o M a r y =.5,8%,85% 4,% 4,7%,% / =. 9,9%,%,%,8%,4% =. 4,99% 4,54% 4,% 4,7% 4,98% =.5 7,5% 8,4% 8,57% 8,7% 9,7% / =.,88%,49%,%,7%,% =.,5%,88%,9% 7,% 7,8% =.5 9,47% 9,9%,%,9%,9% =.,5%,5%,%,8%,94% =. 4,% 4,54% 4,% 4,% 4,85% As observed by Hll and Whe [988] when s negave nsead of a symmerc smle we observe, n mos of he cases, a consan bas where mpled volales of n-he-money opons are hgher han mpled volales of o-of-hemoney opons. However, when mary approaches, we clearly observe a decreasng U-shape smle as volaly ncreases. Therefore, apar he correlaon effec, we shold expec ha he smle effec ncreases as he mary approaches and decreases as volaly ncreases. We fel ha some emprcal work cold be done n erms of showng how some of hese varables cold affec observed volaly paerns, n parclar for opons wren on eqes. Ths paper explores a daa se of eqy opons and ess wheher me and volaly confrms wh he expeced paerns sggesed n he lerare.

11 . Daa The orgnal daabase consss of 7,8 daly call opon qoes on nne heavly raded socks, raded on he London Inernaonal Fnancal Fres and Opons Exchange (LIFFE) beween Ags 99 o December 99. Then we seleced a smaller sample of 979 call opons sng a fler rle reqrng a leas 5 exercse prces per mary. The frms seleced were Amsrad, Brsh Arways, Brsh Gas, Brsh Perolem, Brsh Telecom, Fore, General Elecrc, Hanson and Rolls Royce. The opon premms were colleced from Daasream, he nderlyng sock prces smlaneos wh he closng qoaons were rereved from he Daly Offcal Ls pblshed by he Sock Exchange. The opon premms are average closng qoed prces (bd + ask)/. The dvdends were assmed known and he necessary adjsmens were done sng he Black [975] model. As call opons on socks raded n LIFFE are Amercan syle, we exclded any opons where early exercse cold be opmal. The daabase was, on average, an a-he-money opon (wh a moneyness degree of ) and wh a me o mary of 5 days. Wh he avalable daa we cold, heorecally, draw,9 smles, whch gves, on average, 5.75 exercse prces per mary. However, as he avalable exercse prces are no cenred on he a-he-money opon and, as reqred by he mehodology, we waned o selec opons wh a moneyness of.94 and., some observaons were dropped for some regressons. Therefore, we obaned 74 n-he-money half-smles, 9 o-of-he-money half-smles and 547 complee smles. 4. Mehodology As a resl of he analyss of eqaon 4, hs paper explores wo man possble emprcal hypoheses. Frs we conjecre ha he smle has an exercse prce bas permanenly observed nl opons mare. However s shape changes as expraon approaches. Secondly we beleve ha he smle shape s also relaed o he volaly of he nderlyng asse. Fnally we combne boh ndependen varables (me and

12 volaly) o observe how sgnfcan s he resdal bas on he smle afer exracng he mary and he volaly effec. The frs hypohess s based on Clewlow and X [99], Heynen [994], Taylor and X [994] and Kraka [998]. Accordng o he dscsson of eqaon 4 and accordng o hese ahors, mpled volales shold ncrease n exercse prce bas, as opons approach mary. Therefore we se p he frs hypohess sang ha he smle magnde ncreases as mary approaches. However, n oher sdes, no specal aenon s gven o he symmery of he smle, whch we explore. In order o measre he smle magnde ndependen of he exercse prce bas, we wold lke o esmae a seres of he mpled volaly for opons a fxed moneyness degrees. We defne he moneyness degree of an opon (Mnss) as: Mnss S NPVDv rt Xe Eq. 5 where S represens he nderlyng sock prce, X s represens he exercse prce, r he rsk free neres rae, T- represens he me o mary and NPV(Dv) he ne presen vale of he dvdends pad on he sock nl expraon. The moneyness degree of ahe-money opons eqals, for o-of-he-money call opons wll be lower han and for n-he-money opons wll be greaer han. We are parclarly neresed n esmang he mpled volaly of hree specfc moneyness degrees, sch as: Mnss =.; Mnss =. and Mnss =.94. As he nderlyng sock prce moves sochascally and he exercse prces are consan, daly observed moneyness of qoed opons for he avalable exercse prces change. Therefore, s mpossble o exrac daly mpled volales wh hose exac moneyness degrees, drecly from he avalable opon premms. In order o overcome hs problem, we sed a B-splne crve n s cbc verson o esmae he desred mpled volales 4. B-splne crves are helpfl nsrmens o draw non-lnear fncons from an rreglar daa se. In order o draw a pece of he desred fncon we need 4 daa pons. Wh hese 4 daa pons, he cbc verson of he B-splne enables s o draw a crve for he nerval conaned beween he nd and he rd pon. Takng n daa pons, where Please see Dqe [994] for adonal deals on hs daabase. 4 Clewlow and X [994] had already sed hese fncons for drawng smoohed crves when sdyng he smle effec.

13 each pon y x p, wh n,...,,, he cbc B-splne for he nerval, p p wh,..., n, s k k k p b B where b, b, b, b, Eq. For each, x- and y-coordnaes, we oban a ransformed vale whch becomes: 4 x x x x x Eq. 7 4 y y y y y Eq. 8 When sdyng he smle effec, he x- and y-coordnaes become, respecvely, he moneyness degree and he mpled volaly. From a se of fve or more avalable exercse prces, we comped he esmaed mpled volaly accordng o he cbc B- splne fncon for hree dfferen moneyness degrees, for each day and for each mary. Assmng ha money he n, mp - denoes he esmaed n-he-money mpled volaly of opon, sng a cbc B-splne crve for a moneyness degree of.94,

14 mp, a he money - denoes he esmaed a-he-money mpled volaly of opon, sng a cbc B-splne crve for a moneyness degree of, mp, o of he money - denoes he esmaed o-of-he-money mpled volaly of opon, sng a cbc B-splne crve for a moneyness degree of.. We defned wo measres for he smle magnde. The frs measre (U n ) accons for he n-he-money bas whle he second (U o ) accons for he o-of-he-money bas: Un n he money a he money mp, mp, Eq. 9 Uo o of he money a he money mp, mp, Eq. Boh measres gve he absole dfference beween wha we wold expec o be he mpled volaly of he a-he-money opon wh a moneyness degree of and wha we wold expec o be he mpled volaly of an n(o)-he-money opon, assmng he exercse prce bas follows a cbc B-splne. As boh measres are paral measres of he smle, only acconng for one of s sdes, we call hese measres redced measres of magnde. As prevosly explaned, we have a daabase of 74 U n (nhe-money half-smles ) and 9 U o (o-of-he-money half-smles ). As an llsrave example, we seleced he opon seres qoed on Fore on he rd of Janary of 99, exprng on he h of March, wh 8 days o mary. There were 7 exercse prces wh acve qoes and posve open neres. Table shows he fgres of he mpled volales exraced from he qoed opons for all avalable exercse prces. Table - Moneyness degree and observed mpled volales for sock opons on Fore, exprng on March, 99 and calclaed on Janary, 99 Moneyness degree Impled Volaly As we prevosly noced, hese are nlkely o be opons wh a moneyness degrees of exacly., and.94! In sch a saon mpled volales for hose 4

15 moneyness degrees shold be esmaed from he 7 observed mpled volales. Therefore, he esmaed mpled volales sng a cbc B-splne crve adjsmen are:.. 59 mp,.. 45 mp, mp,. Ths example also shows how a-he-money mpled volaly may no be he absole mnmm of he smle crve. Fgre, represens he dfferences, js noed, beween smoohed esmaed mpled volales obaned from he cbc B-splne adjsmen and he observed mpled volales, exraced from marke daa. The crve represens he esmaed pons sng a cbc B-splne crve drawn from some known daa pons. From he crve, s possble o nfer mpled volales for opons wh specfc moneyness raos. Fgre Observed and Esmaed Impled Volales The dos represen he observed mpled volales whle he pons of he crve represen esmaed mpled volales sng a B-splne crve fng. We also comped wo addonal esmaes for he smle magnde: U a..94 mp, mp, mp,. Eq. 5

16 U G. mp,.94, mp, x Eq. mp, Eqaon gves a measre for he average deph of he smle, akng no accon boh he n-he-money and he o-of-he-money esmaed mpled volales, and was comped as an addonal measre for he smle magnde. As akes no consderaon boh sdes of he smle, we call hs, he complee measre of magnde. Eqaon was comped n order o sdy he smle skewness 5. If U G he smle s symmerc, oherwse, wll be eher asymmerc posve (when U G ) or asymmerc negave (when U G ). The second se of hypohess was also based on he heorecal mplcaons of eqaon 4. Accordngly, we rased he hypohess ha volaly has an mpac on he smle effec. Ths means ha when he nderlyng sock prce becomes more nsable, he exercse prce bas wll vary. Fnally, we hypohessed ha boh me o mary and he volaly of he nderlyng sock explan he U-shape forma of he smle. 5. Emprcal Resls 5.. Mary and Smle Effec We sared by esng wheher me o mary s relaed o he smle. In order o do so, we regressed me on all measres of magnde prevosly defned by eqaons 9, and. These regressons are expressed by eqaons, 4 and 5. U U U o, (T ) Eq. n, (T ) Eq. 4 a, (T ) Eq. 5 5 We chose UG snce hs measre was sed by Gemmll [99].

17 The resls are presened n Table 4. T-ess presened n Table 4 are already correced for heeroscedascy accordng o Whe [98] esmaor for he varance marx of he leas sqares esmaors. Table 4 - The smle magnde and me o mary Uo, (T ) Un, (T ) Ua, (T ) rao P-vale E-5* -.4E-5* -.5E-5* -rao P-vale... F Sa * 7.94*.55* R * - These resls are sascally sgnfcan a a 95% confdence level. Ths able shows ha all coeffcens are sascally sgnfcan wh for all he ess. As me o mary redces, he magnde of he smle ncreases. Ths paern s common o he hree measres of magnde. Addonally, we conclde ha he bas for o-of-he-money opons ncreases faser wh he decrease of mary han for n-he-money opons. Ths can be obaned from esng wheher from regresson eqaons and 4 sascally dffer from each oher. As a conclson, we fond emprcal sppor for he heorecal developmens repored n he lerare, where opons seem o de smlng. However, as js noced, he exercse prce bas seems o have a dfferen behavor accordng o he opons me o mary. Therefore n order o deermne wheher he asymmery s relaed o mary, we regressed me o mary on he symmery smle measre gven by U G. U G, (T ) Eq. If he slope of hs regresson eqaon s posve hen, when mary approaches, o-of-he-money opon mpled volales ncrease relave o n-hemoney opon mpled volales. Table 5 shows he resls. 7

18 Table 5 - The smle skewness and me o mary UG, (T ) rao P-vale..* -rao.44 P-vale.9 F Sa. 4.4* R.58 * - These resls are sascally sgnfcan a a 95% confdence level. As hypohessed, he mary of he opons end o be assocaed wh an ncrease of he exercse prce bas for boh n and o-of-he-money opons, b o-ofhe-money opons end o become more based. Addonally, we also observe ha when me o mary s greaer han 5, s expeced n-he-money mpled volales wll be greaer han o-of-he-money mpled volales. B for shorer me perods, mpled volales of n-he-money opons are lower han o-of-hemoney opons mpled volales. Therefore, a possble explanaon for prevos emprcal fndngs on he smle shape s me o mary. In fac, he mary approach changes he opons smle asymmery, converng a wry grn ypcal for longer erm seres no a reverse grn for nearly exprng opons, wh a more or less symmerc smle for mddle erm opons. Fgre gves an dea of he possble shape for he smle n a dmensonal doman. Insde he same range of moneyness [.94;.] he bas changes accordng o me o mary. 8

19 Fgre The Smle Asymmery along Tme o Mary For longer-erm opons (T-=7 days) he U-shape paern of he smle shows mpled volales of n-he-money opons hgher han mpled volales of o-of-he-money opons. For shorer-erm opons (T-=4) he smle paern reverses: mpled volales of o-of-he-money opons are hgher han mpled volales of n-he-money opons. Alhogh he paerns fond seem o dffer from he paern sggesed by Hll and Whe [988], some smlares may be observed. In fac as mary approaches he smle becomes deeper wh o-of-he-money mpled volales hgher han n-hemoney mpled volales (see Table ). However, even for longer erm opons he smle sll persss, b n-he-money opons have hgher mpled volales. 5.. Volaly and Smle Effec Accordng o Table, he smle s expeced o ncrease wh he decrease of he nderlyng sock prce volaly. Ths shold be a conseqence of eqaon 4. As wh prevos ess, we examned wheher relave and complee measres for he smle magnde, are lnearly relaed o he volaly of he nderlyng sock prce. As a proxy for he nderlyng sock prce, we seleced he esmaed mpled volaly for he a-hemoney seres assmng he cbc B-splne fng. Uo, mp, a - he - money Eq. 7 9

20 Un, Ua, mp, a - he - money Eq. 8 mp, a - he - money Eq. 9 n Table. The resls dd no show sgns of heeroscedascy and he fgres are presened Uo, Table - The smle magnde and volaly mp, ATM U ATM U ATM n, mp, a, mp, rao P-vale *.547* rao P-vale F Sa..57* *.75 R * - These resls are sascally sgnfcan a a 95% confdence level. The resls show ha when redced measres of he smle magnde are sed, we fnd a sascally sgnfcan posve relaon beween he nderlyng sock volaly and he smle. Ths means ha when volaly ncreases, he exercse prce bas ends o ncrease, for boh n and o-of-he-money opons. Ths s he oppose of wha we shold expec from eqaon 4 when he smlaons gven n Table are consdered. However, we dd no fnd a sascally sgnfcan dfference beween he slopes of regresson eqaons 7 and 8, wh a sgnfcance level of 5%. Therefore, apparenly, we can no expec a sgnfcan asymmery when comparng he exercse prce bas of n and o-of-he-money opons, as we observed for he mary effec. In order o confrm hs sspcon, we sed he symmery es, as prevosly, b adaped o he volaly effec on he smle. UG, mp, a - he - money Eq.

21 If he slope of hs regresson eqaon s posve, hen when volaly ncreases, n-he-money opon mpled volales ncrease relave o he o-of-he-money opon mpled volales. Table 7 shows he resls. Table 7 - The smle skewness and volaly UG, mp, ATM rao P-vale..85* -rao P-vale. F Sa. 7.59* R.489 * - These resls are sascally sgnfcan a a 95% confdence level. The resl, slghly nexpeced, shows a sgnfcan effec of nderlyng sock volaly on he ncrease of he exercse prce bas. For hgh volales, he relave dfference beween n-he-money and o-of-he-money esmaed mpled volales ends o be hgher han for lower volales. The resls also enable s o conclde ha Û G (he expeced asymmery measre for a gven volaly) only rns posve when volaly reaches 5.4% whch s a qe hgh vale. Therefore, seems ha n mos of he cases of or daabase, o-of-he-money esmaed mpled volales are hgher han n-he-money esmaed mpled volales. These fndngs are also a possble explanaon for prevos nconssences n mpled volaly analyss. The fndngs show an oppose effec o he expeced bas gven by eqaon 4 and consderng he daa of Table. The relave dfference beween n and o-of-he-money opons decreases as volaly ncreases for all of he me-o-mares observed.

22 5.. Tme o Mary, Volaly and Smle Effec As a resl of he conclsons obaned n secons 5. and 5. we hypohessed ha boh varables (me o mary and he volaly of he nderlyng sock) cold explan he smle n a mlple lnear regresson model: Uo, T a - he - money mp, Eq. Un, Ua, T - a - he - money mp, Eq. T - a - he - money mp, Eq. n Table 8. The resls dd no show sgns of heeroscedascy and he fgres are presened Table 8 - The smle magnde and volaly Uo, T ATM Un, T ATM Ua, T ATM rao P-vale E-5* -.87E-5* -.5E-5* -rao P-vale *.4* rao P-vale F Sa. 4.89* *.974* R * - These resls are sascally sgnfcan a a 95% confdence level. The resls do no dffer sgnfcanly from wha we were expecng, mplyng ha for boh n and o-of-he-money opons, he mary and he volaly effec do no offse each oher. They boh seem o conrbe o explanng he smle effec. I s also neresng o observe ha, as we wold expec, o-of-he-money opons seem o be more sensve o changes n volaly and n me o mary han n-he-money opons.

23 . Conclsons The smle effec s a resl of an emprcal observaon of he opons mpled volaly wh he same expraon dae, across dfferen exercse prces. I ypcally descrbes a U-shape form showng hgh mpled volaly paerns for n and o-of-hemoney opons and low volaly fgres for a-he-money opons. We may fnd emprcal evdence for hs exercse bas n prevos research. Some lerare presens heorecal reasons for sch a bas as Hll and Whe [987], Taylor and X [994] or Heynen [994]. Reasons were sochasc volaly, rader s behavor, ransacon coss, and he effec of dvdends on prcng Amercan opons. B he mos recen lerare seems o conclde ha he sophscaon of fnancal modellng for opon prcng s no enogh o remove he smle. We seleced he Hll and Whe [998] opon prcng model, ha assmes volaly s a sochasc varable followng a mean reverson paern, o search for heorecal relaons ha shold exs beween he smle shape, me-o-mary and volaly. We sared by exracng a sample of call opons qoed on LIFFE on nne heavly raded socks reqrng a leas 5 exercse prces for each mary. Then, we fed B-splne crves o he exsen observed mpled volales o esmae nobserved opon mpled volales on specfc moneyness degrees. Ths paper refnes he emprcal analyss of he smle, ryng o esablsh sascally sgnfcan lnks beween some varables whch mpac on opon valaon and he shape of he smle. We red o analyse emprcally how mpled volaly smles vary wh he approach of expraon as well as wh changes n volaly. We conclded ha as me o mary redces, he magnde of he smle ncreases. Boh he absole and he relave dfference beween mpled volaly of n and a-he-money opons or o and a-he-money opons rse as expraon approaches. Frher, hs mary bas seems o be more evden for o-of-he-money opons han for n-he-money opons, showng an asymmerc paern. As a frs conclson, we fond emprcal sppor for he heorecal developmens repored n he lerare, where opons seem o de smlng, however, dfferenly, accordng o he moneyness. When observng longer me o mary opons, we fond n-he-money mpled

24 volales larger han o-of-he-money mpled volales. B for shorer me perods, mpled volales of n-he-money opons become smaller han o-of-hemoney opons mpled volales. In fac, he mary approach changes he opons smle asymmery, converng a wry grn ypcal for longer erm seres no reverse grns for exprng opons, wh a symmerc smle for mddle erm opons. Ths conclson dffers from wha cold be expeced when Hll and Whe [998] model s consdered, especally when longer-erm opons are consdered. Then we checked for possble evdence on he volaly effec. In he lerare we fond developmens for a bas eqaon whch shows ha smles shold be volaly dependen, decreasng when volales ncrease. Or resls, n conras, show ha, when redced measres of he smle magnde are sed, we fnd a sascally sgnfcan posve relaon beween he nderlyng sock volaly and he smle. When volaly ncreases, he exercse prce bas end o rse, boh for n and o-of-he-money opons. As observed wh he mary effec, we also recognse an asymmerc behavor for n and o-of-he-money opons. For hgh volales, n-he-money opons end o evdence hgher esmaed mpled volales han for lower volales. Fnally we fond ha when me and volaly are combned we ncrease he explanaory power of he smle bas. Ths paper confrms prevos fndngs on he mary and volaly mpac on he exercse prce bas, also called he smle effec. However, addonally, we show emprcal evdence for an asymmerc behavor. Tme o mary and volaly seem o have a dsnc nflence on he smle effec when n and o-of-he-money opons are observe. We beleve ha n descrbng he smle and s assocaon wh oher varables, we conrbe o he refnemen of fre opon prcng models. 4

25 References BAKSHI, G. and Z. CHEN (997), An Alernave Valaon Model for Conngen Clams, Jornal of Fnancal Economcs, 44, pp. -5. BATES, D. (99), The Crash of 87: Was I Expeced? The Evdence from Opons Markes, Jornal of Fnance, 4, N., Jly, pp BATES, D. (99), Jmps and Sochasc Volaly: Exchange Rae Processes Implc n PHLX Deschemark Opons, Revew of Fnancal Sdes, 9, pp BLACK, F. (975), Fac and Fanasy n he Use of Opons, Fnancal Analyss Jornal,, pp. -7. BLACK, F. and M. SCHOLES (97), The Valaon of Opon Conracs and a Tes of Marke Effcency, Jornal of Fnance, 7, pp BLACK, F. and M. SCHOLES (97), The Prcng of Opons and Corporae Lables, Jornal of Polcal Economy, 8, N., May/Jne, pp BOOKSTABER, R. (99), Opon Prcng and Invesmen Sraeges, rd edon, Probs Pblshng, Chcago. CLEWLOW, L. and X. XU (99), A Revew of Opon Prcng wh Sochasc Volaly, FORC Preprn, 9/5, Unversy of Warwck. CLEWLOW, L. and X. XU (99), The Dynamcs of Sochasc Volaly, Workng paper, Fnancal Opons Research Cenre, Warwck Unversy, May. CONSTANTINIDES, G. M. (997), Transacon Coss and The Volaly Impled by Opons Prces, Workng Paper, Unversy of Chcago, Ocober. CORRADO, C. and T. SU (99), Skewness and Kross n S&P 5 Index Rerns Impled by Opon Prces, The Jornal of Fnancal Research, Vol. 9, N., Smmer, pp COX, J. (99), The Consan Elascy of Varance Opon Prcng Model, The Jornal of Porfolo Managemen, specal edon, pp CHRISTIE, A. A. (98), "The Sochasc Behavor of Common Sock Varances", Jornal of Fnancal Economcs,, December, pp DUMAS, B., J. FLEMING and R. WHALEY (998), Impled Volaly Fncons: Emprcal Tess, Jornal of Fnance, Vol. 5, N., December, pp DUQUE, J. (994), The Meanng of Impled Volaly n Prcng Sock Opons Traded n Opons Markes, PhD Dsseraon, Unversy of Mancheser. DUQUE, J. and D. PAXSON (994), Impled Volaly and Dynamc Hedgng, The Revew of Fres Markes,, N., pp GEMMILL, G. (99), Dd Opon Traders Ancpae he Crash? Evdence from Volaly Smles n he U. K. wh U.S. Comparsons, The Jornal of Fres Markes,, N. 8, pp GEMMILL, G. and N. KAMIYAMA (997), Inernaonal Transmsson of Opon Volaly and Skewness: When yo re smlng, does he whole world smle?, Proceedngs of he 4 h Annal Meeng of Eropean Fnance Assocaon, Venna, 7- Ags. GESKE, R. and R. ROLL (984), On Valng Amercan Call Opons wh he Black- Scholes Eropean Formla, Jornal of Fnance, 9, N., Jne, pp GKAMAS, Dmros and D. PAXSON (999), "Whch Sochasc Volaly Model for Opon Prcng?", Workng Paper, Mancheser Bsness School. GROSSMAN, S. and Z. ZHOU (99), Eqlbrm Analyss of Porfolo Insrance, Jornal of Fnance, 5, n. 4, Sepember, pp

26 HEYNEN, R. (994), An Emprcal Invesgaon of Observed Smle Paerns, The Revew of Fres Markes,, N., pp HULL, J. and A. WHITE (987), The Prcng of Opons on Asses wh Sochasc Volales, Jornal of Fnance, Vol. 4, n., Jne, pp. 8-. HULL, J. and A. WHITE (988), An Analyss of he Bas n Opon Prcng Cased by a Sochasc Volaly, Advances n Fres and Opons Research, Vol., pp. 9-. JOHNSON H. and D. SHANNO (987), Opon Prcng When Varance s Changng, Jornal of Fnancal and Qanave Analyss,, N., Jne, pp KON, S. J. (984), "Models of Sock Rerns - a comparson", Jornal of Fnance, 9, March, pp KRATKA, M. (998), No Mysery Behnd he Smle, Rsk, Aprl, pp MACBETH, J. and L. MERVILLE (979), An Emprcal Examnaon of he Black- Scholes Call Opon Prcng Model, Jornal of Fnance, December, pp MADAN, Dlp B., Peer P. CARR and Erc C. CHAN (998), The Varance Gamma Process and Opon Prcng, Eropean Fnance Revew, Vol., n., pp MERTON, R. (97), Opon Prcng When Underlyng Sock Rerns are Dsconnos, Jornal of Fnancal Economcs,, pp PEÑA, I., G. SERNA and G. RUBIO (997), Why Do We Smle? On he Deermnans of he Impled Volaly Fncon., Unversy Carlos III, Ocober. RUBINSTEIN, M. (985), Nonparamerc Tess of Alernave Opon Prcng Models Usng All Repored Trades and Qoes on he Mos Acve CBOE Opon Classes from Ags, 97 hrogh Ags, 978, Jornal of Fnance, 4, N. Jne, pp SCOTT, L. (987), Opon Prcng When Varance Changes Randomly: Theory, Esmaon and an Applcaon, Jornal of Fnancal and Qanave Analyss,, N. 4, December, pp STEIN, E. and J. STEIN (99), Sock Prce Dsrbons wh Sochasc Volaly: An Analyc Approach, Revew of Fnancal Sdes, 4, Wner, pp TAYLOR, S. and X. XU (994), The Magnde of Impled Volaly Smles: Theory and Emprcal Evdence for Exchange Raes, The Revew of Fres Markes, N., pp WHITE, H. (98), A Heeroskedascy-Conssen Covarance Marx and a Drec Tes for Heeroskedascy, Economerca, 48. WIGGINS, J. (987), Opon Vales Under Sochasc Volaly: Theory and Emprcal Esmaes, Jornal of Fnancal Economcs, 9, pp. 5-7.