SAMUELSON S HYPOTHESIS IN GREEK STOCK INDEX FUTURES MARKET

Transcription

1 154 Invesmen Managemen and Financial Innovaions, Volume 3, Issue 2, 2006 SAMUELSON S HYPOTHESIS IN GREEK STOCK INDEX FUTURES MARKET Chrisos Floros, Dimirios V. Vougas Absrac Samuelson (1965) argues ha fuures prices become increasingly volaile as fuures conracs approach mauriy. However, available evidence does no provide clear suppor for Samuelson s hypohesis (or mauriy effec) for index fuures. This paper applies linear regressions and GARCH (volailiy) models o examine he mauriy effec of index fuures conracs using daily daa from he Ahens Derivaives Exchange (ADEX). To he bes of our knowledge, his is he firs empirical invesigaion of he Greek fuures markes. Our resuls sugges ha volailiy series depend on ime o mauriy. When daa for he nearby monh conracs are considered, employed GARCH models show a sronger suppor o he hypohesis han radiional linear regressions. In general, i is concluded ha Samuelson s hypohesis is valid, and herefore, volailiy of he series increases as expiry approaches. These findings are helpful o risk managers dealing wih Greek sock index fuures. Key words: Fuures, ADEX, Samuelson, GARCH. JEL Classificaion: G13, G Inroducion I is apparen ha early in a fuures conrac s life, lile is known abou he fuure spo price, bu as he conrac approaches mauriy he rae of informaion acquisiion increases (Kolb, 1994). This is he basic heory behind he mauriy effec. In oher words, when here is a long ime o delivery, new informaion may affec he delivery price. Neverheless, when delivery is abou o ake place, here is no enough ime for furher informaion acquisiion, and he new informaion has a definie effec on he delivery price. In view of his, several heories aemp o explain he relaionship beween conrac mauriy and volailiy. Firs, here is srong evidence ha volailiy is higher when informaion abou he harves of some good (commodiies) is reaching he marke. Hence, volailiy depends primarily on he ime of he year. Secondly, volailiy may depend on he day of he week or oher seasonal effecs and no so much on he conrac s expiraion. More specifically, in empirical sudies, Samuelson s hypohesis is saed as explaining he relaionship beween fuures prices and mauriy. The negaive relaionship beween volailiy and days o mauriy is ofen referred o as he mauriy effec. The alernaive hypohesis saes ha volailiy is relaed o informaion flows. Samuelson (1965) saes ha volailiy of fuures prices should increase as he conrac approaches expiraion (a mauriy dae). He uses he assumpion ha fuures price equals expeced fuure spo price. Tha is, Samuelson s model implies ha he fuures price should be more volaile near expiraion. Hence, he variance 1 of fuures prices increases as he fuures conrac approaches mauriy. Therefore, fuures prices reac more srongly o informaion as hey approach expiraion han oherwise 2. Then, i could be possible o have an increase in margins 3, as he margin size is a posiive funcion of he volailiy of fuures prices. Therefore, he cash balances will also rise, and hen, he correlaion beween spo and fuures prices will decline. Theoreically, Samuelson (1965, p. 786) explains he role of harves paerns as follows: The presen heory can conribue an elegan explanaion of why we should expec far-disan fuures o move more sluggishly han near ones. Is explanaion does no lean a all on he undoubed fac ha, during cerain 1 Ruledge (1976) suggess ha he variance of he fuures price is an empirical problem. 2 I should be noed ha conrac mauriy has no effec on he flow of valuable informaion for socks. 3 Margin is defined as he cash balanced (or securiy deposi) required from a fuures rader. Chrisos Floros, Dimirios V. Vougas, 2006

2 Invesmen Managemen and Financial Innovaions, Volume 3, Issue 2, pre-harves periods when socks are normally low, changes in spo prices can hemselves be expeced o experience grea volailiy. More recenly, Johnson (1998, p. 3) explains he behaviour of fuures raders and fund managers in relaion o Samuelson hypohesis as follows: If his is he case hen a fund manager wishing o keep fuures price volailiy o a minimum would need o rade in he longes daed conrac. If, however, price volailiy is unrelaed o he lengh of ime o mauriy, fund managers can iniiae a fuures posiion in he mos convenien conrac, han in any oher conrac currenly rading. This may enable a fuures rader o remain wihin a single conrac and no have o make a decision regarding he appropriae ime o roll over ino he nex o maure conrac. Samuelson (1965) uses wo assumpions, namely ha: (i) each fuures price equals he rading dae expecaion of he delivery dae spo price 1, and (ii) he spo price is saionary following an AR(1). In general, he heory of Samuelson s hypohesis requires wo condiions: sysemaic increase in spo reurn volailiy near each fuures expiraion dae or negaive covariaion beween spo reurns, and he slope of he fuures erm srucure. In addiion, according o Sucliffe (1993), he expeced value of he spo price a delivery exceeds he curren index fuures price, and herefore, here is a risk premium. So, Samuelson s argumens do no apply o index fuures. Also, Bessembinder e al. (1996) argue ha he firs condiion is implausible, while for he second condiion hey sae ha he hypohesis is no correc in markes where spo prices follow a random walk. This case depends upon he behaviour of he spo prices. Furhermore, according o Bessembinder e al. (1996, p. 3), i is cerainly possible ha a sysemaic clusering of informaion flows near fuures delivery daes could cause commensurae increases in price change variances a hose imes, consisen wih he Samuelson hypohesis. However, Bessembinder e al. (1996) conclude ha he informaion flows condiion is no necessary as he hypohesis holds where informaion flows do no cluser near delivery daes. In general, hey show ha he Samuelson hypohesis is successful wihou he wo condiions: he clusering of informaion flows near delivery daes, and ha fuures price is an unbiased forecas of he delivery dae spo price. They conclude ha he hypohesis is correc in markes where equilibrium spo prices are mean revering. Barnhill e al. (1987) argue ha Samuelson s hypohesis is a monoonic mauriy hypohesis. This resuls from he assumpion of saionary, auoregressive process for spo prices. In addiion o ha, Anderson and Danhine (1983) poin ou ha Samuelson s hypohesis is a hypohesis abou he uncerainy resoluion 2. They sugges ha fuures price volailiy is high when we have large amouns of supply and demand. However, Milonas (1986) poins ou ha, far from mauriy, fuures conracs conain much more uncerainy. As a resul, prices will respond less srongly o new informaion. Recenly, Hong (2000) argues ha Samuelson s effec is a price elasiciy effec. This is due o he fac ha he price elasiciy of he fuures conrac increases and so is reurn volailiy rises, as i approaches he expiraion dae. Also, Hong (2000) saes ha Samuelson s effec does no hold when he informaion asymmery is large, while in markes where he informaion asymmery is small, here is a definie effec. This paper examines he relaionship beween daily volailiy and ime o expiraion (mauriy) in he Greek fuures markes. For his, a range of volailiy regressions is employed. Our findings are imporan since no previous work has examined Samuelson s hypohesis in he Greek fuures markes. The paper is organised as follows. Secion 2 provides a deailed lieraure review, while Secion 3 oulines he mehodology. Secion 4 describes he daa, and Secion 5 presens empirical resuls from various economeric models. Finally, Secion 6 concludes he paper and summarizes he findings. 1 Kolb (1994) saes ha fuures prices follow maringales. This implies ha fuures prices equal he expeced fuure spo prices. 2 According o Anderson and Danhine (1983), Samuelson s hypohesis is a case of he sae variable hypohesis, where uncerainy resoluion is greaes before expiraion, see also Allen and Cruickshank (2000).

3 156 Invesmen Managemen and Financial Innovaions, Volume 3, Issue 2, Lieraure Review A wide area of research examines fuures price volailiy in a number of markes. I is a common belief ha facors such as rading volume and mauriy may cause price volailiy. Firs, Segall 1 (1956) suggess ha changes in ineres raes could have a greaer effec on he price of conracs furher from mauriy and ha various facors (e.g. he uncerainy facor or he speculaive effec 2 ) and evens may affec he value of he conrac during is life. Anderson (1985) and Kenyon e al. (1987) find ha seasonaliy is an imporan facor ha influences fuures price volailiy. Furher, Sucliffe (1993) argues ha he approach of delivery has a predicable effec on he absolue size of he basis 3. In addiion, Khoury and Yourougou (1993) argue ha ime o expiraion is one of he facors ha cause fuures price volailiy. However, according o Moosa and Bollen (2001) he empirical evidence on he mauriy effec in fuures prices is mixed. Usually, fuures prices are more volaile when hey are close o expiry because high price volailiy indicaes more informaion. As he conrac approaches mauriy, he rae of informaion increases, and herefore, here is an increase in volailiy. In general, Moosa and Bollen (2001, p. 693) sugges ha here are wo conclusions abou he hypohesis of mauriy effec in fuures prices: (i) he seasonal effec is more imporan han he mauriy effec for agriculural commodiies, and (ii) he mauriy effec plays a significan role in deermining he volailiy of fuures prices for commodiies (bu no for hese which he cos-of-carry model of fuures prices works well). Previous empirical sudies find some suppor for Samuelson s hypohesis. Firs, Ruledge (1976) discusses ha boh silver and cocoa conracs fuures prices suppor he hypohesis, bu for whea and soybean oil here is no suppor of he hypohesis. Also, Dusak-Miller (1979) and Caselino and Francis (1982) find srong evidence for commodiy conracs. Anderson (1985) suppors Samuelson s hypohesis for nine commodiies and argues ha he seasonal effec is he mos imporan facor affecing volailiy. He argues ha he hypohesis is no valid where spo prices are non-saionary. Milonas (1986) also finds suppor for Samuelson s hypohesis in 10 of 11 commodiies (i.e. agriculural, financial and meal conracs). Furhermore, Grammaikos and Saunders (1986) argue ha he variance of he price in fuures decreases wih he ime o mauriy of he conrac. In addiion, Karpoff (1987) repors ha he ime o delivery of fuures conracs affecs he price variabiliy. Bessembinder e al. (1996) show ha markes for real asses suppor Samuelson s hypohesis. They argue ha he hypohesis is srongly suppored in agriculural markes, where here is a large mean revering componen in spo prices. Also, in sudying commodiy fuures from Chicago Board of Trade (CBOT), Hennessy and Wahl (1996) explain he seasonaliy and mauriy effec and find suppor o he Samuelson hypohesis. From previous sudies, i is known ha he mauriy effec in financial fuures markes is no very srong. Firs, Barnhill e al. (1987) apply Samuelson s hypohesis o he U.S. Treasurybond fuures marke and find ha a significan effec does exis. For UK marke, Chamberlain (1989) uses daily daa of he FTSE-100 index fuures conracs o examine Samuelson s hypohesis. His resuls are mixed. The March 1985 conrac indicaes no relaionship beween volailiy and mauriy, while he June 1985 conrac shows a definie suppor for Samuelson s hypohesis. However, Board and Sucliffe (1990) show lile suppor o Samuelson s hypohesis for he period from May 1984 o Augus In a recen sudy, Bessembinder e al. (1996) argue ha in financial fuures here is no relaionship beween volailiy and ime o expiry, since here is no significan relaion beween prices and he fuures erm slope. Also, Johnson (1998) examines he relaionship beween price volailiy and lengh of ime o mauriy for SPI fuures using several volailiy regressions. He finds no evidence of he hypohesis in Ausralian share price index fuures. In addiion, Johnson (1998) finds ha volailiies of near and far conracs are highly correlaed. For NYSE Composie and S&P 500 indices, Park and Sears (1985) show a significan posiive effec of mauriy on volailiy. Consisenly, Han and Misra (1990) find no suppor for Samuelson s hypohesis, also using he S&P 500 index. The resuls afer he sock marke crash in 1 Segall (1956) uses whea fuures o examine he relaionship beween mauriy and price volailiy. He does no find evidence o suppor he uncerainy effec. 2 See Hong (2000). 3 See Sucliffe (1993, p. 135) for he behaviour of he basis facor during he life of a conrac.

4 Invesmen Managemen and Financial Innovaions, Volume 3, Issue 2, Ocober 1987 show a posiive relaionship beween volailiy and fuures. Recenly, Moosa and Bollen (2001) examine he mauriy effec using he S&P 500 fuures conracs. To es for he mauriy effec, hey use a simple OLS model (including a dummy variable for he mauriy effec), which measures he volailiy. Moosa and Bollen (2001) find ha volailiy is independen of he ime o mauriy. Also, Galloway and Kolb (1996) find ha he prices of financial fuures conrac do no exhibi a mauriy effec. This finding is in line wih he resuls obained by Moosa and Bollen (2001), Leisikow (1989), and Han e al. (1999). Leisikow (1989) finds no suppor for Samuelson s hypohesis in commodiy fuures, and also, Han e al. (1999) find no mauriy effec for currency fuures, since he relaionship beween volailiy and days o mauriy is found o be posiive. Serleis (1991) finds a saisically significan mauriy effec for a number of oil fuures conracs. In anoher paper, Serleis (1992) finds ha he inroducion of volume rading as an explanaory variable significanly reduces he power of ime unil mauriy for energy fuures. His resuls show ha energy fuures prices are more volaile and rading volume increases as fuures conracs approach mauriy. Also, Bessembinder e al. (1996) sugges srong negaive relaions beween crude oil prices and fuures erm slopes, indicaing ha Samuelson s hypohesis is likely o hold. Recenly, Walls (1999) invesigaes he elecriciy fuures markes (NYMEX, COB and PV) by looking a he volailiy and mauriy effec. In general, he resuls proposed by Walls (1999) differ subsanially from he resuls obained by Serleis (1992), as Walls (1999) suggess ha price volailiy increases for 86% as he fuures conracs approach mauriy (Serleis (1992) resuls show 70%), and ha he mauriy effecs seem o be sronger han for he oher energy fuures. Noice ha, Herber (1995) finds ha he volume raher han mauriy explains beer variance price volailiy in naural gas fuures. A paper ha sudies Samuelson s hypohesis, using ARCH/GARCH models, is by Yang and Brorsen (1993). They find ha only 6 of he 15 conracs suppor he hypohesis, while 9 of he 15 conracs show ha seasonaliy is imporan. In addiion, Chen e al. (1999) use a bivariae GARCH model o examine he mauriy effecs for he Nikkei-225 sock index and index fuures. Chen e al. (1999) argue ha he mauriy and GARCH effecs are simulaneously presen. They find ha he condiional variance of fuures price decreases if is mauriy is shorened (as he conracs approach expiraion). Recenly, Allen and Cruickshank (2000) apply Samuelson s hypohesis o commodiy fuures conracs on hree inernaional exchanges: he Sydney Fuures Exchange (SFEX), he London Inernaional Fuures and Opions Exchange (LIFFE), and he Singapore Inernaional Moneary Exchange (SIMEX). They conclude ha Samuelson s hypohesis holds in mos of he fuures conracs. Adrangi and Charah (2003) examine he non-linear dependence in fuures reurns of coffee, cocoa and sugar using ARCH-ype models. More specific, o examine he impac of any ime-o-mauriy (TTM) effecs on he ess for non-lineariy hey consider GARCH specificaions wih and wihou a dummy variable in he variance equaion. The resuls show a negaive TTM coefficien, indicaing a srong suppor o he Samuelson hypohesis. Also, according o Adrangi and Charah (2003, p. 254) TTM plays a role as a conrol variable in he ess of chaos. Recenly, Akin (2003) examines he volailiy dynamics of 11 financial fuures reurns by looking a he Samuelson hypohesis. The resuls show a srong ime-o-mauriy effec for currency fuures, and mixed evidence in equiy index and ineres rae fuures. 3. Mehodology As discussed earlier, Samuelson (1965) assumes he following assumpions: (i) Spo prices ( S ) mus follow a saionary firs order auoregressive process and, (ii) Fuures prices ( F ) are unbiased predicors of he expiraion price. Recall ha he hypohesis requires eiher a negaive covariaion beween spo reurns and he slope of he fuures erm srucure, or sysemaic increases in spo reurn volailiy near fuures expiraion dae. Tha is, S as 1 u or S S 1 a( S 1 S2 ), (1) 2 where, E ( u ) 0 and E ( u 2 ). If 1, he variance of he change in fuures prices will rise as a conrac approaches expiry.

5 158 Invesmen Managemen and Financial Innovaions, Volume 3, Issue 2, 2006 Ruledge (1976) proposes a model where he spo price is generaed hrough a fis order auoregression of he change in spo prices. Tha is, S S 1 ( S 1 S2 ) u. (2) If 0, fuures prices variaion will acually decrease as expiry approaches. The heoreical relaionship beween a sock index fuures price and is underlying asse is given by he cos-of-carry model. F S exp ( r d)( T ), (3) where S is he sock index price a ime, F is he value of he fuures price, r is he risk-free rae of reurn, d is he dividend yield, and T- is he mauriy dae for he fuures conrac, see Brooks e al. (2001). I is apparen ha he value of he index a any ime depends on he price of he sock. The idea is o inves F exp( r d)( T ), now, and ge S back a mauriy of he fuures conrac. In a seminal paper, Bessembinder e al. (1996) show ha he variance of fuures price changes (in relaion o he cos-of-carry model) is given by 2 Var( f ) Var( u ) Var( s ) 2Cov( u, s ), (4) P 1 where f ln( F 1, T ) ln( F, T ), s s1 s, and u ln( ). P is he E ( P 1) spo price a dae, E (.) denoes he expecaion, and F, T denoes he fuures price a dae, for delivery a dae T. Also, Var and Cov sand for variance and covariance, respecively. Under Equaion 4, Samuelson s hypohesis shows an increase in Var( f ) as he number of periods unil he ime o expiraion (delivery) decreases. According o Bessembinder e al. (1996, p. 8), he Samuelson hypohesis holds if and only if he variance of unexpeced spo reurns ( u ) increases as he delivery dae approaches. In addiion, hey sae ha he hypohesis may rely on he negaive covariaion beween spo reurns and changes in he fuures erm slope. Therefore, hey conclude ha i is imporan o focus on he fuures erm slope variaion raher han in spo reurn variances close o expiraion daes. Anoher heory, explaining he fuures price, is he general equilibrium model from Hemler and Longsaff (1991). They argue ha, he model shows an increase or decrease o he variance of fuures reurns as he delivery dae approaches. Samuelson s hypohesis does no hold when he covariances beween changes in he spo index and changes in he volailiy of sock reurns are zero. For empirical examinaion of Samuelson s hypohesis and volailiy measuremen in fuures markes, several echniques may be applied. Johnson (1998) and Allen and Cruickshank (2000) use hree ways o esimae fuures price volailiy. Also, Bessembinder e al. (1996) use he concep of nearby o define volailiy of he conrac closes o expiry (i.e. Nearby 1). Therefore, nearby 2 is he second closes o expiry and so on. Following his mehod, Allen and Cruickshank (2000) calculae he average volailiy by nearby analysis and find suppor for Samuelson s hypohesis (i.e. 10 of he 12 commodiies show srong evidence in he hypohesis). A second mehod of examining Samuelson s hypohesis is hrough a regression of daily fuures volailiy on days o expiry and spo volailiy. Allen and Cruickshank (2000) esimae daily volailiy on days o expiry using enire and reduced daases (i.e. ses used for spo volailiy regressions). Again, 10 of he 12 commodiies show srong suppor for Samuelson s hypohesis, as he coefficien on he days o expiry is found o be negaive and saisically significan from zero. Following Bessembinder e al. (1996), a regression of daily spo prices can be employed wih near fuures prices used as proxies for spo prices. This is so, because from heory, close- oexpiry fuures prices should approach spo prices. Using his regression, Allen and Cruickshank

6 Invesmen Managemen and Financial Innovaions, Volume 3, Issue 2, (2000) show ha he coefficiens of he spo volailiies are posiive and significan as expeced. Our mehodology is based on early works by Moosa and Bollen (2001), Walls (1999), Johnson (1998) and Allen and Cruickshank (2000). According o Johnson (1998), Anoniou and Holmes (1995), Grammaikos and Saunders (1986) and Allen and Cruickshank (2000), here are several measures of volailiy 1. For example, Ruledge (1979) uses he absolue log change from one rading day o he nex, while Tauchen and Pis (1983) use he square of he firs difference of he fuures price of adjacen periods. In addiion, Karpoff (1987) uses he absolue value of he firs difference o measure volailiy. In his paper, we invesigae he effec of ime unil mauriy on price volailiy and esimae volailiy as follows: Vola = ln F ln 100. (5) F 1 Equaion (5) measures he absolue value of he coninuously compounded rae of reurn muliplied by 100. Firs, we measure he mauriy effec using he following linear regression equaion: Vola = a ln, (6) where volailiy is defined by Equaion (5), M is he number of days o mauriy and is an independenly and idenically disribued random disurbance wih mean zero and finie variance, i.e. ~ iid (0, ). Noice ha, i would also be possible o use he number of calendar 2 days remaining unil delivery of fuures conrac akes place, because informaion arrives during non-rading periods (e.g. weekends). However, here we use he number of rading days remaining, following he work of Moosa and Bollen (2001) and Walls (1999). According o Walls (1999), he inercep of Equaion (6) should be posiive and saisically significan, since i reflecs price volailiy a conrac mauriy, when fuures and spo prices are close. For Samuelson s hypohesis o be correc, 0 and saisically significan, so ha price volailiy increases as he number of days unil conrac mauriy decreases, see Walls (1999) and Herber (1995). In oher words, as we move closer o he delivery dae of he conrac, he mauriy effec index becomes smaller in value. We also apply sochasic volailiy models o our daa for modelling he variance of each series. Since Samuelson s hypohesis holds when models show evidence of greaer variance, we use various GARCH specificaions. Wih hese specificaions, we es wheher here is a change in he volailiy of he series by including a dummy variable in he mean equaion, following Allen and Cruickshank (2000); see also McMillan and Speigh (2004). Specifically, we capure financial ime series characerisics by employing a GARCH(1,1) model, and is EGARCH(1,1) and TGARCH(1,1) exensions. All models have a dummy variable D ) in he mean equaion. Tha is, ( Vola D M a. (7) The dummy variable D akes he value 1 for he final 10 rading days from he conrac expiry, and 0 for all oher days. In his approach, when is posiive and saisically significan, hen Samuelson s hypohesis holds, implying a greaer change in volailiy over his period. Finally, we es Samuelson s hypohesis using wo differen linear regressions. Equaion (8) esimaes volailiy of a near conrac and a far conrac. Following Johnson (1998), he volailiy of near conrac ( Vola ) could be expressed in erms of volailiy of he far conrac ( Vola 1 ). Tha is, Vola a Vola 1. (8) 1 Sucliffe (1993, p. 176) presens some of he definiions of price volailiy.

7 160 Invesmen Managemen and Financial Innovaions, Volume 3, Issue 2, 2006 In his case, Samuelson shows a bea greaer han one ( 1) and saisically significan, implying a greaer volailiy in conracs closes o mauriy (Johnson, 1998, p. 17). Similarly, we apply he regression for he volailiy of a far and a near conrac. In oher words, he volailiy of far conrac is now expressed in erms of he volailiy of he near conrac (i.e. reverse regression). Vola1 a Vola. (9) Equaions (8) and (9) highligh he relaionships beween near and far conracs and depend on he volailiy (see Johnson (1998) for deails). 4. Daa Descripion Daily closing prices for he FTSE/ASE-20 sock index fuures are used over he period of Augus 1999-Augus For he FTSE/ASE Mid 40 sock index fuures, daily closing prices are used over he period of January 2000-Augus The sample is examined boh as a whole and spli ino smaller sub-periods (i.e. monhly conracs). According o Sucliffe (1993), i is necessary o spli up he daa for analysing he mauriy effec. We consider (i) daa wih one series for each index, and (ii) daa for he nearby monh conracs ha leaves us wih 20 or 25 observaions per conrac (monhly conracs). In he firs case, fuures series are consruced by splicing ogeher conracs near mauriy. More specific, he firs closing price (from he curren monh) was picked up in order o make he fuures series for boh indices. Our series have 516 and 406 observaions for FTSE/ASE-20 and FTSE/ASE Mid 40 respecively. In he second case, we have 24 conracs for FTSE/ASE-20 index and 19 conracs for he FTSE/ASE Mid 40 index. This echnique has been used by Herber (1995), Serleis (1991), and Najand and Yung (1991). All daa are colleced from he official web page of he Ahens Derivaives Exchange ( Table 1 has saisical informaion for volailiy. The resuls for boh indices show posiive skewness and high kurosis coefficien. This means ha, he disribuion is skewed o he righ, and also ha he pdf is lepokuric. The Jarque-Bera saisics ess are very high, rejecing normaliy. This is in line wih Kolb (1994) who suggess ha he disribuion of fuures price is non-normal (lepokuric). In addiion, Pagan (1996) shows ha reurns of mos financial asses have semi-fas ails. Also, our resuls suppor he sudy of Allen and Cruickshank (2000) who show evidence of non-normaliy and excess kurosis for several commodiy fuures conracs. Saisics for Volailiy Series FTSE/ASE-20 FTSE/ASE Mid 40 Number of Obs MEAN MEDIAN MAXIMUM MINIMUM STD. DEV SKEWNESS KURTOSIS JARQUE-BERA PROB Table 1 5. Empirical Resuls We begin he empirical analysis by examining he saionariy condiion, which is required for he exisence of he mauriy effec in he price of he fuures conracs, under he heory of uni roo and inegraion (see Floros and Vougas, 2004).

8 Invesmen Managemen and Financial Innovaions, Volume 3, Issue 2, Bessembinder e al. (1996) explain furher he saionariy condiion required for Samuelson s hypohesis and sae ha i is no sricly required. However, according o Allen and Cruickshank (2000), he main condiion for Samuelson s hypohesis is price saionariy. Anderson (1985) suggess ha Samuelson s hypohesis is no correc when spo prices are non-saionary. So, our iniial concern is o es for non-saionariy in our daa. We employ he Augmened Dickey Fuller (ADF) and Philips Perron (PP) ess o es saionariy. The resuls of saionariy es confirm ha volailiy series do no conain uni roos 1. This is consisen wih he findings of Allen and Cruickshank (2000). Furher, he resuls sugges ha Spo series for boh FTSE/ASE-20 and FTSE/ASE Mid 40 conain a uni roo. Also, he saionariy ess for mauriy coefficien (M) show ha ime o mauriy series do no conain a uni roo. Hence, he condiion of Samuelson s hypohesis holds, and we conclude ha he hypohesis is correc for Greek fuures markes. The resuls are in line wih resuls from Walls (1999) for elecriciy fuures conracs. Since he volailiy series do no conain uni roos, we proceed by using a range of volailiy regressions o es for mauriy effecs in he daa. From Equaion 1, when a 1, he volailiy of fuures price will increase as mauriy approaches. Table 2 repors he parameer esimae and resuls from he model specified by Equaion 1. The resuls show a coefficien parameer alpha less han one (i.e. a 1) and significanly differen from zero. Hence, he variance of he change in fuures prices for boh FTSE/ASE-20 and FTSE/ASE Mid 40 will rise as mauriy approaches. So, Samuelson s hypohesis is suspeced o be correc. Index Regression resuls (Equaion 1) Table 2 Par A. S as 1 u FTSE/ASE * FTSE/ASE Mid * Par B. S S 1 a( S 1 S2) u FTSE/ASE * FTSE/ASE Mid * * Significan a he 5% level. Table 3 repors resuls for he regression model specified by Equaion 6. Recall ha in Equaion 6, we expec he inercep o be posiive. This reflecs he price volailiy a conrac mauriy. Also, we expec he mauriy coefficien o be negaive, so ha Samuelson s hypohesis holds. Table 3 repors resuls for boh fuures indices. For FTSE/ASE-20 and FTSE/ASE Mid 40 indices, he wo condiions hold. Hence, here is a mauriy effec and Samuelson s hypohesis is valid for he Greek daa, as price volailiy increases. Also, he consan erm is posiive and significanly differen from zero. This finding is consisen wih Walls (1999) for elecriciy fuures. Index Regression resuls: Vola a ln (One Series) M FTSE/ASE * * FTSE/ASE Mid * * * Significan a he 5% level. Table 3 1 The resuls of saionariy ess are available upon reques.

9 162 Invesmen Managemen and Financial Innovaions, Volume 3, Issue 2, 2006 Tables 4 and 5 repor resuls from Equaion 6, when daa for he nearby monh conracs are considered. In his case, we repor 24 conracs for FTSE/ASE-20 (Table 4) and 19 conracs for FTSE/ASE Mid 40 (Table 5). For FTSE/ASE-20, he consan erm is posiive and significanly differen from zero in 17 ou of 24 conracs. Hence, 70.83% of consan erm coefficiens reflec price volailiy a conrac mauriy, when fuures and spo prices are close. For similar equaion, Walls (1999) finds 92.85% (i.e. 13 of he 14 conracs in elecriciy fuures). The coefficien on mauriy ( ) is negaive and significanly differen from zero for 5 of he 24 conracs. This implies ha 20.8% of bea coefficiens suppor Samuelson s hypohesis. However, i mus be noed ha he coefficien on mauriy is found o be negaive only in 13 of he 24 conracs (i.e %). Similarly, for FTSE/ASE Mid 40, resuls show a negaive and significan slope coefficien for 4 of he 19 conracs. This indicaes ha 21% of slope coefficien parameers imply ha price volailiy increases as he number of days unil conrac mauriy decreases. In oal, is negaive in 13 ou of 19 conracs (i.e %). In addiion, for 15 ou of 19 conracs examined, a is posiive and significanly differen from zero. Regression resuls: Vola a ln (FTSE/ASE-20) M CONTRACT OBS. a a Sep ** Oc Nov ** Dec ** Jan Feb Mar ** Table 4 Apr ** * May June ** July ** Aug ** Sep ** Oc ** * Nov ** * Dec ** Jan ** Feb ** Mar ** Apr ** May ** * June ** July ** * Aug ** ** Indicaes ha consans are posiive and significanly differen from zero. * Indicaes ha slope coefficiens are negaive and significanly differen from zero (i.e. Samuelson s hypohesis is correc). Aserisk(s) denoe(s) significance a he 5% or 10% level.

10 Invesmen Managemen and Financial Innovaions, Volume 3, Issue 2, Regression resuls: Vola CONTRACT OBS. a a ln (FTSE/ASE Mid 40) M a Feb ** Mar ** * Apr ** May ** June July ** Aug ** Sep ** * Oc ** Nov Dec ** Jan ** * Feb ** Mar ** Apr May ** June ** July ** * Aug Table 5 ** Indicaes ha consans are posiive and significanly differen from zero. * Indicaes ha slope coefficiens are negaive and significanly differen from zero (i.e. Samuelson s hypohesis is correc). Aserisk(s) denoe(s) significance a he 5% or 10% level. GARCH Modelling Following Allen and Cruickshank (2000), we apply (G)ARCH specificaions o model fuures volailiy, and use diagnosic ess o find wheher ARCH effecs exis in he daa. Calculaed Ljung-Box saisics sugges ARCH processes, alhough he LM ess sugges ha ARCH effecs are no presen in he volailiy series. Furher, Allen and Cruickshank (2000) fi ARCH (1) and GARCH (1,1) models o heir series. Ljung-Box Q-saisics for up o 20 lagged values show evidence ha he null hypohesis of no auocorrelaion in volailiy series is rejeced a 5% level (i.e. p<0.05), bu he acual level of auocorrelaion is very small. Our findings are consisen wih Allen and Cruickshank (2000) for fuures volailiies from hree inernaional commodiy fuures markes. Then, we fi various GARCH models o he daa. As before, we use wo differen echniques. We fi volailiy models for each index and examine monhly conracs. The bes GARCH represenaion is seleced by AIC. The model wih lowes AIC is aken o fi daa bes. For examining Samuelson s hypohesis, we employ GARCH(p,q), EGARCH(1,1), and TGARCH(1,1) models wih a dummy variable ( D ) in he mean equaion (Equaion 7). Firsly, resuls from variance equaions sugges ha condiional variance exhibis reasonably long persisence of volailiy. Also, in some cases, he leverage effec is saisically significan, indicaing exisence of leverage over he period. Hence, a negaive shock increases he condiional variance. These resuls are no presened in his paper, as we focus on he resuls obained from he mean

11 164 Invesmen Managemen and Financial Innovaions, Volume 3, Issue 2, 2006 equaions. Recall ha he mauriy coefficien,, in he mean equaion should be posiive and saisically significan o have a posiive change in volailiy. Table 6 repors resuls from he mean equaion of several GARCH models for he FTSE/ASE-20 index. The EGARCH model shows a lower AIC value, and hus our daa can be beer modelled by an Exponenial-GARCH. However we highligh ha he coefficien of he dummy variable ( ) is always posiive bu no significanly differen from zero. Therefore, here is no mauriy effec o he FTSE/ASE-20 over his period. However, hree models show a significan increase in volailiy. The null hypohesis of no auocorrelaion is rejeced, and Engle s ARCH-LM es saisics suggess no ARCH processes in he residuals. GARCH Resuls: Vola a (One Series- FTSE/ASE- 20) D Table 6 MODEL AIC DUMMY T-RATIO EGARCH GARCH(1,1) TGARCH GARCH(0,1) * GARCH(0,2) GARCH(2,1) * GARCH(1,2) * GARCH(2,2) * Significan a he 10% level. Using monhly series, we obain various resuls for he mauriy effec in he price of FTSE/ASE-20 fuures conracs. In oal we examine 24 conracs for FTSE/ASE-20 index. The resuls of mean equaions from he seleced models are presened in Table 7. The coefficien of dummy variable, which presens he mauriy effec, is posiive and saisically significan for 11 ou of 24 conracs. Therefore, we find ha 45.8% of mauriy effec coefficiens are posiive and saisically significan in Equaion 6 for monhly conracs. Only in five cases bea coefficiens are no significanly differen from zero. GARCH Resuls: Vola a (Monhly Series- FTSE/ASE- 20) D Table 7 CONTRACT OBS. MODEL DUMMY T-RATIO Sep EGARCH Oc GARCH(1,1) Nov EGARCH Dec GARCH(1,1) * Jan GARCH(1,1) Feb GARCH(1,1) Mar EGARCH Apr TGARCH * May EGARCH June GARCH(1,1) * July EGARCH Aug EGARCH

12 Invesmen Managemen and Financial Innovaions, Volume 3, Issue 2, Table 7 (coninuous) CONTRACT OBS. MODEL DUMMY T-RATIO Sep GARCH(1,1) * Oc EGARCH * Nov GARCH(1,1) * Dec EGARCH * Jan EGARCH * Feb EGARCH Mar EGARCH * Apr EGARCH May EGARCH * June TGARCH July GARCH(1,1) * Aug GARCH(1,1) * Indicaes ha slope coefficiens are posiive and significanly differen from zero (i.e. Samuelson s hypohesis is correc). Aserisk(s) denoe(s) significance a he 5% or 10% level. On he oher hand, resuls obained for FTSE/ASE Mid 40 volailiy daa are quie differen. In his case, he seleced model shows a posiive and saisically significan bea coefficien. In oher words, he EGARCH model indicaes ha volailiy depends on he ime o mauriy. Therefore, since he dummy variable in he mean equaion is always posiive, we conclude ha Samuelson s hypohesis holds for FTSE/ASE Mid 40. Table 8 shows resuls from mean equaions. Q-saisics probabiliies indicae rejecion of he null hypohesis of no auocorrelaion, while LM ess sugges no ARCH in he residuals. These findings are consisen wih Allen and Cruickshank (2000). GARCH Resuls: Vola a (One Series- FTSE/ASE Mid 40) D MODEL AIC DUMMY T-RATIO EGARCH * GARCH(1,1) * TGARCH * GARCH(0,1) * GARCH(0,2) * GARCH(2,1) * GARCH(1,2) * GARCH(2,2) * * Significan a he 10% or 5% level. Table 8 Furhermore, we use monhly series o measure volailiy of FTSE/ASE Mid 40 index and find ha 6 conracs ou of 19 conracs suppor Samuelson s hypohesis. In oher words, 31.58% of conracs show ha he dummy coefficien (i.e. bea) is posiive and significanly differen from zero. Therefore, we conclude ha he variance of FTSE/ASE Mid 40 changes posiively indicaing a rise in volailiy a mauriy. Hence, Samuelson s hypohesis is valid. Noice ha 9 conracs show an insignifican bea coefficien. Table 9 repors resuls from mean equaions for FTSE/ASE Mid 40. Variances in June 2000, Augus 2000, Sepember 2000, March 2001 and July 2001 end o be greaer shorly before expiraion.

13 166 Invesmen Managemen and Financial Innovaions, Volume 3, Issue 2, 2006 GARCH Resuls: Vola a (Monhly Series- FTSE/ASE-40) D Table 9 CONTRACT OBS. MODEL DUMMY T-RATIO Feb EGARCH Mar EGARCH Apr GARCH(1,1) May GARCH(1,1) June GARCH(1,1) * July EGARCH Aug GARCH(1,1) * Sep EGARCH * Oc GARCH(1,1) Nov EGARCH Dec TGARCH Jan EGARCH * Feb GARCH(1,1) Mar EGARCH * Apr EGARCH May GARCH(1,1) June EGARCH July GARCH(1,1) * Aug TGARCH * Indicaes ha slope coefficiens are posiive and significanly differen from zero (i.e. Samuelson s hypohesis is correc). Aserisk(s) denoe(s) significance a he 5% or 10% level. Finally, we follow Johnson (1998) abou he relaionship beween volailiy of near and far conracs. He uses daily selemen prices o calculae volailiy on fuures prices. In oher words, o es Samuelson s hypohesis furher, Equaions 8 and 9 are esimaed. In Equaion 8 (9), volailiy of near (far) conrac is expressed in erms of volailiy of far (near) conrac. Johnson (1998) saes ha mus be greaer han one, o have greaer volailiy in conracs closes o mauriy. Table 10 summarises resuls from Equaion 8. In all cases, alhough variables are posiive and significan, volailiy of near conrac is less han ha of he far conrac. In oher words, is always less han one. This resul is consisen wih he findings of Johnson (1998) for SPI fuures conrac. Therefore, he resuls show less volailiy in conracs closes o mauriy. Index Regression resuls: Vola a Vola 1 (One Series) FTSE/ASE * * FTSE/ASE Mid * * * Significan a he 5% level. Table 10 Running he reverse regression (i.e. Equaion 9), we conclude ha volailiy of far conrac is also less han ha of near conrac. The resuls are presened in Table 11. As is sill less han one, here is no evidence supporing Samuelson s (1965) hypohesis. However, in all cases,

14 Invesmen Managemen and Financial Innovaions, Volume 3, Issue 2, he relaionship beween near and far conracs depends on volailiy. Our findings are in line wih resuls obained by Johnson (1998). According o Johnson (1998, p. 3), This being he case index fuures raders are free o rade whaever conrac is he mos convenien and decisions based on price volailiy can be pu aside, as hey have no impac on conrac choice. Index Regression resuls: Vola1 a Vola (One Series) FTSE/ASE * * FTSE/ASE Mid * * Table 11 * Significan a he significance levels. Aserisk(s) denoe(s) significance a he 5% or 10% level. 6. Summary and Conclusions This paper examines he relaionship beween volailiy and ime o expiraion. We empirically invesigae Samuelson s (1965) hypohesis. Criics of Samuelson s hypohesis sugges a greaer volailiy in fuures conracs closer o mauriy. In general, his is a behavioural hypohesis, where fuures price reflecs curren informaion abou he spo price a mauriy (expiry dae). Hence, as mauriy approaches, he amoun of informaion increases, indicaing large changes in price volailiy. Ohers sugges ha Samuelson s hypohesis holds when informaion abou he harves of commodiies is reaching he marke, and herefore, volailiy depends on he ime of he year. In addiion, a number of papers show ha he hypohesis is correc when informaion flows do no cluser near delivery daes, and also, in markes where equilibrium spo prices are mean revering. Overall, empirical resuls of he volailiy-mauriy relaionship for fuures show lile suppor o Samuelson s hypohesis. In mos cases, he mauriy effec in index fuures markes is no srong. For fuures conracs, oher han index fuures (and commodiies) resuls sugges ha Samuelson s hypohesis is srongly suppored (in paricular for agriculural commodiies and energy fuures). Previous empirical sudies for UK and U.S. financial markes do no indicae clearly wheher here is a mauriy effec in he sock index fuures markes. Here, we analyse he mauriy effec in he Ahens Derivaives Exchange (ADEX). We examine he mauriy effec in he price of FTSE/ASE-20 and FTSE/ASE Mid 40 fuures conracs of he ADEX. Using daily daa, he period is examined as a whole and spli ino monhly periods. A range of volailiy models is employed. These include simple linear regressions and GARCH models. For boh indices, he model proposed by Samuelson (1965) shows ha he variance of he change in fuures prices will rise as mauriy approaches. Secondly, simple linear regressions sugges ha here is a mauriy effec, and herefore, Samuelson s hypohesis seems o be correc for boh FTSE/ASE-20 and FTSE/ASE Mid 40 index. In he case of monhly conracs, we find ha 70.83% (17 ou of 24 conracs) of consan erm coefficiens are significan for FTSE/ASE-20, reflecing price volailiy a conrac mauriy. In addiion, 20.8% (5 ou of 24 conracs) bea coefficiens suppor Samuelson s hypohesis. Similarly, for FTSE/ASE Mid 40, resuls sugges ha 21% (4 ou of 19 conracs) of slope coefficien parameers show price volailiy increases as he number of days unil conrac mauriy decreases. On he oher hand, using GARCH models, we arrive a differen conclusions. Firs, for FTSE/ASE-20, here is no evidence o suppor Samuelson s hypohesis over he whole period. However, using monhly series, resuls from he mean equaions sugges ha 45.8% (11 ou of 24 conracs) of mauriy effec coefficiens suppor Samuelson s hypohesis. For FTSE/ASE Mid 40, resuls (over he whole sample) show a posiive and significan coefficien, indicaing ha volailiy depends on ime o mauriy. Also, using monhly series o

15 168 Invesmen Managemen and Financial Innovaions, Volume 3, Issue 2, 2006 measure volailiy, we find ha 31.58% (6 ou of 19) of he conracs indicae a rise in volailiy a mauriy. Finally, following Johnson (1998) abou he relaionship beween volailiy of near and far conracs, we conclude ha he relaionship depends on he volailiy parameer. In oher words, volailiies of near and far conracs are highly correlaed, implying ha he volailiy facor depends on he ime period under examinaion. In summary, his paper finds ha he volailiy series depends on he ime o mauriy. This resul is no consisen wih he findings of Moosa and Bollen (2001) and Galloway and Kolb (1996) for he S&P 500 fuures conrac. In general, our empirical resuls show ha Samuelson s hypohesis is correc when linear regressions are used. Also, when daa for nearby monh conracs are considered, GARCH models show a sronger suppor o Samuelson s hypohesis raher han linear regressions. Our findings are in line wih resuls obained by Allen and Cruickshank (2000), Walls (1999) and Johnson (1998). Furher empirical work should invesigae pairwise relaionships beween volumemauriy, basis-mauriy and volailiy-volume o a wider range of fuures conracs. References 1. Allen, D.E. and Cruickshank, S.N. (2000), Empirical esing of he Samuelson hypohesis: An applicaion o fuures markes in Ausralia, Singapore and he UK, Working Paper, School of Finance and Business Economics, Edih Cowan Universiy. 2. Anderson, R. W. (1985), Some deerminans of he volailiy of fuures prices, Journal of Fuures Markes, 5, Anderson, R.W. and Danhine, J.P. (1983), The ime paern of hedging and he volailiy of fuures prices, Review of Economic Sudies, 50, Anoniou, A. and Holmes, P. (1995), Fuures rading, informaion and spo price volailiy: evidence for he FTSE-100 sock index fuures conrac using GARCH, Journal of Banking and Finance, 19, Barnhill, T.M., Jordan, J.V. and Seale, W.E. (1987), Mauriy and refunding effecs on reasury-bond fuures price variance, Journal of Financial Research, Vol. X, No. 2, Bessembinder, H., Coughenour, J.F., Seguin, P.J. and Smeller, M.M. (1996), Is here a erm srucure of fuures volailiies? Reevaluaing he Samuelson hypohesis, Journal of Derivaives, 4, Board, J.L. G. and Sucliffe, C.M.S. (1990), Informaion, volailiy, volume and mauriy: an invesigaion of sock index fuures, Review of Fuures Markes, Bollerslev, T. (1986), Generalised Auoregressive Condiional Heeroskedasiciy, Journal of Economerics, 30, Bollerslev, T. (1987), A condiional heeroscedasiciy ime series model for speculaive prices and raes of reurn, Review of Economics and Saisics, 69, Bollerslev, T. and Wooldridge, J.M. (1992), Quasi-maximum likelihood esimaion and inference in dynamic models wih ime varying covariances, Economeric Reviews, 11, Brooks, C., Rew, A.G., and Rison, S. (2001), A rading sraegy based on he lead-lag relaionship beween he spo index and fuures conrac for he FTSE 100, Inernaional Journal of Forecasing, 17, Caselino, M.G. and Francis, J.C. (1982), Basis speculaion in commodiy fuures: he mauriy effec, Journal of Fuures Markes, 2, Chamberlain, T.W. (1989), Mauriy effecs in fuures markes: some evidence from he Ciy of London, Scoish Journal of Poliical Economy, 36, Chen, Y-J, Duan, J-C and Hung, M-W (1999), Volailiy and mauriy effecs in he Nikkei index fuures, Journal of Fuures Markes, 19, Dickey, D.A. and Fuller, W.A. (1979), Disribuion of he esimaors of auoregressive ime series wih a uni roo, Journal of he American Saisical Associaion, 74,

16 Invesmen Managemen and Financial Innovaions, Volume 3, Issue 2, Dusak-Miller, K. (1979), The relaion beween volailiy and mauriy in fuures conracs, in Commodiy Markes and Fuures Prices (Ed.) R.M. Leuhold, Chicago Mercanile Exchange, Chicago, Engle, R.F. (1982), Auoregressive condiional heeroskedasiciy wih esimaes of he variance of U.K. inflaion, Economerica, 50, Engle, R.F. and Ng, V. (1993), Measuring and esing he impac of news on volailiy, Journal of Finance, 48, Floros, C. and Vougas, D.V. (2004), Hedge Raios in Greek Sock Index Fuures Marke, Applied Financial Economics, 14(15), Galloway, T. and Kolb, R.W. (1996), Fuures prices and he mauriy effec, Journal of Fuures Markes, 16, Glosen, L.R., Jagannahan, R., and Runkle, D.E. (1993), On he Relaion Beween he Expeced Value and he Volailiy of he Nominal Excess Reurn on Socks, Journal of Finance, 48, Grammaikos, T. and Saunders, A. (1986), Fuures price variabiliy: A es of mauriy and volume effecs, Journal of Business, 59, Han, L.M. and Misra, L. (1990), The relaionship beween he volailiies of he S&P 500 index and fuures conracs implici in heir call opion prices, Journal of Fuures Markes, 10, Han, L-M, Kling, J. M. and Sell, C.W. (1999), Foreign exchange fuures volailiy: Day-ofhe-week, inraday, and mauriy paerns in he presence of macroeconomic announcemens, Journal of Fuures Markes, 19, Hemler, M.L. and Longsaff, F.A. (1991), General equilibrium sock index fuures prices: heory and empirical evidence, Journal of Financial and Quaniaive Analysis, 26, Hennessy, D.A. and Wahl, T.I. (1996), The effecs of decision making on fuures price volailiy, American Journal of Agriculural Economics, 78, Herber, J.H. (1995), Trading volume, mauriy and naural gas fuures price volailiy, Energy Economics, 17, Hong, H. (2000), A model of reurns and rading in fuures markes, Journal of Finance, Vol. LV, Johnson, J. (1998), Does he Samuelson effec hold for SPI fuures?, Working Paper, Deparmen of Accouning and Finance, The Universiy of Wesern Ausralia. 30. Karpoff, J.M. (1987), The relaion beween price changes and rading volume: a survey, Journal of Financial and Quaniaive Analysis, 22, Kenyon, D., Kenneh, K., Jordan, J., Seale, W. and McCabe, N. (1987), Facors affecing agriculural fuures price variance, Journal of Fuures Markes, 7, Khoury, N. and Yourougou, P. (1993), Deerminans of agriculural fuures price volailiies: evidence from Winnipeg commodiy exchange, Journal of Fuures Markes, 13, Kolb, R.W. (1994), Undersanding Fuures Markes, Kolb Publishing Company, 4 h Ediion. 34. Leisikow, D. (1989), Announcemens and fuures price variabiliy, Journal of Fuures Markes, 9, Milonas, N.T. (1986), Price variabiliy and he mauriy effec in fuures markes, Journal of Fuures Markes, 6, Moosa, I.A. and Bollen, B. (2001), Is here a mauriy effec in he price of he S&P 500 fuures conrac?, Applied Economics Leers, 8, Najand, M. and Yung, K. (1991), A GARCH examinaion of he relaionship beween volume and price variabiliy in fuures markes, Journal of Fuures Markes, 11, Nelson, D. (1991), Condiional Heeroscedasiciy in Asse Reurns: A New Approach, Economerica, 59, Pagan, A. (1996), The economerics of financial markes, Journal of Empirical Finance, 3, Pagan, A.R. and Schwer, G.W. (1990), Alernaive models for condiional sock volailiy, Journal of Economerics, 45,

17 170 Invesmen Managemen and Financial Innovaions, Volume 3, Issue 2, Park, H.Y. and Sears, R.S. (1985), Esimaing sock index fuures volailiy hrough he prices of heir opions, Journal of Fuures Markes, 5, Ruledge, D.J.S. (1976), A noe on he variabiliy of fuures prices, Review of Economics and Saisics, 58, Ruledge, D.J.S. (1979), Trading volume and price variabiliy: new evidence on he price effecs of speculaion, in Fuures Markes: Their Esablishmen and Performance (ed. B.A. Goss), Croom Helm, London, 1986, Samuelson, P.A. (1965), Proof ha properly anicipaed prices flucuae randomly, Indusrial Managemen Review, 6, Segall, J. (1956), The effec of mauriy on price flucuaions, Journal of Business, 29, Serleis, A. (1991), Raional expecaions, risk and efficiency in energy fuures markes, Energy Economics, 12, Serleis, A. (1992), Mauriy effecs in energy fuures, Energy Economics, 14, Sucliffe, C.M.S. (1993), Sock Index Fuures: Theories and Inernaional Evidence, Chapman & Hall. 49. Tauchen, G.E. and Pis, M. (1983), The price variabiliy volume relaionship on speculaive markes, Economerica, 51, Walls, W.D. (1999), Volailiy, volume and mauriy in elecriciy fuures, Applied Financial Economics, 9, Yang, S.R. and Brorsen, B.W. (1993), Nonlinear dynamics of daily fuures prices: Condiional heeroskedasiciy or chaos?, Journal of Fuures Markes, 13, Zakoian, J.M. (1990), Threshold heeroskedasic models, Manuscrip, CREST, INSEE, Paris.