Market Risk and the Concept of Fundamental Volatility:

Transcription

1 Marke Risk and he Concep of Fundamenal Volailiy: Measuring Volailiy across Asse and Derivaive Markes and Tesing for he Impac of Derivaives Markes on Financial Markes Soosung Hwang, Deparmen of Applied Economics, Universiy of Cambridge and Sephen E. Sachell * Faculy of Economics and Poliics and Triniy College Universiy of Cambridge Absrac This paper proposes an unobserved fundamenal componen of volailiy as a measure of risk. This concep of fundamenal volailiy may be more meaningful han he usual measures of volailiy for marke regulaors. Fundamenal volailiy can be obained using a sochasic volailiy model, which allows us o filer ou he signal in he volailiy informaion. We decompose four FTSE100 sock index relaed volailiies ino ransiory noise and unobserved fundamenal volailiy. Our analysis is applied o he quesion as o wheher derivaive markes desabilise asse markes. We find ha inroducing European opions reduces fundamenal volailiy, while ransiory noise in he underlying and fuures markes does no show significan changes. We conclude ha, for he FTSE100 index, inroducing a new opions marke has sabilised boh he underlying marke and exising derivaive markes. Keywords: Fundamenal Volailiy, Kalman Filer, Sochasic Volailiy Model, Implied Volailiy JEL Classificaion: C32, G18 * We would like o hank unknown referees, Professor M. J. Chambers, Professor A. C. Harvey, and Professor Peer Spencer for valuable commens and Inquire and he Newon Trus for financial suppor.

2 1. Inroducion Tradiionally, he efficien marke hypohesis views price volailiy as a resul of he random arrival of new informaion which changes reurns. However, empirical sudies such as Shiller (1981), Schwer (1989), and French and Roll (1986) sugges ha volailiy canno be explained only by changes in fundamenals. Significan amouns of volailiy in asse prices come from noise rading of irraional raders. From his poin of view, volailiy may be defined as he sum of ransiory volailiy caused by noise rading and unobserved fundamenal volailiy caused by sochasic informaion arrival. Our modelling of fundamenal volailiy in his paper assumes ha he fundamenal volailiy is an unobserved random variable; i changes hrough ime. There are many volailiies relaed o only one underlying asse which are measurable a a given ime poin: he reurn volailiy of he underlying asse, fuures reurn volailiy on he asse, and call and pu opion implied volailiies over various mauriies and exercise prices, ec. However, i is naural o assume ha here is only one fundamenal volailiy defined over he underlying asse and all is derivaives. This is because informaion which affecs he fundamenals of he underlying asse is he same across all derivaives of he asse and, hus, resuls in he same fundamenal volailiy. Oher facors will also influence his single fundamenal volailiy as well as informaion arrival: he srucure of relaed markes, he disribuion of asses held by invesors, ransacion coss and numerous oher facors in he global economy, including all he macroeconomic informaion available a he ime. This sudy does no address hese oher facors which may be imporan. Our decision o no include hem was driven by unavailabiliy of daa and he difficulies of specifying a plausible model ha covers all hese poins. 1

3 Our sudy proceeds by decomposing he FTSE100 sock index relaed volailiies ino ransiory noise and fundamenal volailiy and uilises he decomposiion o invesigae he effec of he inroducion of derivaives on he volailiy. Using he sochasic volailiy model (SVM) developed by Harvey and Shephard (1993, 1996) and Harvey, Ruiz, and Shephard (1994), we calculae he porion of ransiory noise in he observed volailiy (i.e., signal-o-noise raio), and are able o infer he fundamenal volailiy process and also he relaionship beween ransiory noises of differen volailiies. Our analysis reveals he following resuls. Noise in he opions marke is no correlaed wih noise in he underlying and/or fuures markes. However, he differen noises associaed wih differen opions conracs are correlaed wih each oher, and noise in he underlying marke is correlaed wih ha of he fuures marke. In addiion, fundamenal volailiy has a high degree of persisence, a feaure ofen observed in high frequency financial daa; see Engle and Bollerslev (1986). An ineresing area of sudy for volailiy is o invesigae he effec of he inroducion of derivaives on he underlying asse volailiy. In a fricionless no-arbirage world, derivaives are redundan asses and will no effec he underlying marke. However, in he real world where markes are incomplee, effecs of he inroducion of derivaives markes on he underlying marke exis. Derivaive markes may sabilise underlying markes by more efficien risk allocaion or desabilise underlying markes by increasing speculaion. Our sudy invesigaes he effecs of he inroducion of derivaives on he unobserved fundamenal volailiy and he ransiory noise of he FTSE100 index relaed volailiies. Fuures and American opions on he FTSE100 index were inroduced on 3 May 1984 and European opions were lised on 1 February We are no able o show 2

4 he effecs of he inroducion of fuures and American opions on he FTSE100 index volailiy, since he impac of inroducing wo derivaives a he same ime can no be separaed and he number of daily observaions before he inroducion of he derivaives is relaively small (i.e., 85 observaions). However, we find ha inroducing European opions reduced fundamenal volailiy, while he ransiory noise in he underlying and fuures markes did no show significan changes. On he basis of he evidence, we conclude ha, for he FTSE100 index, inroducing an opions marke sabilised he oher financial markes (ha is, underlying and derivaive markes). 2. Fundamenal and Noise Componens of Volailiy An observed volailiy series may be regarded as a combinaion of ransiory noise and permanen fundamenal volailiy. Empirical sudies such as Shiller (1981), French and Roll (1986), and Schwer (1989) show ha changes in he fundamenal value canno explain all of he price movemens in financial markes. Tha is, he observed volailiy series has noise. We define he volailiy caused by informaion as fundamenal volailiy and he volailiy caused by noise rading as emporary noise. Observed volailiy series may be regarded as a combinaion of ransiory noise and permanen fundamenal volailiy. On a given day many differen volailiies which are relaed o one underlying asse can be calculaed, e.g., underlying asse reurn volailiy (RV), fuures price RVs, opion implied volailiies (IVs). When informaion arrives, permanen componens of all volailiies will move in he same way. On he oher hand, ransiory componens of volailiies caused by noise rading, for example, may no behave in he same way. We shall assume ha here is only one rue permanen componen for he many volailiies which are relaed o one underlying asse, while here are muliple ransiory noises. Our 3

5 inenion is o sudy how hese measures behave. Le us consider opion IV. We expec he IVs of any se of opions on he same underlying asse o be idenical. However, when he Black and Scholes (1973) (BS) opion pricing formula is used, many differen IVs can be observed on he underlying asse for differen ime-o-mauriies and exercise prices. 1 The inconsisency beween heory and empirical findings may be explained by he invalidiy of BS opion pricing model. I migh be argued ha IVs from sochasic volailiy models appear less biased han he IVs from BS models and hus, more appropriae han he IVs from BS models. 2 However, sochasic volailiy opion pricing models also need an assumpion abou an explici volailiy process such as a mean-revering AR(1) specificaion which may no be he rue process. Therefore, he volailiies inferred from a sochasic volailiy model also may be biased due o misspecificaion in he underlying sochasic processes. Oher opion pricing models have a similar model specificaion problem in calculaing IVs. 3 In his sense, any opion pricing model oher han he rue model can no give us 1 Hull and Whie (1987) show ha in he Black and Scholes (1973) (BS) opion pricing formula, he longer he ime o mauriy, he lower he IV. This is a misspecificaion bias ha comes from he assumpion of consan volailiy in he BS model. The mauriy-specific variaion also reveals he erm srucure of IV; see Canina and Figlewski (1993), Resnick, Sheikh, and Song (1993), and Xu and Taylor (1994). For he differen IVs across exercise prices, several alernaive weighed average mehods ha are designed o aggregae he differen IVs ino a single IV have been used; see Laane and Rendleman (1976), Chiras and Manaser (1978), Schmalensee and Trippi (1978), Beckers (1981) and Whaley (1982). Recenly, a-hemoney IV ends o be used more frequenly; see Baes (1995) for a summary. 2 See pp , Campbell, Lo, and MacKinlay (1997) and Baes (1995) for furher discussions on he BS opion pricing model and he sochasic volailiy model. 3 We appreciae a referee for he commens discussed here. 4

6 he rue volailiy process implici in opion prices. In his sudy we use IVs inferred from he BS opion pricing model. We acknowledge ha he BS opion formula is a bes a convenien heurisic, bu all we need in his sudy is a measure of IV which is a proxy of volailiy dynamics and he IVs from BS opion pricing formula are one of he proxies, see Baes (1995). In any case, he IV repored by opion exchanges such as LIFFE is based on BS and is he saisic undersood and aced upon by raders. Besides he problems in he idenificaion of he rue opion pricing model, we also have measuremen errors in IV: inappropriae use of risk-free ineres raes, dividends and early exercise in American opions, non-simulaneous opion and sock price, bid/ask price effec, infrequen rading of he index, ec. For discussion on daa limiaions, see Harvey and Whaley (1991, 1992). Finally, we noe he suggesion of Brenner and Galai (1984) ha he IV based on he las daily observaions may be unreliable. Noing he above caveas, we assume ha a ime an IV of an underlying asse has he following relaionship wih unobserved fundamenal volailiy (FV) IV = FV + Noise, (1) IV The underlying asse reurn volailiy has differen properies from he implied volailiy. Observed implied volailiy is larger han underlying asse reurn volailiy and implied volailiy is smooher han underlying asse reurn volailiy; see secion 3. Laane and Rendleman (1976) show ha he correlaion beween implied volailiy and underlying asse reurn volailiy is no close o 1. In addiion, French and Roll (1986), using he difference in equiy volailiy beween rading and non-rading hours, show ha a significan porion of daily variance is caused by mispricing. Therefore, we represen he reurn volailiy of an underlying asse a ime, RV, as 5

7 RV = FV + NoiseRV, (2) Noice ha implied volailiy has he inerpreaion of an ex ane marke expeced reurn volailiy o opion mauriy, if he opion pricing assumpions are correc. However, since he unobserved fundamenal volailiy in he implied volailiy reflecs informaion which affecs he fundamenals of he underlying asse, we sugges ha he unobserved fundamenal volailiy in he reurn volailiy is he same as he unobserved fundamenal volailiy in he implied volailiy. Tha is, unobserved fundamenal volailiies are assumed o be he same across he underlying asse and is opions. Now, le us consider he relaionship beween he reurn volailiy of an underlying asse and ha of fuures. The no-arbirage fuures price can be denoed as F = S e (r -d ) f, τ, where F is he fuures price a ime, S is he underlying asse price a ime, d is he dividend yield, r f, is he risk-free rae a ime, and τ is he ime-omauriy. Then, upon aking logarihms of he no-arbirage fuures price equaion and differencing, fuures reurn volailiy (squared reurn) a ime, RV fuures,, and he underlying asse reurn volailiy (squared reurn) a ime, RV, have he following relaionship: 2 2 fuures, r, d, RV = RV + σ + σ + Cov = FV + Noise f fuures, (3) 2 where σ d, is he volailiy of changes in dividend yield, σ r 2 f, is he volailiy of changes in he risk-free ineres rae, Cov is he sum of he covariance iems beween underlying asse reurns, changes in dividend yield, and changes in he risk-free ineres rae, and Noise fuures 2 2, = σd, + σr, + Cov + NoiseRV,. Therefore, in his case, RV fuures, has f 6

8 he same common unobserved fundamenal volailiy as in (1) and (2). Furhermore, we would expec he wo noise erms o be correlaed as he fuures noise would conain elemens of underlying asse noise. The explanaion above assumes ha for an underlying asse, we can idenify only one unobserved fundamenal componen bu muliple ransiory noises from many observable volailiies of he underlying asse across differen markes. The seing requires us o use mulivariae models raher han univariae models. More formally, k observed volailiies relaed o one underlying asse can be assumed o have one FV as follows: V = FV e + ε (4) where V = [ V 1 V 2 V k ] k 1 underlying asse, e,,, is a vecor of observed volailiies which are relaed o one k 1 [1 1 1], and ε = [ ε1, ε2, ε, ] = k 1 k is a vecor of ransiory noises of observed volailiies. Equaion (4) is a mulivariae model bu wih only one unobserved process. The model is essenial o our perspecive, since i isolaes our scalar risk measure, i.e., FV. Facor models could be used o conrol oher significan changes in economy; any effec we find on volailiy may be due o macroeconomic facors. 4 In he GARCH class of models, facors can be included as in Engle (1987). However, he facor GARCH models have a large number of parameers, resuling in compuaional problems. Engle, Ng, and Rohschild (1990) and Bollerslev and Engle (1993) sugges simpler mehods o avoid he problem. In SVMs, facors can be included as in Harvey, Ruiz, and Shephard (1994), see 4 We would like o hank he referee for suggesing ha we discuss his approach as a possible exension o our procedure. 7

9 Secion 8.5, Harvey (1989) and Ruiz (1992) for deailed discussion. Leaving aside he compuaional issues, we urn nex o a discussion of macroeconomic daa. Alhough he above suggesion would in principle allow us o relae volailiy direcly o informaional announcemens, we would need o compile a daabase of macroeconomic announcemen over he relevan period. Using macroeconomic informaion wihou considering he announcemen effecs inroduces new problems of frequency; daily reurns and quarerly macroeconomic measures. 3. Daa Four daily volailiy series which are relaed o he FTSE100 sock index are used in his sudy 5 : FTSE100 sock index reurn volailiy, fuures reurn volailiy, American call opion implied volailiy, and European call opion implied volailiy. To invesigae he possible changes in he unobserved fundamenal volailiy and ransiory noise resuling from he inroducion of derivaives, we divide he enire sample period ino hree subperiods: before he inroducion of derivaives (he firs sub-period, from 1 January 1984 o 2 May 1984), afer he inroducion of American opions and fuures bu before he inroducion of European opions (he second sub-period, from 3 May 1984 o 31 January 1990), and afer he inroducion of all hree derivaives (he hird sub-period, from 1 February 1990 o 29 March 1996 ). FTSE100 sock index opion daa (boh American and European) from March 1992 are provided by he London Financial Opions and Fuures Exchange (LIFFE). 5 The FTSE100 FLEX(r) (European syle opion), which was inroduced on 30 June 1995, is no used in his sudy. The implied volailiy of he opion is difficul o obain because of he flexibiliy of he opion. 8

10 American opion price daa from May 1984 o March 1992 and European opion price daa from February 1990 o March 1992 are obained from he Sock Exchange Daily Official Lis. The implied volailiies of boh American and European opions are calculaed from he Black (1976) pricing formula for opions on fuures. Two disinc benefis come from using Black s opion pricing formula on fuures. Firsly, fuures and opions on he FTSE100 sock index have he same closing ime and hus he nonsimulaneous price problem (see Harvey and Whaley, 1991), arising from he difference in closing imes beween he sock marke and he derivaive marke, becomes rivial. Secondly, he expeced marke dividend rae embedded in fuures prices is used insead of he widely used ex-dividend rae. Harvey and Whaley (1992) repor large pricing errors in American opions when coninuous dividends are assumed in he S&P 100 index, suggesing ha discree and seasonal dividend paymens should be considered. However, using he fuures price on he FTSE100 raher han he FTSE100 index iself removes hese pricing errors. Therefore, implied volailiy using fuures prices is likely o be closer o he expeced marke implied volailiy, if such a concep is well defined. Baes (1995) suggess a-he-money implied volailiies as relaively robus esimaes of expeced average variances under a sochasic variance process. However, even hough a-he-money implied volailiy is used, he erm srucure of implied volailiy is difficul o remove, unless here are many available a-he-money opions of differen mauriies. Usually in his case, he volume is so low ha he prices are no longer rusworhy. To minimize he erm srucure effec of implied volailiy, he opions wih he shores mauriy bu wih a leas 15 working days o mauriy are used, as in Harvey 9

11 and Whaley (1991, 1992) 6. Opions which have he March cycle - March, June, Sepember, and December - are used. The Newon-Raphson algorihm on Black s model is used o calculae implied volailiy. We use he hree monh UK Treasury Bill for he risk-free ineres rae. The FTSE100 index fuures series was provided by LIFFE and he daily FTSE100 index series was obained from Daasream. As wih implied volailiy, he March cycle of fuures prices is used and, o remove possible erm srucure effecs in fuures, fuures prices wih he shores mauriy, bu wih a leas 15 working days o mauriy are used. Therefore, all derivaives used in his sudy have he same mauriy. The acual reurn volailiies of he FTSE100 index and fuures are calculaed by squaring he log-reurns of he index and fuures prices muliplied by 250 o conver o an annualized amoun 7. We emphasize ha we use variances, and hence squared reurns raher han sandard deviaions 8. Table 1 repors he saisical properies of each logarihmic volailiy series. Noe ha zero volailiies should be convered o posiive numbers when applying logarihms. The zero volailiies were convered o -15 for index reurn logarihmic volailiy (log-rv) 6 However, he effecs of he erm srucure of implied volailiy canno be removed compleely. This is a weakness in his sudy, alhough we aemp o minimize is impac. By working wih conracs of approximaely he same mauriy we can argue ha our analysis reas mauriy as fixed (cross-secionally) a a poin in ime bu is changing hroughou he cycle. 7 Square of log-reurns will resul in larger volailiies han he square of residuals from any logreurn process. 8 This is for consisency wih he sochasic volailiy model. However, sandard deviaions may also be used in he sochasic volailiy model, as suggesed in Fornari and Mele (1994). 10

12 and -12 for fuures log-rv, which are he minimum log-rvs when zero volailiies are excluded from each log-rv series. As expeced, logarihmic volailiies decrease kurosis and skewness. 9 However, fuures and index log-rvs show negaive skewness because of close-o-zero reurn volailiies. Alhough logarihmic implied volailiies (log-ivs) of he hird sub-period are far from normal (for he normaliy es, a criical value of 5.99 a 5% significance can be used for he Jarque and Bera (1980) (J&B) saisics in he able), applicaion of logarihms make he raw volailiy series closer o normaliy. Therefore, he saisical properies in Table 1 sugges ha log-volailiies migh be beer used in a linear modelling framework han volailiies hemselves. Some ineresing differences beween log-volailiies are found in Table 1. Firs of all, he mean of he log-rvs is smaller han ha of he log-ivs. This means ha he acual opions prices are higher han he opion prices obained by using index reurn volailiy as a volailiy measure. The overpricing phenomenon is found over all sub-periods. Anoher ineresing poin is ha he mean value of he fuures log-rv is larger han ha of he index log-rv. The covariance in equaion (3) is no large enough o offse he volailiy of changes in he risk-free rae and he dividend yield. On he oher hand, he wo log-ivs have almos he same saisical properies. As expeced, he log-ivs are srongly auocorrelaed and heir sandard deviaions are relaively small. The saisical properies are quie differen o hose of log-rvs. This can be explained by Hull and Whie (1987) who argue ha Black-Scholes implied volailiy can be regarded as an ex ane averaged volailiy o mauriy. The averaging procedure removes a large porion of noise, increases 9 Alhough i is no repored, all reurn volailiies are posiively skewed, lepokuric, and fail o show signs of normaliy. 11

13 he auocorrelaion, and makes he averaged process smooher han he unaveraged one. 4. Sochasic Volailiy Model Decomposiion of volailiies ino one fundamenal volailiy and noises can be carried ou wih GARCH models or sochasic volailiy models (SVMs). 10 We expec ha here is no significan difference in our analysis beween he wo models since consisen esimaes of a sochasic volailiy model can be obained wih GARCH models under cerain condiions, see Nelson and Foser (1994), Nelson (1996). However, he wo models are differen in he sense ha he SVM has been developed in erms of informaion arrival and is known o be consisen wih diffusion models for volailiy, while he GARCH model has been predominanly used o describe some sylised facs of volailiy, see Taylor (1994) and Ghysels, Harvey, and Renaul (1996). Noe ha SVM is a discree-ime srucural model of he geomeric diffusion process used by Hull and Whie (1987), where hey generalise he Black-Scholes opion pricing model o allow for sochasic volailiy. In his sudy, he SVM developed by Harvey and Shephard (1993, 1996) and Harvey, Ruiz, and Shephard (1994) is used o decompose observed volailiy ino unobserved fundamenal volailiy and ransiory noise as represened in secion 2. As explained above, we explain volailiy in erms of informaion arrivals in his sudy. In addiion, changes in he level of he fundamenal volailiy which are used for he invesigaion of he effecs of inroducion of derivaive markes, are hard o idenify in GARCH models, because a non-negaive ime rend included in he condiional volailiy equaion of GARCH models is usually no significanly differen from zero. The SVM suggesed by Harvey and Shephard (1993) may be represened by 10 See Taylor (1994) for a comparaive sudy on hese wo models. 12

14 u = σξ e 0. 5FVP FVP = φfvp 1 + η where u represens observed random residuals of a series (e.g., log-reurn series), σ is a posiive scale facor, ξ is an independen, idenically disribued random disurbance series, FVP is unobserved fundamenal volailiy process, and η is a series of independen disurbances wih mean zero and variance σ 2 η. When we ake logarihms of he squared residuals, he SVM can be represened as 11 log( u 2 ) V FVP = φfvp = log( σ + η 2 = µ + FVP 1 ) + FVP + ε + log( ξ ) 2 (5) 2 2 where V is a logarihmic value of he squared residual a ime, µ = log σ + E(log ξ ), 2 2 and ε = log ξ E(log ξ ) is a zero mean whie noise. The disurbance erm, ε, in (5) is no normal unless ξ is log-normal. When ξ is sandard normal, he mean and variance of logξ 2 are and In general, he disribuion of ε is no known, and i is no possible o represen he likelihood funcion in closed form. However, quasi-maximum likelihood (QML) esimaors of he parameers in (5) can be obained using he Kalman filer by reaing ε and η as normal. Ruiz (1994) suggess ha for he kind of daa ypically encounered in empirical finance, he QML for he SVM has good finie-sample properies. Equaion (5) assumes ha he fundamenal volailiy process follows an AR(1) process wihou a rend. Insead of a rend, we inroduce a consan, µ, which represens 11 We use he ime invarian SVM in his sudy. The ime invarian SVM is a SVM which has ime invarian parameers, bu whose value changes hrough ime. 13

15 he level of expeced volailiy in he measuremen equaion. This should no be misinerpreed as an assumpion of consan fundamenal volailiy. As menioned by a referee, he fundamenal volailiy may include a rend. Alhough our model does no accommodae his since equaion (5) only provides levels, i is esimaed over sub-periods which allow changes in he level. The changing volailiy levels over sub-periods can parly accommodae a rend in volailiy level. In addiion, he impacs of he inroducion of derivaive markes on he financial markes can also be invesigaed wih changes in volailiy levels over sub-periods. Therefore, he fundamenal volailiy (FV ) in secion 2 can be furher decomposed ino a volailiy level (µ) and a fundamenal volailiy (mean zero) process (FVP ) as in equaion (5). Noe ha we have only one fundamenal volailiy process in each period, while volailiy levels are differen across he four volailiy series used in his sudy. Precise mahemaical deails of our SVM processes (i.e., mulivariae SVMs and idenifiabiliy of he models) are given in Appendix. We presen resuls for AR(1), AR(2), and ARMA(2,1) exensions of equaion (5). I is assumed hroughou his paper ha FVP and ε are uncorrelaed. A referee has raised he poin ha in realiy he correlaion beween hese wo would be non-zero and probably posiive. We noe ha in hese srucural ime series models, i is possible o consider his case, see chaper 3 of Harvey (1989). Ineresingly, Harvey (1989) presens a ransformaion procedure which allows one o redefine ransformed signal and noise ha are uncorrelaed. If correlaion is presen, we inerpre our signal and noise as being hese ransformed variables, since our variables are unobservable. (We hank he referee for clarifying his poin). 14

16 5. Empirical Resuls 5.1 Esimaes of he SVM Esimaed SVMs using he FTSE100 sock index log-rv (univariae model) are in panel A of Table The firs sub-period shows quie a differen fundamenal volailiy process compared wih hose of sub-periods 2 and 3. The fundamenal volailiy process before he incepion of derivaives shows mean-reversion, while afer he incepion of derivaives, he process is highly persisen. In addiion, ransiory noises in sub-periods 2 and 3 are relaively larger han he permanen innovaion and hus, he signal-o-noise (STN) raios for he AR(1) model are and in sub-periods 2 and 3, respecively 13. On he oher hand, in he firs sub-period, he STN raios are quie differen for he models used. The unsable STN raios seem o come from he small sample (85 observaions) in he firs sub-period. Panels B and C of Table 2 represen he esimaed mulivariae SVM during subperiods 2 and 3. Three log-volailiies (i.e., FTSE100 index log-rv, fuures log-rv, and American call opions log-iv) for he second sub-period and four volailiies (i.e., FTSE100 index log-rv, fuures log-rv, American and European call opions log-ivs) for he hird sub-period are used in he mulivariae SVM of equaion (A2) in he Appendix. Alhough he coefficiens of he fundamenal volailiy processes in he mulivariae SVMs are differen from hose of he univariae SVM of panel A, all fundamenal volailiy 12 We also used volailiy series in he sae-space form under he assumpion of an addiive process. As expeced in he previous secion, using volailiy raher han log-volailiy in sae-space models is no preferable. Skewness, kurosis, and pormaneau saisics are poor compared wih hose obained by using he SVM. 2 2 η σ ε 13 The signal-o-noise (STN) raio is defined as STN = σ /. 15

17 processes excep he firs sub-period have srong persisence. However, he STN raios are differen beween he volailiies. 14 During he second sub-period, he STN raios are 0.003, 0.003, and for he FTSE100 index, Fuures, and American opions, respecively. In addiion, in he hird sub-period, he STN raios are 0.001, 0.001, 4, and 3 for he FTSE100 index, Fuures, American opions, and European opions, respecively. Our resuls sugges ha log-ivs have relaively more signal han noise, while log-rvs have relaively more noise han signal. Noice ha maximum likelihood values are no significanly differen beween models over all sub-periods. Therefore, an AR(1) model will be used for he sae equaion for he res of his sudy. 5.2 Properies of Fundamenal Volailiies and Relaionship beween Transiory Noises of Differen Volailiies We now invesigae he changes in he unobserved fundamenal volailiy resuling from he inroducion of derivaives. The decomposiion of observed volailiy ino fundamenal volailiy and ransiory noise gives a new perspecive on he invesigaion of he effec of derivaive lising on volailiy. To obain he unobserved fundamenal volailiy, FV i, we use a smoohing algorihm 15. An inference abou FV i using he full se of informaion, i defined as FV T /, is called he smoohed esimae of FV i, which can be represened as i i i FVT / = E( FV / Ψ T) (6) where Ψ T i = ( V ', V 1',, V 1 ')' and i=ftse100, Fuures, American opions, and it, it, i, European opions. 14 The sandard deviaion of ransiory noises, σ ε, can be inferred from he STN raios, since σ η is given in panels B and C of Table This is a fixed-inerval smoohing algorihm; see Harvey (1989), p

18 Using he smoohing echnique for he AR(1) plus noise model, we obain smoohed esimaes of FV i for each sub-period, and hus a ransiory noise series. Figure 1 shows he unobserved fundamenal sandard deviaion of FV FTSE (i.e., exp(0.5µ FTSE +0.5FVP ), where µ FTSE is he level of FTSE100 sock index volailiy). 16 The fundamenal volailiy process shows srong persisence in he second and hird subperiods and a random walk may be he rue process for he fundamenal volailiy. Smoohed fundamenal volailiy process was re-esimaed using an AR(1) model for each sub-period. Dickey-Fuller ess rejec he hypohesis of a uni roo a he 1% level for all hree sub-periods 17. Therefore, he fundamenal volailiy in all hree sub-periods seems o be highly persisen bu no an inegraed process 18. Table 6 repors correlaion marices for observed log-volailiies and ransiory noises. As expeced, he correlaion beween observed volailiies is posiive. In paricular, he correlaion beween he FTSE100 index and fuures reurn volailiies is high. We also find high correlaion beween he American and European call opion implied volailiies. However, he correlaion beween reurn volailiies and implied volailiies is relaively 16 Noe ha he FTSE100 index and fuures reurn volailiies, and American and European opions implied volailiies have he same fundamenal volailiy process, FVP = φ FVP 1 + η, bu he levels of he fundamenal volailiy, µ i, are differen across he four volailiy series. See equaion (5) for furher discussion. 17 Dickey-Fuller saisics (criical values a 1% level) of he smoohed fundamenal volailiy for he firs, second and hird sub-periods are (-13.2), (-13.8), and (-13.8), respecively. 18 We should pay aenion o he inerpreaion of he Dickey-Fuller es resuls. Harvey, Ruiz, and Shephard (1994) argue ha he Dickey-Fuller ess are poor when he auoregressive parameer is close o 1 and he STN raio is very small as in our sudy. In his case, he Dickey-Fuller ess rejec he null hypohesis of a uni roo oo ofen. 17

19 low. Panel B of Table 6 repors he correlaion beween ransiory noises of volailiies. The ransiory noises of reurn volailiies are highly correlaed and ransiory noises of implied volailiies are highly negaively correlaed, bu ransiory noises beween reurn volailiies and implied volailiies do no seem o be correlaed. Therefore, ransiory noises may be grouped ino wo major facors: a noise facor in reurn volailiy and a noise facor in implied volailiy. Ineresingly, ransiory noises in American and European opion implied volailiies are srongly negaively correlaed (-0.906), while observed American opion implied volailiy is highly posiively correlaed (0.987) wih observed European opion implied volailiy. 5.3 Effecs of he Inroducion of Derivaive Markes on he Volailiy of he FTSE100 Index and Is Derivaives In radiional pricing heories such as he Black-Scholes, derivaives are redundan. They can be replicaed wih he underlying asse and a riskless bond. However, ouside he fricionless non-arbirage world, he inroducion of derivaives may have wo opposing effecs on he underlying marke: sabilising and desabilising effecs. Theoreical and empirical invesigaions of he effecs of a fuures lising on he underlying asse are inconclusive 19. Recen sudies such as Lee and Ohk (1992) and Anoniou and Holmes (1995) claim ha he underlying marke becomes more efficien as a resul of he inroducion of he fuures marke. On he oher hand, heoreical and empirical sudies on he effecs of an opion lising refer o an increase in he underlying asse price and a decrease in he volailiy of he underlying asse reurn. Deemple and Selden (1991) underake heoreical analysis of he effecs of he inroducion of an opion in an incomplee marke wih a sock, a call 19 See Board and Sucliffe (1993) for a summary. 18

20 opion on he sock, and a riskless bond. They show ha he inroducion of he opion resuls in an increase in he sock price and a decrease in he volailiy of he sock rae of reurn because of invesors differen assessmens abou he downside poenial of he sock in a quadraic uiliy seing. Mos empirical sudies suppor he heoreical resuls; see Trennepohl and Dukes (1979), Skinner (1989), Conrad (1989), Deemple and Jorion (1990), Damodaran and Lim (1991), Haddad and Voorheis (1991), Wa, Yadav and Draper (1992), Chamberlain, Cheung and Kwan (1993), and Gjerde and Sæem (1995) 20. Some empirical sudies use marke models and find ha he sysemaic risk of he underlying asse changes lile, while unsysemaic risk decreases. In addiion, opion rading seems o make he underlying asse adjus more rapidly o new informaion, and rading volume ends o be increased by opion rading. Table 1 shows ha here are changes in observed volailiies beween sub-periods. By decomposing observed volailiy ino fundamenal volailiy and noise, we can furher analyze changes in volailiy resuling from he inroducion of derivaives. As a preliminary es, he T-es and he Mann-Whiney-Wilcoxon es are used. Panel A of Table 3 shows he -es resuls. The FTSE-100 index reurn volailiy, he fuures reurn volailiy, and he American opion implied volailiy show significan changes coinciding wih he lising of European opions. However, since volailiies have a long ail, a nonparameric es seems o be more appropriae. For his purpose, he Mann-Whiney- Wilcoxon es resuls are repored in panel B of Table 3. The resuls of he -es and he Mann-Whiney-Wilcoxon es are similar. The FTSE100 index reurn volailiy and he 20 Chamberlain, Cheung and Kwan (1993) and Gjerde and Sæem (1995) repor lile change in underlying asse volailiy. 19

21 American opion implied volailiy are changed by he lising of European opions, while here is no significan change in he FTSE100 index reurn volailiy wih he inroducion of he FTSE100 fuures and he American opions. The effecs of he inroducion of derivaives on fundamenal volailiy are repored in Table 4. The inroducion of American opions and fuures significanly increases FV FTSE and he AR coefficien in he fundamenal volailiy process. On he oher hand, he inroducion of European opions significanly decreases all hree FV FTSE, FV fuures, and FV American, while i does no change he fundamenal volailiy process. Table 5 repors he changes in he ransiory noises. The ransiory noise of he FTSE100 index reurn volailiy is decreased significanly by he incepion of American opions and fuures. The inroducion of European opions reduces he noise of he American opion implied volailiy significanly. Our resuls in Tables 4 and 5 can be discussed ogeher wih Table 3. In he preliminary es, here is no significan change in he FTSE100 index reurn volailiy as a resul of he incepion of he American opions and fuures. However, by he decomposiion, we find ha FV FTSE increases significanly and ransiory noise decreases significanly wihou significan impac on he observed volailiy. Wih he incepion of he European opions, he FTSE100 index reurn volailiy and he American opions implied volailiy decrease significanly. The majoriy of he decrease seems o come from FV FTSE, and FV American, because here is lile change in he ransiory noise of he FTSE100 index reurn volailiy, and he significan decrease in he ransiory noise of he American opion implied volailiy is quie a bi smaller han he decrease of FV American. In addiion, while he fuures reurn volailiy does no show any changes wih he inroducion of European opions, FV fuures decreases significanly. This 20

22 means ha he exising marke becomes more efficien wih he lising of he European opions. We canno separae he effecs of he inroducion of fuures from hose of American opions, because boh derivaives were inroduced a he same ime. In addiion, because of he small number of observaions of he firs sub-period, he changes in fundamenal volailiy and observed volailiy beween sub-period 1 and 2 fails o provide convincing evidence for or agains a change in volailiy. 6. Conclusion Using sochasic volailiy models, we decomposed four differen volailiies, he FTSE100 index reurn volailiy, he reurn volailiy for fuures on he FTSE100 index, and he FTSE100 index American and European call opion implied volailiies, ino wha we call unobserved fundamenal volailiy and ransiory noise. For he reurn volailiies such as he FTSE100 index and is fuures, ransiory noise is much larger han he fundamenal volailiy, while implied volailiies of European and American call opions consis of fundamenal volailiy raher han ransiory noise. In addiion, ransiory noises of he FTSE100 index reurn volailiy and fuures reurn volailiy are correlaed wih each oher, and ransiory noises of FTSE100 American and European call opion implied volailiies are also correlaed wih each oher. However, ransiory noises of reurn volailiies are no correlaed wih hose of implied volailiies, suggesing ha rading noise in opions markes is differen from ha in an underlying marke or fuures marke. We have obained wo ypes of volailiy changes: changes in levels, and changes in he underlying dynamic process which correspond o a change in overall persisence of all he markes. Whils boh are ineresing o asse managers or regulaors, we feel ha 21

23 large changes in he laer should be of paricular ineres as hey reflec he fac ha shocks may accumulae raher han die away. Unforunaely, we canno reach a firm conclusion on he effec of he inroducion of he fuures or American opions on he fundamenal volailiy and ransiory noise, since here is only a small number of observaions prior o he American opions and fuures on he FTSE100 index and heir simulaneous inroducion. The finding ha persisence increases as a resul of he inroducion of derivaives needs o be suppored by more daa and analysis in oher markes. This may reflec beer risk managemen whereby anicipaed shocks are spread ou over longer periods hrough he use of derivaives. However, following he inroducion of European opions, we find ha he level of fundamenal volailiy is reduced bu here is no significan change in he fundamenal volailiy process. Furhermore, he ransiory noise of American call opions decreased significanly, while oher ransiory noises do no show significan change. Our sudy proposes ha fundamenal volailiy may be he correc measure of risk for he oal marke. Changes in fundamenal volailiy raher han observed volailiy may be more appropriae for marke regulaors when hey invesigae he sysemaic effec of he inroducion of derivaives on he marke or he curren sae of he marke. Regulaors who currenly compue he risk-neural densiy of reurns implied by opion prices may wish o consider our procedure as a complimenary calculaion o assess changes in he riskiness of marke. 22

24 Appendix A more generalised SVM is used in his sudy, where he sae equaion in (5) is allowed o follow an ARMA(p,q) model. In his generalised model, a sae-space represenaion for a univariae model is V = µ + FVP + ε FVP = φ FVP φ FVP 2 2 +,..., + φ FVP p p + θ η θ η 2 2 +,..., + θ η q q (A1) + η where he φ s and θ s are AR and MA coefficiens. The auoregressive and moving average lags, defined as p and q, are allowed o ake values of up o 2 in his sudy. Therefore, a oal of nine SVMs can be considered. A mulivariae k equaion SVM for (4) can be represened as V = µ + FVPe + ε FVP = φ FVP + φ FVP + θ η + θ η + η (A2) where V = [ V 1, V 2, V k, ], µ = [ µ µ µ ] k 1 k k, ε = [ ε1, ε2, ε k, ], k 1 e k 1 = [ 1 1 1], E( εε Ω τ ) = κ κ 0 for = τ, E( ηη σ τ ) = for τ 0 2 η for = τ, and for τ E( ε η ) =0 for all and τ. Noice ha even hough V is mulivariae, FVP is univariae. τ Unobserved FVP which is relaed o he underlying asse can be obained by considering all volailiy series relaed o ha asse. Alhough he fundamenal volailiy process is assumed o follow only one unobserved process, we allow via he vecor µ differen volailiy levels for each volailiy o reflec he differen volailiy levels in Table 1. Therefore, he fundamenal volailiy of he observed volailiy i, FV i, is he sum of he fundamenal volailiy process and he volailiy level of he observed volailiy i. The above SVM can be represened as 23

25 V = µ + Θ FVP FVP = Θ FVP 1 + ε + Ξ (A3) where FVP [ FVP, FVP, FVP ], [,θ, ] Ξ [, 0, ] = 0 η = 1 2 φ φ Θ = 1 1 θ2, Φ = 1 0 0, and Noe ha he marix represenaion of SVM in (A3) can be applied o he univariae SVM as well as he mulivariae SVM; for he univariae SVM (i.e., k=1), [ ] V =, µ = [ µ ], and ε = ε ]. V 1, [ 1, We now address he issue of idenifiabiliy of he sae-space models, see pp , Harvey (1989). When here exiss any non-singular 3 3 marix H which can saisfy he following sae-space model, we say ha he FVP is no idenifiable. V * = µ + Θ FVP FVP * = Θ * * * FVP 1 + ε + Ξ * (A4) where, FVP = FVP, 1 Θ * * = Θ Η, H * Φ = ΗΦΗ 1 *, and Ξ = HΞ τ. However, o make FVP follow he ARMA model, all elemens excep θ * 1, θ * 2, φ * 1, φ * 2, and η * in he Θ, Φ, and Ξ mus be he same as hose of Θ and Φ in (A3). The only marix ha saisfies his resricion is he ideniy marix. Therefore, as long as he FVP follows ARMA models, here is only one FVP in he SVM of equaion (A3) and he FVP is idenifiable for all p and q. Noe ha his argumen applies o boh he univarie and mulivariae SVMs. The non-exisence condiions for a non-singular 3 3 marix H are he idenifiabiliy condiions of he FVP. However, his does no necessarily mean ha he SVMs of equaions (A1) and (A2) are idenifiable; alhough he FVP is idenifiable for all p and q, here are many ses 24

26 of he parameers which make he SVM have he same FVP. We need addiional condiions for he idenifiabiliy of he SVM; he order condiion for idenifiabiliy requires p q+1 under he assumpion ha he fundamenal volailiy process is saionary and inverible, see pp , Harvey (1989) for furher discussion. Therefore, among he nine SVMs o be considered in his sudy, he SVMs ha saisfy hese condiions for he sae equaion are ARMA(1,0), ARMA(2,0), and ARMA(2,1) models. 25

27 References Amihud, Y. and H. Mendelson, 1987, Trading Mechanisms and Sock Reurns: An Empirical Invesigaion, Journal of Finance 42, Anoniou, A. and P. Holmes, 1995, Fuures Trading, Informaion and Spo Price Volailiy: Evidence for he FTSE100 Sock Index Fuures Conrac Using GARCH, Journal of Banking and Finance 19, Baes, D. S., 1995, Tesing Opion Pricing Models, mimeo, Wharon School, Universiy of Pennsylvania. Beckers, S., 1981, Sandard Deviaions Implied in Opion Prices as Predicors of Fuure Sock Price Variabiliy, Journal of Banking and Finance 5, Black, F., 1976, The Pricing of Commodiy Conracs, Journal of Financial Economics 3, Black, F. and M. Scholes, 1973, The Pricing of Opions and Corporae Liabiliies, Journal of Poliical Economy 81, Board, J. and C. Sucliffe, 1993, The Effecs of he Exisence of Index Fuures on UK Share Price Volailiy, Universiy of Souhampon, Deparmen of Accouning and Managemen Science Discussion Paper No Bollerslev, T., 1986, Generalized Auoregressive Condiional Heeroskedasiciy, Journal of Economerics 31, Bollerslev, T. and R. F. Engle, 1993, Common Persisence in Condiional Variances, Economerica 61, Brenner, M. and D. Galai, 1984, On Measuring he Risk of Common Sock Implied by Opions Price: A Noe, Journal of Financial and Quaniaive Analysis 19, 4, Canina, L. and S. Figlewski, 1993, The Informaion Conen of Implied Volailiy, Review of Financial Sudies 6, 3, Chiras, D. P., and S. Manaser, 1978, The Informaion Conen of Opion Prices and a Tes of Marke Efficiency, Journal of Financial Economics 6, Chamberlain, T. W., C. S. Cheung, and C. Y. Kwan, 1993, Opion Lising, Marke Liquidiy and Sock Behaviour: Some Canadian Evidence, Journal of Business Finance and Accouning 20, 5, Conrad, J. 1989, The Price Effec of Opion Inroducion, Journal of Finance 44, Damodaran, A., 1991, The Effecs of Opion Lising on he Underlying Socks Reurn Processes, Journal of Banking and Finance 15, Damodaran, A. and J. Lim, 1991, The Effecs of Opion Lising on he Underlying Socks Reurn Process, Journal of Banking and Finance 15, Deemple, J. and P. Jorion, 1990, Opion Lising and Sock Reurns: An Empirical Analysis, Journal of Banking and Finance 14, Deemple, J. and L. Selden, 1991, A General Equilibrium Analysis of Opion and Sock Marke Ineracions, Inernaional Economic Review 32, 2, Engle, R. F. and T. Bollerslev, 1986, Modelling he Persisence of Condiional Variances, Economeric Reviews 5, Engle, R. F., V. K. Ng and M. Rohschild, 1990, Asse Pricing wih a Facor ARCH Covariance Srucure: Empirical Esimaes for Treasury Bills, Journal of 26

28 Economerics 45, Fornari, F. and A. Mele, 1994, A Sochasic Variance Model for Absolue Reurns, Economics Leers 46, French, F. and R. Roll 1986, Sock Reurn Variances: The Arrival of Informaion and he Reacion of Traders, Journal of Financial Economics 17, Ghysels, E., A. C. Harvey, and E. Renaul, (1996), Sochasic Volailiy, in: G. S. Maddala and C. R. Rao, eds., Handbook of Saisics, Vol. 14 (Amserdam, New York, and Oxford: Elsevier, Norh-Holland) Gjerde, O. and F. Sæem, 1995, Opion Iniiaion and Underlying Marke Behaviour: Evidence from Norway, Journal of Fuures Markes 15, 8, Haddad, M. M. and F. L. Voorheis, 1991, Iniial Opion Trading and Securiy Risk and Reurn, Journal of Business Finance and Accouning 18, Harvey, A. C., 1989, Forecasing, Srucural Time, Series Models and he Kalman Filer (Cambridge Universiy Press). Harvey, A. C. and J. Durbin, 1986, The Effec of Sea Bel Legislaion on Briish Road Casualies: A Case Sudy in Srucural Time Series Modelling, Journal of Royal Saisical Sociey Associaion 149, Par 3, Harvey, A. C. and N. Shephard, 1993, Esimaion and Tesing of Sochasic Variance Models, Economerics discussion paper EM/93/268, London School of Economics. Harvey, A. C. and N. Shephard, 1996, Esimaion of an Asymmeric Sochasic Volailiy Model for Asse Reurns, Journal of Business and Economic Saisics 14, 4, Harvey, A. C., E. Ruiz and N. Shephard, 1994, Mulivariae Sochasic Variance Models, Review of Economic Sudies 61, Harvey, C. R. and R. E. Whaley, 1991, S&P 100 Index Opion Volailiy, Journal of Finance 46, 4, Harvey, C. R., and R. E. Whaley, 1992, Dividends and S&P 100 Index Opion Valuaion, Journal of Fuures Markes 12, 2, Hull, J. C., 1993, Opions, Fuures, and Oher Derivaive Securiies, Second Ediion, (Prenice Hall, Inc.). Hull, J. C. and A. Whie, 1987, The Pricing of Opions on Asses wih Sochasic Volailiies, Journal of Finance 42, 2, Jarque, C. M. and A. K. Bera, 1980, Efficien Tess for Normaliy, Homoskedasiciy and Serial Independence of Regression Residuals, Economics Leers 6, Laane, H. A. and R. J. Rendleman, Jr. 1976, Sandard Deviaions of Sock Price Raios Implied in Opion Prices, Journal of Finance 31, 2, Lee, S. B. and K. Y. Ohk, 1992, Sock Index Fuures Lising and Srucural Change in Time-varying Volailiy, Journal of Fuures Markes 12, 5, Ljung, G. M. and G. E. P. Box, 1978, On a Measure of Lack of Fi in Time Series Models, Biomerika 65, 2, Nelson, 1996, Asympoically Opimal Smoohing wih ARCH Models, Economerica 64, Nelson, D. B. and D. P. Foser, 1994, Asympoic Filering Theory for Univariae ARCH Models, Economerica 62, Resnick, B. G., A. M. Sheikh and Y. Song, 1993, Time Varying Volailiy and Calculaion of he Weighed Implied Sandard Deviaion, Journal of Financial and Quaniaive Analysis 28,

29 Robinson, G., 1993, The Effec of Fuures Trading on Cash Marke Volailiy: Evidence from he London Sock Exchange, Bank of England Working Paper Series. Ruiz, E., 1992, Heeroscedasiciy in Financial Time Series, Unpublished Ph.D Disseraion, Universiy of London. Ruiz, E., 1994, Quasi-maximum Likelihood Esimaion of Sochasic Volailiy Models, Journal of Economerics 63, Schmalensee, R. and R. R. Trippi, 1978, Common Sock Volailiy Expecaions Implied by Opion Premia, Journal of Finance 33, Schwer, G. W., 1989, Why Does Sock Marke Volailiy Change Over Time?, Journal of Finance 44, Skinner, D. J., (1989), Opion Markes and Sock Reurn Volailiy, Journal of Financial Economics 23, Taylor, S. J., 1994, Modeling Sochasic Volailiy: A Review and Comparaive Sudy, Mahemaical Finance 4, Trennepohl, G. L. and W. P. Dukes, 1979, CBOE Opions and Sock Volailiy, Review of Business and Economic Research 18, Wa, W. H., P. K. Yadav and P. Draper, 1992, The Impac of Opion Lising on Underlying Sock Reurns: he UK evidence, Journal of Business Finance and Accouning 19, 4, Wei, S. J. and J. A. Frankel, 1991, Are Opion-Implied Forecass of Exchange Rae Volailiy Excessively Variable?, Naional Bureau of Economic Research Working Paper No Whaley, R. E., 1982, Valuaion of American Call Opions on Dividend-paying Socks, Journal of Financial Economics 10, Xu, X. and S. J. Taylor, 1994, The Term Srucure of Volailiy Implied by Foreign Exchange Opions, Journal of Financial and Quaniaive Analysis 29,

30 Table 1 Summary Saisics for he Daily Logarihmic Reurn Volailiies of he FTSE100 Index and he FTSE100 Index Fuures and he Daily Logarihmic Implied Volailiies of American and European Call Opions on he FTSE100 Index A. FTSE100 Index Log-Reurn Volailiy Mean Sd. Skew- Kurosis J&B Auocorrelaions Pormaneau Saisic Dev. ness Saisics Q(10) Q(100) Enire Period *** *** Sub-Period Sub period *** Sub period *** *** Noes: Enire sample period : 1 January March 1996, 3098 observaions. Sub-period 1 : 1 January May 1984, 85 observaions. Sub-period 2 : 3 May January 1990, 1454 observaions. Sub-period 3 : 1 February March 1996, 1559 observaions. *** represens significance a 1 % level. B. FTSE100 Fuures Log-Reurn Volailiy Mean Sd. Skew- Kurosis J&B Auocorrelaions Pormaneau Saisic Dev. ness Saisics Q(10) Q(100) Enire Period *** *** Sub period *** *** Sub period *** *** Noes: Enire sample period : 4 May March 1996, 3012 observaions. Sub-period 2 : 4 May January 1990, 1453 observaions. Sub-period 3 : 1 February March 1996, 1559 observaions. *** represens significance a 1 % level. C. Logarihmic Implied Volailiy of American Call Opions on he FTSE100 Index Mean Sd. Skew- Kurosis J&B Auocorrelaions Pormaneau Saisic Dev. ness Saisics Q(10) Q(100) Enire Period *** *** Sub period *** *** Sub period *** *** Noes: Enire sample period : 3 May March 1996, 3013 observaions. Sub-period 2 : 3 May January 1990, 1454 observaions. Sub-period 3 : 1 February March 1996, 1559 observaions. *** represens significance a 1 % level. D. Logarihmic Implied Volailiy of European Call Opions on he FTSE100 Index Mean Sd. Skew- Kurosis J&B Auocorrelaions Pormaneau Saisic Dev. ness Saisics Q(10) Q(100) Enire Period *** *** Noes: Enire sample period : 1 February March 1996, 1559 observaions. *** represens significance a 1 % level.