The Impact of SGX MSCI Taiwan Index Futures on the Volatility. of the Taiwan Stock Market: An EGARCH Approach

Transcription

1 The Impac of SGX MSCI Tawan Index Fuures on he Volaly of he Tawan Sock Marke: An EGARCH Approach Phlp Hsu, Asssan Professor, Deparmen of Fnance, Naonal Formosa Unversy, Tawan Yu-Mn Chang, Asssan Professor, Deparmen of Insurance and Fnance, SHU-TE Unversy, Tawan ABSTRACT Ths arcle examnes he mpac of SGX MSCI Tawan Index Fuures on he volaly of he Tawan sock marke. The emprcal work s conduced wh he use of weekly sock reurns from 995 o 998 and by applyng an expanded EGARCH model. Our fndngs show ha here s no srucural change on eher he condonal or he uncondonal varance afer he nroducon of ndex fuures conracs. I mples ha here s no sgnfcan nfluence of he nroducon of ndex fuures marke on he volaly of he Tawan sock marke. The robusness of he emprcal fndngs s also esed on he bass of dagnoscs performed on he esmaed sandardzed resduals. All ess fal o show any evdence of msspecfcaon. Keyword: Fuures Marke, Sock Marke Volaly, EGARCH Model INTRODUCTION Snce he nroducon of ndex fuures conracs on February 4, 98, here has been much debae over he role of ndex fuures radng n he volaly of he underlyng sock marke. In parcular, he sock marke crashes n Ocober 987 and n Ocober 989 led academcs and regulaory auhores o focus on he possble damagng role of he ndex fuures marke n creang excess sock marke volaly. A popular belef s ha fuures radng encourages msnformed raders and rsk-lovng speculaors, whch desablzes he sock marke and nduces a hgher sock marke volaly. Wheher he ndex fuures ncreases he volaly of he sock marke s dffcul o answer from eher heorecal or emprcal aspecs. One vew s ha fuures radng has a sablzng nfluence on sock marke volaly. I can reduce he sock marke volaly by provdng low cos sae-conngen sraeges, whch enable nvesors o mnmze porfolo rsk, by nroducng posve nformaon exernales and by ransferrng speculaors from sock markes o fuures markes. For nsance, Edwards (988b) fnds ha sock reurn volaly of he S & P 5 has no rsen afer he nroducon of ndex opons and ndex fuures radng for he perod of In fac, he fnds ha he volaly n he S & P 5 was even hgher n 973-8, before fuures radng began. Anoher vew s ha he hgher volaly n he fuures marke, caused by speculaors, may be he man facor n ncreasng he volaly of he sock marke. Subrahmanyam (996) develops a radng model ha ncorporaes nformed speculaors as well as nvesors, who possess ncorrec expecaons

2 abou asse values. He shows ha he nroducon of an ndex fuures marke, by smulang addonal msnformed speculaon, ncreases he volaly of sock marke. Maberly, Allen, and Glber (989) demonsrae ha he volaly of he sock marke rses wh he nroducon of ndex fuures. Harrs (989) shows ha he volaly of he spo marke s sascally hgher due o fuures radng. Besdes hese wo vewpons, some researchers fnd ha he nroducon of ndex fuures has no nfluence on he volaly of he sock marke. Turnovsky (983) presens an equlbrum model o show ha he effec of fuures radng on spo marke volaly s ambguous. There are also many sudes ha fnd no sgnfcan changes n he volaly of spo marke afer nroducon of ndex fuures, such as Forune (989), Becke and Robers (99), Kamara, Mller, and Segel (99), Percl and Koumos (997) and so on. Snce here s no heorecal or emprcal evdence o show wheher he nroducon of ndex fuures has nfluence on he volaly of he sock marke, one wonders f he newly publshed ndex fuures conracs based on sock ndex, whch s famous for s hgh volaly, has any nfluence on he volaly of he underlyng sock marke. The purpose of hs arcle s o examne he mpac of he nroducon of Sngapore Exchange (SGX) MSCI Tawan Index Fuures on he volaly of he underlyng sock ndex. The model used s he expanded EGARCH model nroduced by Percl and Koumos (997). Despe he smlares, hs arcle dffers n he followng ways: frs, he objec of he research s dfferen; second, he properes of sock reurns, for example he dsrbuon, he auocorrelaon, and he saonary of he reurn seres, are examned o ensure he f of he model. Ths arcle s organzed as follows: The frs secon nvesgaes he sascal properes of reurn dsrbuons of he Tawan sock ndex and res o fnd ou he class of reurn-generang sochasc processes ha are conssen wh hese properes. In he second secon, he possble processes he ARCH famles wll be nroduced, and her advanages and dsadvanages wll be compared. The hrd secon presens he expanded exponenal generalzed auoregressve condonal heeroscedasc (expanded EGARCH) model. The fourh secon carres ou he sascal ess of he fness of EGARCH process ha are used o verfy wheher he volaly of sock reurns n he Tawan sock marke changes afer he nroducon of ndex fuures. Fnally, he ffh secon concludes he arcle. DATA AND METHODOLOGY The daa used n hs sudy are weekly ndces of he Tawan sock marke obaned from he AREMOS/UNIX economc and sasc daabase sysem of he Mnsry of Educaon Tawan. They conan 7 weekly ndces coverng he perod from July, 995, o July, 998. Snce he nroducon of ndex fuures boh n CME and SGX s on January 9, 997, and he local ndex fuures conrac of TAIMEX (Tawan Inernaonal Mercanle Exchange Corp.) s publshed on July, 998, he perod of weekly sock ndces would be separaed no pre-ndex fuures perod (from July, 995, o January 8, 997) and pos-ndex fuures perod (from January 9, 997, o July, 998). The reurn s defned as he frs dfference n he naural logarhm of prce ndces: ( I I ) R = log, where I s he ndex of me.

3 Sascal Analyss A common assumpon n models of sock prces s ha he sock prce movemens can be represened by lnear whe-nose processes wh ndependen ncremens. Before a model for he analyss of he sock prce behavor s consruced, hs common assumpon should frs be esed. The purpose of hs secon s herefore o examne wheher hs common assumpon can adequaely descrbe he sock prce movemens n he Tawan sock marke and o ry o dscover a process ha can appropraely represen he properes of he sock prce movemens. To examne he properes of sock prce movemens n he Tawan sock marke, s necessary o carry ou some sascal ess, whch are for he saonary, for he ndependence and auocorrelaon, and for he normaly. Tradonal ess for sascal nference presume he use of saonary daa. I s necessary o examne f he daa used are saonary before researchng furher no he maer, snce he regresson of non-saonary varables ono each oher can lead o poenally msleadng nferences abou he esmaed parameers and he degree of assocaon. A common es for he saonary s he un roo es, whch provdes an easy mehod of esng wheher a seres s saonary, so ha rejecon of he un roo hypohess s necessary o suppor saonary. The un roo ess used n hs sudy are he Augmened Dckey-Fuller (ADF) es, he Phllps-Perron (PP) Tes, and he Bayesan Un Roo Tes of Sms. In almos all of he popular models of sock reurns, s requred ha reurns are ndependen random varables. In order o es he hypohess of ndependence, mus be calculaed boh n he frequency doman (Durbn's cumulaed perodogram) and n he me doman (Ljung-Box Q es). If he hypohess of ndependence s rejeced, he sample auocorrelaon funcons mus also be analyzed n order o deermne he lags of auocorrelaon. For he es of null hypohess of normaly, he Jarque-Bera sasc s used. Sascal fndngs Table shows prelmnary sascs for he weekly reurns, ncludng he followng dsrbuonal parameers: mean, varance, skewness, kuross, medan, he Jarque-Bera sasc for he null hypohess of normaly, and he sascs for un roo ess. Also ncluded are sascs o es he null hypohess of src whe nose boh n he frequency doman (Durbn's cumulaed perodogram for seral correlaon) and n he me doman (Ljung-Box Q es). Table : Sample Sascs on Weekly Reurn Seres Sasc Perod 95/7/ - 97//8 97//9-98/7/ 95/7/ - 98/7/ Sample sze Mean SE of Mean (mean = ) Varance Sandard Error Skewness Kuross.6665* Medan Jarque-Bera 8.393*

4 Cum. Perodogram.7*.795*.9* LB(6).7643* * LB() * LB(4) * 38.9* ADF( γ µ ) * * * Z( γ µ ) * -69.9* * γ * * * * denoes sgnfcance a he.5 level. The crcal value of he Jarque-Bera sasc s 5.99 for sgnfcance levels of.5. The crcal values of he cumulaed perodogram es a level.,.5, and. are.78,., and.44, respecvely. LB( n ) means he Ljung-Box sasc a lag n, whch s dsrbued as a χ varae wh n degrees of freedom. ADF( γ µ ) means ADF un roo es sascs. The crcal values a levels.,.5, and. are -.57, -.86, and -3.47, respecvely. γ µ Z( ) means PP un roo es sascs. The crcal values a levels.,.5, and. are -.3, -4., and -.7, respecvely. γ means he Sm's sascs. If γ >, hen he null hypohess would no be rejeced. As can be seen, all of he hree sascs for un roo ess n all hree perods are sascally sgnfcan even a level.. The null hypoheses of un roo are rejeced n all cases. A ransformaon of he daa for saonary s herefore no necessary. Excep he sample momens n he pre-ndex fuures perod, whch ndcaes ha he emprcal dsrbuon has a sharp peak a he cener compared o he normal dsrbuon, he sample momens n he whole perod and n he pos-ndex fuures perod reveal ha he emprcal dsrbuons are normal dsrbuons. Also he Jarque-Bera sascs show he same resuls. Only n he pre-ndex fuures perod are he null hypohess of normaly rejeced. The perodogram of each seres s esmaed, and he es sascs are calculaed, whch ndcae ha he hypohess of ndependence s rejeced n all perods. The Ljung-Box Q es sasc s calculaed for lags up o 6 weeks, and hose for lags 6,, and 4 are lsed n Table. The null hypohess of src whe nose s rejeced n he whole perod. The concluson mus be ha weekly reurn seres are no made up of ndependen varaes. Implcaon for model consrucon The nonlnear dependence n weekly reurn seres could be explaned by he well-documened fac of changng varances. The changng varance s ofen relaed o he level of radng acvy, he rae of nformaon arrvals, and he corporae fnancal and operang leverage decsons, whch hen end o affec he level of sock prce. A naural way o consruc a model o descrbe such phenomenon s o represen he reurn dsrbuons as dsrbuons of sochasc momens or a mxure of dsrbuons. Meron (98) and many ohers have proposed models of hs ype. Unforunaely, all of hese models assume ha he observaons are ndependen random varables and he reurn seres are src whe nose processes, whch are no conssen wh he emprcal evdence repored here. Therefore, hese models are no compable wh he nonlnear dependence srucure observed here. A possble way of solvng hs problem s o ransform he reurn seres o an uncorrelaed resdual seres, whch could be obaned by usng ordnary leas squares esmaon of he followng AR() regresson: R + = Φ + ΦR e () 4

5 Table : The Regresson Model and Resduals Sascs Perod Sasc 95/7/ - 97//8 97//9-98/7/ 95/7/ - 98/7/ A Esmaes of he model: R = Φ + ΦR + e Φ (.88) (.56) (.7) Φ.737*.4856*.5835* (.43) (.44) (3.45) Durbn s h Sasc B Sample sascs on he resdual seres e Sample sze Mean SE of Mean (mean = ) Varance Sandard Error Skewness Kuross Medan Jarque-Bera Cum. Perodogram LB(6) LB() LB(4) ADF( γ µ ) -8.44* * -.879* Z( γ µ ) * -9.97* -68.4* γ * * * Table repors he OLS esmaes of regresson model () and a number of sascs descrbng he dsrbuon of he resduals. The esmaes of Φ are sascally sgnfcan greaer han zero, ndcang he presence of frs-order auocorrelaon n reurn seres. The dsrbuon of he resduals s shown o be normally dsrbued. Jarque-Bera sascs are also all nsgnfcan. The Durbn s h es sascs could no rejec he null hypohess even a. sgnfcance level. I ndcaes ha here s no frs-order auocorrelaon n he resdual seres. Ths confrms ha an AR() ransformaon of reurns gves an uncorrelaed seres of resduals as desred. The un roo ess are sgnfcan n all cases, mplyng ha he seres n each perod may be generaed by a saonary random walk, whch s conssen wh he prevous fndngs. In order o es he hypohess of ndependence for he resdual seres, he Durbn's cumulaed perodogram and Ljung-Box es sascs are calculaed. Table shows ha n all cases hese ess fal o rejec he hypohess ha he resdual seres s src whe nose. Therefore, he frs-order auocorrelaon n he reurn seres could be modeled as a lnear process of he form AR(). The nonlnear dependence of he reurn seres, whch may be affeced by he changng varance, could be represened by a nonlnear process, whch ncludes funcons of pas values of e. Ths nonlnear process would allow he probably dsrbuon of he reurn seres o depend on pas realzaons. Unforunaely, as dscussed by Presley (98), he sascal esmaon of nonlnear processes s ofen nracable. An alernave way of modelng he nonlnear dependence s he model 5

6 nroduced by Engle (98) under he name of he auoregressve condonal heeroscedascy (ARCH) model, whch can closely approxmae he second-order nonlnear process. The ARCH models nroduced by Engle (98) make he condonal varance of he me predcon error a funcon of me, sysem parameers, exogenous varables, lagged endogenous varables, and pas predcon errors. e R = R Ω φ φ R ~ N( µ, ) µ = φ + φ R = α + where p >, p = α e, α >, α, =, L, p. Ω s he se of all nformaon avalable a me and p s he lag of he process. An ARCH model wh p lags s called ARCH( p ) model. Emprcal applcaons ofen need a relavely long lag n he condonal varance equaon n he ARCH model. The esmaon of a fxed lag srucure s ypcally mposed n order o avod he problem wh negave varance parameer. In order o reduce he lags n he condonal varance equaon, Bollerslev (986) generalzed he ARCH model and allowed lagged pas condonal varances o ener he model and o deermne he condonal varance as follows: p q = + + α α e β j j = j= where p >, q, α >, α, β, =, L, p, j =, L, j q, If q =, hen he GARCH model s an ARCH( p ) process. If = q =. p, hen R s a smple whe nose. The GARCH model s capable of capurng he hree mos emprcal feaures observed n sock reurn daa: lepokuross, skewness, and volaly cluserng. I s found o be more approprae han oher sandard sascal models because s conssen wh a reurn dsrbuon, whch s lepokurc. Besdes hs advanage, he GARCH model also allows for a long-erm memory n he varance of he condonal reurn dsrbuons. Alhough he GARCH model has s advanages by descrbng he phenomenon n he fnancal markes, here are sll some drawbacks. Nelson (99) poned ou hree drawbacks of he ARCH famles. The frs one s ha he (G)ARCH model assumes only he magnude and no he posvy or he negavy of unancpaed excess reurns, whch deermnes feaures of condonal varance of predcon error. Snce he change n varance s condonally uncorrelaed wh he excess reurns under he assumpon of symmercal dsrbuon of sandardzed resduals, ndcaes ha a model, whch assumes he deermnaon of 6

7 hrough he posve and negave resduals asymmercally, mgh be more approprae for he asse prcng applcaons. Second, he (G)ARCH model s usually enforced o add nonnegave consrans o ensure he nonnegavy of he condonal varance of predcon error, (.e., he consan erm and he varance parameer of he condonal varance equaon mus be so defned ha he nonnegavy of he condonal varance s guaraneed). These consrans rule ou random oscllaory behavor n he process because he ncrease all he me when he sandardzed resduals ncrease. A he same me, hese consrans make dffcul o esmae he GARCH model because n order o preven some of he α coeffcens from becomng negave, s always mposed o consruc a lnearly declnng srucure on he α. The hrd drawback concerns he nerpreaon of perssence of shocks o condonal varance. Nelson (99) shows ha he IGARCH(,) model s acually a naural model of perss shocks, and herefore does no behave lke a random walk. The shocks may perss n one norm and de ou n anoher n he GARCH(,) model, so he condonal momens of he model may explode even when he process self s srcly saonary and ergodc. A model nroduced by Nelson, he exponenal ARCH model, can avod hese drawbacks and may be more suable for modelng condonal varance. Nelson red o use anoher devce for ensurng nonnegave, by makng ln( ) lnear n some funcon of me and lagged sandardzed resduals. By modelng he naural logarhm of he varance, he mposon of unduly parameer resrcon can hen be elmnaed. The exponenal ARCH model s formulaed as follows: R Ω ~ f ( z, υ k + = µ = φ φ ) R p = exp α + α g( z = where ), z s he sandardzed resdual and g( z ) θ z + δ [ z - E z ]. The erm z ) ( g s he asymmerc funcon of pas sandardzed resduals. I allows he condonal varance o respond asymmercally o posve and negave values of he pas sandardzed resduals. The erm δ [ z - E z ] measures he sze effec and represens a magnude effec n he spr of he GARCH models. The erm θ z measures he sgn effec. Smlar o he relaonshp beween he GARCH and ARCH models, he exponenal ARCH model can also be generalzed and convered no an exponenal GARCH (EGARCH) model as follows: 7

8 R Ω µ = φ + ~ f ( z, υ) k = φ R = exp[ α + p = α g( z ) + q j= β ln( j j )]. The Model R Ω The model used here s he expanded EGARCH model nroduced by Percl and Koumos (997). ~ f ( z, υ ), k + φ, F DF + φ = µ = φ R () = exp[ α + α DF + ( α DF + α ) g( z ) + ( β DF β ) ln( (3) ( ), F, F, F + g z θ z + z - E z, where he varable DF s a dummy varable represenng he pos-ndex fuures perod for he Tawan sock ndex. The uncondonal mean of equaon () s φ + φ, F µ =, for =, L, k. k φ ln( ) = The logarhm of he uncondonal varance of equaon (3) s α + α, F β β =., F The esmaon s done by maxmum lkelhood. The dsrbuon for he normalzed resduals could be assumed as a suden, or a generalzed error dsrbuon (GED). The GED was orgnally used wh he EGARCH model by Nelson (99) and could be employed here snce can accommodae faer als and peakedness. The densy funcon of he GED s as follows: f υ z υ exp λ z, υ) =, where + υ λ Γ υ ( υ Γ υ. λ 3 Γ υ υ s an endogenously esmaed scale parameer, whch conrols he shape of he GED dsrbuon, and Γ ( ) s he gamma funcon. j )] If υ =, hen a GED dsrbuon s equvalen o a normal dsrbuon. For υ <, s a dsrbuon wh excess kuross, (.e., faer als). For he suden, s equvalen o a normal dsrbuon as υ, bu ges very close for > 3 υ or so. 8

9 L The log-lkelhood funcon s gven by T ( Θ p q) =, ln f (, υ ). = z The order k of he auoregressve process n equaon () s decded on he bass of log-lkelhood rao ess. Afer esmaon of he coeffcens of he expanded EGARCH model, he normalzed resduals (.e., he resduals dvded by he square roo of he condonal varance) and he squared normalzed resduals wll be esed for seral correlaon. The lnear and nonlnear ndependence wll be esed by means of he Ljung-Box sasc. In order o deermne he answer o he queson regardng how well he model capures he mpac of posve and negave nnovaons on volaly, he dagnoscs proposed by Engle and Ng (993) wll be used. The dagnoscs nclude hree ess: he sgn bas es, he negave sze bas es, and he posve sze bas es. These ess are based on a new mpac curve of he ARCH-ype model and suppose f he volaly process s correcly specfed, hen he squared sandardzed resduals should no be predcable on he bass of observed varables. RESULTS The resuls of he maxmum-lkelhood esmaes are repored n Table 3. The consan n he condonal mean equaon and he coeffcen of he dummy for he pos-ndex fuures sub-perod φ, F are boh sascally nsgnfcan a he.5 level. On he bass of log-lkelhood rao ess, he order k of he auoregressve process n equaon () s shown as uny, and he esmae of φ s sgnfcanly greaer han zero. Boh of hese confrm he presence of frs-order auocorrelaon n he me seres { R }. Pas reurns, up o he second lag, are nsgnfcan deermnans of he presen reurns. The shape of he dsrbuon of he esmaed resduals s sll normal. A smple es could no rejec he υ = hypohess even a he. sgnfcance level. The esmaed coeffcens for he condonal varance reveal ha he condonal varance s me dependen. The value of θ s negave as expeced, bu sascally nsgnfcan. Ths ndcaes ha a negave realzaon has he same effec on he volaly as a posve realzaon wh he same magnude wll do. The erm α, F, whch s desgned o measure wheher he sensvy of he condonal varance o pas nnovaons n he pos-ndex fuures perod has changed, s sascally nsgnfcan. Ths mples ha no change of he condonal varance of he weekly reurns n spo marke has occurred afer he nroducon of ndex fuures conracs n SGX. The erm β s desgned o capure wheher any change has occurred n he perssence of he, F condonal varance n he perod followng he nroducon of ndex fuures conracs. The nsgnfcance of 9

10 hs erm ndcaes ha he nroducon of ndex fuures conracs s no relaed o he perssence of he condonal varance. The degree of perssence, β, s sascally sgnfcan, mplyng ha would ake me for a shock o de ou. If he half-lfe of a shock could be measured as log(.5)/log( β ), hen would ake approxmaely.7 weeks for a shock o de ou. The nsgnfcance of coeffcens α, F,, F α, and β, F mples ha here s no evdence of any sgnfcan srucural change n he condonal varance process over he enre sample perod. As for he model-mpled uncondonal sandard devaons based on equaon (3), he values for he pre- and he pos-ndex fuures sub-perod are.3 and.38, respecvely. An F-es could no rejec he null hypohess of he equaly of hese wo uncondonal varances. The unchange of he uncondonal varance n pre- and pos-ndex fuures perod mples ha he ndex fuures does no conrbue o any ncremenal change n he volaly process. Table 3: Maxmum-Lkelhood Esmaes for Weekly Reurn. Esmaon Perod: 995/7/-998/7/ Condonal Mean Equaon Parameer Coeffcens Sandard Error -sasc φ φ ,F φ.57795* υ Condonal Varance Equaon Parameer Coeffcens Sandard Error -sasc α α ,f α α ,f β * β ,f θ The resul of he model-mpled uncondonal sandard devaons and he resdual-based dagnoscs s shown n Table 4. The sascs of he sgn bas es, he posve sze bas es, and he negave sze bas es as well as he F sasc of he jon es are all sascally nsgnfcan. Ths mples ha he squared sandardzed resduals could no be predced on he bass of observed varables. Therefore, can be sad ha he expanded EGARCH model wh GED can capure he second momen dynamcs of weekly reurns que well. Table 4: The Model-mpled Uncondonal Sandard Devaon and he Resdual-based Dagnoscs E( z ) -.79 LB(6) Sandard Error.5 LB() E( z ).997 LB (6) 3.7

11 Sandard Error.357 LB () Sgn Bas ( es).999 Posve Sze Bas ( es).583 Negave Sze Bas ( es).87 Jon Tes ( F es; F[3,69]).3686 SUMMARY AND CONCLUSION Ths arcle nvesgaes he mpac of ndex fuures conracs publshed n Sngapore on he volaly of weekly reurns of he Tawan sock marke. The emprcal work s conduced wh he use of weekly sock reurns from 995 o 998 and by applyng an expanded EGARCH model. Resuls ndcae ha here s no srucural change on eher he condonal or he uncondonal varance afer he nroducon of ndex fuures conracs. To ensure he approprae specfcaon of he model, he robusness of he emprcal fndngs s also esed on he bass of dagnoscs performed on he esmaed sandardzed resduals. The resuls ndcae ha all ess fal o show any evdence of msspecfcaon. REFERENCES Becke, S. and Robers, D.J. (99). Wll ncreased regulaon of sock ndex fuures reduce sock marke volaly?, Federal Reserve Bank of Kansas Cy, Nov./Dec., Bollerslev, T. (986). Generalzed auoregressve condonal heeroscedascy, Journal of Economercs, 3, Edwards, F. R. (988). Does fuures radng ncrease sock marke volaly?, Fnancal Analyss Journal, Jan./Feb., Engle, R. F (98). Auoregressve condonal heeroscedascy wh esmaes of he varance of Uned Kngdom Inflaons, Economerca, 5, Engle, R. F. and Ng, V. K. (993). Measurng and esng he mpac of news on volaly, Journal of Fnance, 48, Forune, P. (989). An assessmen of fnancal marke volaly: Blls, bonds and socks, New England Economc Revew, Nov./Dec., 3-7. Harrs, L. (989). S & P 5 cash sock prce volales, Journal of Fnance, 44, Kamara, A., Mller, T. W., and Segel, A. F. (99). The effec of fuures radng on he sably of Sandard and Poor 5 reurns, The Journal of Fuures Markes,, Maberly, E., Allen, D., and Glber, R. (989). Sock ndex fuures and cash marke volaly, Fnancal Analyss Journal, Nov./Dec., Meron, R. (98). On he mahemacs and economcs assumpons of connuous me models, Fnancal Economcs: Essays n Honor of Paul Cooner, Engelwood Clffs, NJ: Prence-Hall, 9-5. Nelson, D. B. (99). ARCH models as dffuson approxmaons, Journal of Economercs, 45, Nelson, D. B. (99). Condonal heeroscedascy n asse reurns: A new approach, Economerca, 59, Percl, A., and Koumos, G. (997). Index fuures and opons and sock marke volaly, The Journal of Fuures Markes, 7, Presley, M. B. (98). Specral analyss and me seres, Academc Press, New York. Subrahmanyam, A., (996). On speculaon, ndex fuures markes, and he lnk beween marke volaly and nvesor welfare, The Fnancal Revew, 3, Turnovsky, S.J. (983). The deermnaon of spo and fuures prces wh sorable commodes, Economerca, 5,