Asymmetric Futures Price Distribution and Bid-Ask Quotes *

Transcription

1 Asia-Pacific Journal of Financial Sudies (009) v38 n6 pp89-94 Asymmeric Fuures Price Disribuion and Bid-Ask Quoes * Lars Nordén Sockholm Universiy, Sockholm, Sweden Received February 009; Acceped 4 July 009 Absrac This sudy presens a model for esimaing he asymmery of he fuures price wih respec o he fuures bid-ask spread. Analysis of Daa from he Swedish OMXS 30 index fuures marke shows clear evidence of fuures price asymmery, where he fuures price in general ends o be closer o he bid han o he ask quoe. Moreover, in a fuures marke environmen wih a relaively low liquidiy, he fuures price ends o be closer o he bid quoe, whereas he fuures price is virually symmerically locaed wihin he fuures spread when liquidiy is relaively high. Keywords: Fuures Bid-ask Spread; Asymmery; Liquidiy; Fuures Hedging Performance; Mauriy JEL Classificaion: G3 * This paper has benefied from commens and suggesions by an anonymous referee, Hyoung-jin Park, and seminar paricipans a he Third Inernaional Conference on Asia- Pacific Financial Markes in Seoul, 008. The paper was honored wih he Hana Daeoo Securiies Co., Ld. Ousanding Paper Award a he Third Inernaional Conference on Asia- Pacific Financial Markes in Seoul, 008. The auhor is graeful o he Jan Wallander and Tom Hedelius foundaion and he Tore Browaldh foundaion for research suppor. ** Corresponding Auhor. Address: Professor of Finance, Sockholm Universiy School of Business, S-06 9 Sockholm, Sweden; ln@fek.su.se; Tel: ; Fax:

2 Asymmeric Fuures Price Disribuion and Bid-Ask Quoes. Inroducion Traders a fuures markes and researchers in finance show a common ineres in equilibrium fuures prices. However, observed fuures prices are no necessarily equal o corresponding rue equilibrium prices. In general, i is reasonable o assume ha he fuures price a which an invesor can ake a shor (long) fuures posiion is lower (higher) han he equilibrium fuures price. The difference beween he fuures price for he shor posiion (bid price) and he fuures price for he long posiion (ask price) consiues he fuures bid-ask spread, which reflecs he ransacions coss of rading he fuures conrac. ) From he marke microsrucure lieraure, several models can be used o explain he presence of bid-ask spreads in fuures markes. Invenory models (see Soll, 978; Amihud and Mendelson, 980, 98; Ho and Soll, 98) moivae he spread as marke makers compensaion for bearing he risk of holding undesired invenory. In asymmeric informaion models (see Copeland and Galai, 983; Glosen and Milgrom, 985; Easley and O Hara, 987; Foser and Viswanahan, 994), marke makers migh have an informaional disadvanage and quoe spreads wide enough o compensae for losses from rading wih poenially informed raders (adverse selecion). For derivaives markes, Cho and Engle (999) propose a derivaive hedge heory, according o which derivaives bid-ask spreads are relaed o he liquidiy of he underlying marke. ) If fuures marke makers are able o perfecly hedge heir posiions in he underlying marke, hey will no be exposed o invenory risk or informed rading in he fuures marke. Insead, fuures bid-ask spreads arise from he illiquidiy of he underlying marke, and he widh of fuures spreads reflecs he presence of informed rading in he underlying marke. However, if eiher or boh of he fuures and he underlying marke exhibi low liquidiy, a perfec hedge migh be difficul o obain, and he fuures bid-ask spreads reflec invenory risk and adverse selecion coss of he fuures marke iself. In addiion o no being readily observable, he rue equilibrium fuures price is no necessarily symmerically locaed inside he bid-ask spread. In fac, several reasons ) The fuures bid-ask spread can be regarded as he mos radiional measure of fuures marke liquidiy since i corresponds o a round-rip ransacion cos under normal condiions, i.e. for a given level of fuures rading volume. See Harris (990) for a general discussion of financial markes liquidiy. ) Cho and Engle (999) develop heir derivaives hedge heory for opions. This sudy inegraes heir heory ino a framework for fuures bid-ask spreads. 89

3 Asia-Pacific Journal of Financial Sudies (009) v38 n6 for asymmery can be found in he marke microsrucure lieraure. For insance, in he invenory models, a marke maker alers he bid-ask spread relaive o he rue price in order o arac orders ha would even ou he marke maker s invenory posiion. Hence, if he marke maker faces an excess of buy (sell) orders, he ask (bid) commission will be in excess of he bid (ask) commission, i.e. he rue value will be relaively closer o he bid (ask) quoe. Bossaers and Hillion (99) show ha in foreign exchange markes, possible governmen inervenion gives rise o skewness in he fuure spo exchange rae disribuion. Hence, bid-ask quoes a he currency forward marke are no symmeric around he rue forward prices. Furhermore, Bessembinder (994) repors ha in foreign exchange markes, locaion of bid-ask quoes relaive o he rue asse value is sensiive o dealer invenory-conrol variables. Anoher reason for asymmery, given in Anshuman and Kalay (998) wih reference o he sock marke, is discreeness of quoes. 3) This sudy invesigaes he asymmery of sock index fuures prices wih respec o he fuures bid-ask spread, and conducs an empirical analysis of fuures price asymmery wih daa from he Swedish marke for OMXS 30 index fuures. The analysis conribues o previous research in wo imporan aspecs. Firs, his sudy is he firs o invesigae index fuures price asymmery. While several earlier sudies (see e.g. Wang e al., 997; Aiken e al., 004; Chen and Locke, 004; Nordén, 006) examine how he size of he fuures bid-ask spread is affeced by various explanaory variables, his sudy focuses on he locaion of he fuures spread around he rue equilibrium fuures price. 4) In addiion, he model in Bossaers and Hillion (99) and Bessembinder (994) is inegraed ino he classical fuures hedging framework according o Ederingon (979). The resuling exended model enables an esimaion of he degree of fuures price asymmery in relaion o he fuures bid-ask spread. In addiion, he model provides a framework for esimaing he opimal hedge raio used for fuures hedging, while aking fuures price asymmery ino accoun. Second, he analysis of fuures price asymmery is relaed o fuures marke liquid- 3) Chan and Chung (999) develop a model for equiy opion price asymmery. They find ha equiy opion values are closer o bid quoes han o ask quoes, and ha he degree of asymmery is relaed o opion moneyness, where he asymmery is more pronounced for ou-of-he-money opions han in-he-money opions. 4) In a relaed sudy, Engle and Paon (004) invesigae he imporance of he bid-ask spread as an errorcorrecion erm in he join dynamics of bid and ask quoes for individual socks. While no focusing on he locaion of he rue sock value wihin he bid-ask spread, hey find ha sock buys (sells) have a greaer impac on ask (bid) prices han on bid (ask) prices. 893

4 Asymmeric Fuures Price Disribuion and Bid-Ask Quoes iy, considering he hypohesis ha asymmery of he fuures price wihin he fuures bid-ask spread is more pronounced when liquidiy is low. The analysis considers wo dimensions of liquidiy: widh of he bid-ask spread, and he abiliy o hedge he fuures conrac wih a posiion in he underlying index socks. Thus, asymmery is relaed o wo characerisics of he fuures marke srucure ha are associaed wih he widh of he bid-ask spread and hedging performance: fuures conrac mauriy and he aleraion of he selemen srucure, from forward syle o fuures syle selemen, i.e. from selemen a mauriy only o daily mark-o-marke of he posiions, a he OMXS 30 fuures marke. Accordingly, Nordén (006) finds ha longerm fuures conracs have wider bid-ask spreads, and are associaed wih larger ou-of-sample hedging errors, han shor-erm conracs. Also, Nordén (008) finds narrower fuures bid-ask spreads and smaller hedging errors afer he implemenaion of fuures syle selemen. The resuls show clear evidence of fuures price asymmery, where fuures prices in general are closer o bid han o ask quoes. Furhermore, significan differences are found wih respec o fuures marke liquidiy. Firs, long-erm fuures conracs show a higher degree of asymmery han conracs which are close o expiraion. This resul is consisen wih he idea ha fuures price asymmery is more pronounced when liquidiy is low, since long-erm fuures conracs have significanly wider bidask spreads and are associaed wih larger hedging errors han shor-erm conracs. Second, during he period before (afer) he change in fuures selemen srucure, when he fuures bid ask spreads are relaively wide (narrow), fuures prices end o be close o he bid (midpoin) quoes. These resuls imply ha represening a fuures price wih he bid-ask midpoin resuls in a bias, paricularly in an illiquid fuures marke, overesimaing he rue fuures price. The asymmery of fuures prices wih respec o fuures bid-ask spreads has several imporan implicaions. For example, using he midpoin of he observed fuures bid-ask spread as a proxy for he unobserved rue fuures price migh produce misleading conclusions in empirical fuures marke research if he rue fuures price is closer o he bid han o he ask quoe. In addiion, on a pracical noe, daily selemen in a fuures conrac is ofen based on he daily change in he midpoin of he closing fuures bid and ask quoes. 5) Thus, given he resuls of his sudy, using he 5) A he OMXS 30 index fuures marke, daily selemen in a given conrac is based on calculaing he daily gains or losses by comparing he midpoin of he closing fuures bid and ask quoes wih he corresponding midpoin from he previous rading day. 894

5 Asia-Pacific Journal of Financial Sudies (009) v38 n6 midpoin fuures closing price migh no only inroduce a bias in daily mark-omarke adjusmens due o asymmery, bu also an uninended, and even unwarraned, dependence on fuures marke liquidiy and mauriy of he fuures conrac. The res of he sudy is organized as follows. Secion presens he srucure of he OMXS 30 fuures marke and he daa used in his sudy. Secion 3 oulines he mehodology of his sudy, developing a framework for invesigaing and esing for asymmery in fuures prices. Secion 4 conains he empirical resuls of he sudy. Finally, he sudy ends in secion 5 wih some concluding remarks.. Fuures Marke Srucure and Daa In Sepember 986 he OMX Nordic Exchange (OMX) inroduced he OMXS 30 index. 6) I consiss of a value-weighed combinaion of he 30 mos acively raded socks a he Sockholm Sock Exchange (SSE). The purpose of he inroducion was o use he OMXS 30 index as an underlying securiy for rading in sandardized opions and fuures conracs. Since he inroducion, he marke for OMXS 30 index fuures has grown subsanially. Presenly, i is ranked among he en larges index fuures markes in he world. 7) All lised derivaive securiies a OMX are raded wihin a fully compuerized sysem. The rading sysem consiss of a limi order book managed by he exchange, and all rading is conduced via exchange members. A member is eiher a dealer or a marke maker. Thus, he rading seing is a combinaion of an elecronic maching sysem and a marke making faciliy. Marke makers are likely o endorse derivaives marke liquidiy by posing bid-ask spreads on a coninuous basis. The rading sysem a he SSE is based on he same kind of limi order book as a he OMX. However, here are no marke makers a he SSE. The OMXS 30 index fuures marke consiss of conracs wih differen mauriies. A any ime hroughou a calendar year, rading is possible in a leas hree fuures conracs, wih up o one, wo and hree monhs lef o mauriy, respecively. On he 6) A he ime of he inroducion he OMXS 30 index was called he OMX index. As of November 5, 004, he OMX alered he index name, wihou alering any aspecs of index calculaion or derivaives conracs specificaion. See OMX Press Release ; OMX-index changes name. In May 007, NASDAQ acquired OMX, and he newly merged company was renamed he NASDAQ-OMX Group upon compleion of he deal on February 7, ) In 006, he OMX was ranked among he 0 larges fuures exchanges in he world (all caegories), see he laes issue of Fuures Indusry Magazine; hp:// =

6 Asymmeric Fuures Price Disribuion and Bid-Ask Quoes fourh Friday each monh, when he exchange is open for rading, one se of conracs expires and a new one wih ime o mauriy equal o hree monhs is iniiaed. For insance, owards he end of Sepember, he Sepember conracs expire and are replaced wih newly issued December conracs. A he same ime, conracs mauring in Ocober (wih one monh lef o mauriy) and November (wih wo monhs lef o mauriy) are also lised. In addiion o he basic mauriy cycle, mauriies wih up o wo years exis. These long conracs expire in January and are included in he basic mauriy cycle when hey have less han hree monhs lef o mauriy. All OMXS 30 index fuures are cash seled a mauriy. On February 4, 005 he OMX inroduced a daily selemen srucure for he OMXS 30 index fuures conracs. Before his dae, he fuures were seled a mauriy only. This forward syle selemen srucure is very unusual for sandardized fuures conracs. In fac, he world s major sock index fuures conracs have a daily selemen scheme for gains and losses. The inroducion of daily selemen of gains and losses also enabled he OMX o simplify he margin rules for mainaining a fuures posiion. Before as well as afer February 4, 005 a fuures rader mus deposi an iniial margin in order o esablish a fuures posiion. In addiion, before February 4, 005 raders had o deposi an addiional variaion margin o back up he iniial margin requiremen. 8) Wih he daily selemen srucure, in which conracs are markedo-marke on a daily basis, he OMX does no require raders o deposi any variaion margin during he remaining ime o mauriy of he fuures conrac. This sudy uses a daa se, which consiss of all OMXS 30 index fuures conracs. The sample conains daily closing daa beween January, 00 and Sepember, 007. The daa are obained from OMX and include informaion of fuures quoes (closing bid and ask quoes, and daily high and low ransacion prices) for each conrac. Addiional daa, also obained from he OMX, include daily OMXS 30 index values for he sample period. The nearby fuures conracs, i.e. he series closes o mauriy, are always used in he empirical analysis excep during expiraion weeks. Each Thursday before he expiraion week, he curren fuures posiion is rolled over o he nex conrac. 8) On he implemenaion dae of he new selemen rules, he OMX reained he forward syle selemen srucure for he currenly lised conracs. A he same ime, new conracs wih he same ime o mauriy were esablished, adoping he new fuures syle selemen srucure. Hence, for a ransiion period, unil April 5, 005, fuures conracs are lised wih boh ypes of selemen. Since mauriy of hese duallised conracs, all new conracs are lised wih fuures syle selemen only. 896

7 Asia-Pacific Journal of Financial Sudies (009) v38 n6 3. Mehodology This secion provides a framework for esing he degree of fuures prices asymmery relaive o he fuures bid-ask spread. One major obsacle in an analysis of his kind is ha rue fuures prices are no direcly observable. Bossaers and Hillion (99) as well as Bessembinder (994) develop and use a model for esimaing he locaion of bid-ask quoes of foreign exchange spo raes relaive he rue spo values. This sudy amends he model for an analysis of fuures price asymmery, while aking he derivaive hedge heory according o Cho and Engle (999) ino accoun. Throughou he analysis, he rue sock index value is assumed o be observable and equal o he index value repored by he OMX. Denoe he rue unobservable equilibrium fuures price a ime as F, he corresponding observable fuures bid and ask quoe as B f, and A f, respecively, and he concurren rue observable sock index value as S. The rue fuures price is assumed o be locaed beween he prevailing fuures bid and ask quoe according o: F = θb + ( θ) A () f, f, where θ (0 θ ) is a parameer measuring he locaion of he fuures price relaive he bid-ask quoes. The fuures price is closer o he bid (ask) quoe if θ > ½ (θ < ½). Moreover, if θ = ½ he fuures price is symmerically locaed in he middle of he bid-ask spread. Equaion () can be expressed in erms of a change in he rue fuures price as: Δ F = θδ B + ( θ) Δ A () f, f, By denoing Δ SPDf, = ΔAf, Δ Bf, as he change in he fuures bid-ask spread, and by dividing each erm wih S, equaion () can be rearranged ino: ΔAf, ΔSPDf, ΔF = θ + (3) S S S In equaion (3), he change in ask quoes ( Δ A f, ) and he change in quoed bid-ask spread ( Δ SPD f, ) are readily observable, whereas he rue fuures price change ( Δ F ) 897

8 Asymmeric Fuures Price Disribuion and Bid-Ask Quoes is no. 9) However, following he derivaives hedge heory from Cho and Engle (999), and sandard exbook arbirage heory, a fuures price change mus a any ime be aligned wih a corresponding change in he underlying sock index. Moreover, he classical porfolio hedging model according o Johnson (960), Sein (96) and Ederingon (979) can be used o express a relaive sock index change as a funcion of he corresponding normalized fuures price change. In he regression framework, he hedging model can be wrien as: ΔS S ΔF = + + (4) a b u S where Δ S is he observable sock index price change a ime, a is a consan erm, b is he opimal hedge raio, or he amoun of fuures conracs i akes o hedge a given posiion in he sock index, and u is a residual erm. By solving equaion (4) for Δ F / S and subsiuing ino equaion (3), he following regression expression is obained: ΔA ΔSPD ΔS = β + β + β + ε (5) S S S f, f, 0 where β = a/ b, β = θ, β = / b, and ε = ( / bu ). 0 Equaion (5) consiues a regression of normalized fuures ask quoe changes on normalized fuures spread changes and sock index reurns. The regression coefficien β is an esimaor of he locaion parameer θ. A es for symmery of he rue fuures price wih respec o he bid-ask spread boils down o esing he null hypohesis of no fuures price asymmery: H : β = ½ (6) agains he alernaive hypohesis ha β ½. The coefficien β is an esimaor of he inverse of he opimal fuures hedge raio. Hence, he model in equaion (5) can be used o evaluae he fuures hedging performance. In paricular, he framework enables an analysis of he imporance of fu- 9) All changes are expressed in relaive erms, by dividing hrough by he lagged sock index value, o ensure saionariy a he esimaion sage. 898

9 Asia-Pacific Journal of Financial Sudies (009) v38 n6 ures price asymmery for fuures hedging performance. By imposing he resricion β = ½ in equaion (5), he coefficien β becomes an esimaor of /b from he classical fuures hedging framework in equaion (4), under he assumpion ha he rue fuures price equals he midpoin of he fuures bid and ask quoes. The model in equaion (5) is furher exended o ake ino accoun he possibiliy ha fuures marke liquidiy affecs fuures price asymmery. The analysis highlighs wo fuures marke srucure characerisics ha are known from previous sudies o be associaed wih liquidiy, measured as boh he widh of he fuures bid-ask spread and fuures hedging performance. Firs, Nordén (006) finds ha fuures conrac mauriy is a significan deerminan of fuures bid-ask spreads. According o his resuls, long-erm fuures conracs have significanly wider spreads han shor-erm conracs. For he purpose of his sudy, a long-erm (shor-erm) fuures conrac is classified as a conrac wih a ime o mauriy longer han (shorer han or equal o) days. 0) Accordingly, a dummy variable D, ( D, ) is defined o be equal o one if he curren fuures conrac is long-erm (shor-erm), and zero oherwise. The second fuures marke specific characerisic wih poenial influence on liquidiy and marke qualiy is he aleraion of he selemen srucure a he OMXS 30 index fuures marke. Following he conrac redesign, he OMXS 30 index fuures conrac is from April 8, 005, and onwards subjec o daily selemen in a similar manner as on major index fuures exchanges all over he world. Nordén (008) argues for and finds some empirical evidence of a more liquid fuures marke, wih beer hedging performance, afer he change. Hence, he sample used in his sudy is divided ino wo pars, before and afer he change in fuures selemen srucure, called Period I and Period II respecively, and he dummy variable Q, ( Q, ) akes on a uniy value before (afer) he change, and zero oherwise. The dummy variables are used o exend he model in equaion (5), o allow for differen degrees of fuures price asymmery under differen fuures marke condiions according o: 0) Remember ha only he nearby fuures conrac is used in he analysis and ha each Thursday before he expiraion week, he curren fuures posiion is rolled over ino he nex conrac. Hence, he acual shor-erm mauriy is in he range six o days, whereas long-erm mauriy is beween and 4 days. 899

10 Asymmeric Fuures Price Disribuion and Bid-Ask Quoes ΔA ΔSPD ΔS (7) S S S f, f, = β0 + β (, i j) Di, Qj, + β + ε ij, = where β (, i j) consiues an esimaor of he locaion parameer θ for each of he four possible combinaions of long- and shor-erm fuures mauriy ( Di,, i =, ) and before and afer he change in selemen srucure ( Q, j, j =, ). The specificaion in equaion (7) enables he formulaion of several ess of fuures price asymmery. An overall es of fuures price asymmery under any ype of fuures marke srucure boils down o he es of he join null hypohesis: H : β (, ) = ½, β (, ) = ½, β (, ) = ½, β (, ) = ½ (8) agains he alernaive hypohesis ha a leas one of he four equaliies does no hold. The es of Hypohesis in equaion (8) is a more general es han he one in connecion wih Hypohesis in equaion (6). A es of wheher he level of fuures price asymmery varies wih differen degrees of fuures marke liquidiy corresponds o he es of he following null hypohesis: (,) = (,) = (,) = (,) (9) H : 3 β β β β agains he alernaive hypohesis ha a leas one of he equaliies is no saisfied. The es of Hypohesis 3 in equaion (9) is a es of wheher he exension of he model in equaion (5), o he model wih dummy variables in equaion (7), is worhwhile. In oher words, if he null hypohesis can be rejeced, he degree of fuures price asymmery is significanly differen wih respec o eiher, or boh, fuures mauriy and he periods before and afer he change in fuures selemen srucure. An isolaed es of wheher fuures price asymmery is differen for long- and shorerm fuures conracs is obained by esing he following join hypohesis: H : 4 β(, ) = β(, ) β β (, ) = (, ) (0) agains he alernaive hypohesis ha a leas one of he wo equaliies in equaion (0) does no hold. Similarly, a parial es of difference in asymmery level over Period I (low liquidiy) and Period II (high liquidiy) is implemened using he following null hypohesis: 900

11 Asia-Pacific Journal of Financial Sudies (009) v38 n6 (, ) = (, ), β(, ) = β(, ) () H : 5 β β where one significan inequaliy is enough o rejec he hypohesis. 4. Empirical Resuls 4. Summary Saisics Table repors summary saisics for daily changes in he variables from equaion (5); fuures ask quoe changes, fuures bid quoe changes, changes in he fuures bidask spread, and corresponding relaive changes in he underlying index value (index Table. Summary Saisics, Fuures Bid-ask Quoes Changes Table conains summary saisics for daily changes in he fuures ask quoe ( Δ A f, / S ), bid quoe ( Δ Bf, / S ), bid-ask spread ( Δ SPDf, / S ), and sock index value ( Δ S / S ). Saisics are obained for all daa and subsamples divided wih respec o fuures conrac mauriy ( days and > days respecively), and wo periods; where Period I is beween January, 00, and April 5, 005, and Period II is beween April 8, 005, and Sepember, 007. The augmened Dickey-Fuller es (Fuller, 996) is used o es he null hypohesis ha each ime series has a uni roo over he enire sample period. For each series, a MacKinnon (996) onesided p-value under each null hypohesis is repored. The rows labeled Tes for difference conain p-values from a -es for equaliy beween means, and an F-es for equaliy beween variances beween subsamples. Δ A f, Δ Bf, ΔSPDf, ΔS Saisic All daa Mean 3.8e e e e-4 (n =,430) Sandard deviaion.39e-.39e- 8.69e-4.36e- S S S S Uni roo es, p-value Period I Mean.97e-4 4.4e-4 -.8e e-5 (n = 83) Sandard deviaion.58e-.57e-.0e-3.53e- Period II Mean 7.4e e-4 -.0e-5 7.6e-4 (n = 607) Sandard deviaion.08e-.09e- 3.93e-4.0e- Tes for difference -es, p-value (means) F-es, p-value (variances) Fuures mauriy days Mean.75e-4.65e-4 9.8e e-4 (n = 73) Sandard deviaion.35e-.34e- 7.8e-4.33e- Fuures mauriy > days Mean 3.64e e-4-4.4e e-4 (n = 698) Sandard deviaion.43e-.43e- 9.95e-4.40e- Tes for difference -es, p-value (means) F-es, p-value (variances)

12 Asymmeric Fuures Price Disribuion and Bid-Ask Quoes reurns). Using he daa over he enire sample period, a uni roo es is firs performed o ensure saionariy of each variable used in he empirical analysis. An augmened Dickey-Fuller es (see Fuller, 996) is used o es each individual null hypohesis ha he ime series has a uni roo. Using he p-values from MacKinnon (996), i is possible o rejec each null hypohesis of a uni roo a any reasonable significance level. Hence, all variables can be considered o be saionary. Table presens mean and sandard deviaion for each variable over he enire sample period, and for wo subsample periods during which where Period I is beween January, 00 and April 5, 005 and Period II is beween April 8, 005 and Sepember, 007. The sample pariion is moivaed by a change in he specificaion of he OMXS 30 index fuures conrac fully implemened from April 8, 005, and onwards. Before he change, he OMXS 30 index fuures conrac was seled a mauriy only, much like a forward conrac, and was no marked-o-marke on a daily basis. Afer he change, he fuures conrac is marked-o-marke on a daily basis. The F-ess for differences in variances beween Period I and Period II indicae a significanly higher sandard deviaion of fuures ask quoe changes, bid quoe changes, spread changes, and underlying index reurns, respecively, during Period I relaive Period II. Hence, he wo periods can be regarded as wo differen volailiy regimes, wih a significanly higher (lower) volailiy during Period I (II). In Table, variable means and sandard deviaions are also displayed separaely for days when he nearby fuures conrac has less or more han days lef o mauriy. According o he F-ess in Table, boh fuures ask quoe changes, and bid quoe changes are significanly more volaile when fuures mauriy is longer han days, a leas a he en percen significance level, han when mauriy is shorer. In addiion, changes in he fuures bid-ask spread are significanly more volaile, a any reasonable significance level, for long-erm fuures conrac han for shor-erm conracs. The fac ha he corresponding sandard deviaion of sock index price changes is no significanly differen wih respec o he lengh of fuures conrac mauriy suppors he noion ha he disribuion of fuures quoe changes depends on conrac mauriy, afer aking he corresponding underlying sock index reurn disribuion ino accoun. Alogeher, he summary saisics in Table sugges ha i migh be imporan o ake fuures mauriy and he differen periods, peraining o dissimilar fuures conracual specificaions, ino accoun in he analysis of fuures price asymmery. 90

13 Asia-Pacific Journal of Financial Sudies (009) v38 n6 Table. Summary Saisics, Fuures Liquidiy and Marke Qualiy Measures Table conains summary saisics for relaive fuures bid-ask spread (Spread), Parkinson s (980) measure of volailiy (Volailiy), and he ou-of-sample fuures hedging errors (Hedge Error), which are based on a daily naïve hedge model wih a uniy hedge raio, and a daily rollover sraegy using acual fuures bid and ask quoes. Saisics are obained for all daa and subsamples divided wih respec o fuures conrac mauriy ( days and > days respecively), and wo periods; where Period I is beween January, 00, and April 5, 005, and Period II is beween April 8, 005, and Sepember, 007. The augmened Dickey-Fuller es (Fuller, 996) is used o es he null hypohesis ha each ime series has a uni roo over he enire sample period. For each series, a MacKinnon (996) one-sided p-value under each null hypohesis is repored. The rows labeled Tes for difference conain p-values from a -es for equaliy beween means, and an F-es for equaliy beween variances beween subsamples. Saisic Spread Volailiy Hedge Error All daa Mean 7.5e-4 9.5e-3-7.e-4 (n =,430) Sandard deviaion 7.45e e-3.55e-3 Uni roo es, p-value Period I Mean 9.e-4.e- -9.8e-4 (n = 83) Sandard deviaion 8.9e-4 9.0e-3.9e-3 Period II Mean 4.35e-4 7.6e e-4 (n = 607) Sandard deviaion 3.e-4 4.3e-3.90e-3 Tes for difference -es, p-value (means) F-es, p-value (variances) Fuures mauriy days Mean 6.74e e e-4 (n = 73) Sandard deviaion 6.74e e-3.36e-3 Fuures mauriy > days Mean 7.58e e-3-8.0e-4 (n = 698) Sandard deviaion 8.e e-3.73e-3 Tes for difference -es, p-value (means) F-es, p-value (variances) Table conains summary saisics for variables measuring differen aspecs of fuures marke liquidiy and qualiy; he fuures relaive bid-ask spread, i.e. he absolue spread divided by he midpoin beween he fuures bid quoe and ask quoe, Parkinson s (980) high-low inraday measure of fuures volailiy, which equals h l (ln F ln F ) / ln, where ln F h and ln F l is he highes and lowes observed fuures ransacion price on day, respecively, and an ou-of-sample daily fuures hedge error, which is based on holding he index sock porfolio over one day and hedging wih a shor posiion in one fuures conrac. The hedging sraegy consiues a naïve hedge model wih a uniy hedge raio, and daily roll-over using acual fuures bid and ask quoes. Togeher wih sample mean and sandard deviaion for each variable, for he enire 903

14 Asymmeric Fuures Price Disribuion and Bid-Ask Quoes sample and subsamples divided ino Period I and Period II, and wih respec o shorand long-erm fuures conrac mauriy as in Table, Table presens he resuls from a -es, and an F-es of each hypohesis of equal variable mean, and sandard deviaion respecively across separae subsamples. Enclosed are also resuls from a uni roo es for saionariy of each variable over he enire sample period. According o he p-values from MacKinnon (996), i is possible o rejec each null hypohesis of a uni roo a any reasonable significance level. Hence, he variables displayed in Table are considered saionary. The relaive fuures bid-ask spread shows a significan decrease in mean level from Period I o Period II a a very low level of significance. Evidence of a concurren increase in he mean fuures volailiy is found using Parkinson s (980) measure, where according o he -es, he hypohesis of an equal mean volailiy across Period I and Period II can be rejeced. In addiion, he ou-of-sample hedging errors of he fuures conrac are on average significanly lower and show significanly lower variabiliy in Period II han in Period I. Hence, he summary saisics indicae a more liquid and higher qualiy fuures marke due o a significanly narrower bid-ask spread and a significanly beer hedging performance of he fuures conrac during Period II following he implemenaion of fuures syle selemen. The summary saisics in Table also show ha he shor-erm fuures conracs have a significanly narrower (and less variable) fuures bid-ask spread, as he -es (F-es) of he hypohesis of an equal bid-ask spread mean (variance) for shor- and long-erm fuures conracs alike can be rejeced a he five percen (any) significance level. Moreover, fuures hedging performance is significanly beer when he shorerm conracs are used. A a very low significance level, he hypohesis ha he variance of ou-of-sample fuures hedging errors is he same for shor- and long-erm conracs can be rejeced. Thus, aken ogeher, he summary saisics lead o an idenificaion of wo saes of fuures marke liquidiy and qualiy, wih high (low) liquidiy and qualiy during Period II (Period I) and for shor-erm (long-erm) conracs, consisen wih he resuls from Table. Table 3 conains more deailed summary saisics for he fuures bid-ask spread and hedging error. Each variable s mean and sandard deviaion is calculaed for each of he four possible combinaions of fuures conrac mauriy and Period I and Period II, respecively. The resuls are presened in a coningency able forma. Accordingly, a row-wise es of equaliy beween periods, condiional on mauriy, and a 904

15 Asia-Pacific Journal of Financial Sudies (009) v38 n6 Table 3. Summary Saisics for Fuures Bid-ask Spread and Hedging Error Table 3 conains summary saisics for relaive fuures bid-ask spread (Panel A), and he ouof-sample fuures hedging errors (Panel B), which are based on a daily naïve hedge model wih a uniy hedge raio, and a daily roll-over sraegy using acual fuures bid and ask quoes. Saisics are obained for subsamples divided wih respec o fuures conrac mauriy ( days and > days respecively), and wo periods; where Period I is beween January, 00, and April 5, 005, and Period II is beween April 8, 005, and Sepember, 007. For each lengh of mauriy, p-values are repored row-wise from a -es for equaliy beween means, and an F-es for equaliy beween variances beween Period I and Period II. Likewise, for each period I and II respecively, p-values are repored column-wise from a -es for equaliy beween means, and an F-es for equaliy beween variances beween shor- and long-erm conracs. Panel A: Fuures Bid-Ask Spread Saisic Period I Period II Tes for difference Mean 8.57e-4 4.4e-4 -es, p-value = Fuures mauriy days Sandard deviaion 7.98e-4.89e-4 F-es, p-value = n Mean 9.90e e-4 -es, p-value = Fuures mauriy > days Sandard deviaion 9.79e e-4 F-es, p-value = Tes for difference n es, p-value (means) F-es, p-value (variances) Panel B: Fuures Hedge Error Saisic Period I Period II Tes for difference Mean -0.80e e-4 -es, p-value = Fuures mauriy days Sandard deviaion.7e-3.73e-3 F-es, p-value = n Mean -.05e-3-5.0e-4 -es, p-value = Fuures mauriy > days Sandard deviaion 3.3e-3.06e-3 F-es, p-value = Tes for difference n es, p-value (means) F-es, p-value (variances) column-wise es of equaliy beween shor- and long-erm mauriies, condiional on period, is performed for each variable mean and sandard deviaion. Focusing on mean bid-ask spreads (Panel A), for each caegory of mauriy, he mean fuures bidask spread is significanly higher during Period I relaive Period II, a any reasonable significance level. Moreover, during Period I, long-erm fuures conracs have significanly higher mean bid-ask spread han shor-erm conracs do a he five percen significance level, whereas he corresponding Period II level of significance is only en percen. As illusraed in Figure, he fuures bid-ask spread is on average wides for long-erm conracs in Period I, marked by he dummy variables D and 905

16 Asymmeric Fuures Price Disribuion and Bid-Ask Quoes Q from equaion (6), and narrowes for he shor-erm conracs in Period II, idenified by D and Q, respecively. Figure. Fuures Bid-ask Spread Under Differen Fuures Marke Condiions Figure displays he average relaive fuures bid-ask spread under differen fuures marke condiions; long-erm conracs in Period I (DQ), long-erm conracs in Period II (DQ), shor-erm conracs in Period I (DQ), and shor-erm conracs in Period II (DQ). Longerm (shor-erm) fuures conracs have a mauriy longer han (less han or equal o) days. Period I is beween January, 00, and April 5, 005, and Period II is beween April 8, 005, and Sepember, 007. Fuures Bid-Ask Spread (imes,000) DQ DQ DQ DQ Figure. Hedging Error Sandard Deviaion Under Differen Fuures Marke Condiions Figure displays he average sandard deviaion of ou-of-sample fuures hedging errors, which are based on a daily naïve hedge model wih a uniy hedge raio, and a daily roll-over sraegy using acual fuures bid and ask quoes, under differen fuures marke condiions; long-erm conracs in Period I (DQ), long-erm conracs in Period II (DQ), shor-erm conracs in Period I (DQ), and shor-erm conracs in Period II (DQ). Long-erm (shorerm) fuures conracs have a mauriy longer han (less han or equal o) days. Period I is beween January, 00, and April 5, 005, and Period II is beween April 8, 005, and Sepember, 007. Hedge Error Sandard Deviaion (imes 0,000) 3 0 DQ DQ DQ DQ 906

17 Asia-Pacific Journal of Financial Sudies (009) v38 n6 Using sandard deviaion of hedging errors as he preferred fuures hedging performance measure, Table 3 (Panel B) shows a significanly lower sandard deviaion, and hus beer hedging performance during Period II relaive Period I, irrespecive of conrac mauriy and significance level. In addiion, during boh Periods I and II, he shor-erm fuures conracs are significanly beer hedging vehicles han he long-erm conracs, a leas a he one percen significance level. Hence, boh mauriy and he selemen srucure maers significanly for he hedging performance of he fuures conrac. As can be seen in Figure, he sandard deviaion of he fuures hedging errors is highes for long-erm fuures conracs during Period I (as indicaed by D and Q ), and lowes for shor-erm conracs during Period II ( D and Q ). 4. Fuures Price Asymmery Resuls Table 4 repors he esimaion resuls from he regression model according o equaion (5), where he relaive fuures ask quoe change is regressed on he normalized fuures bid-ask spread change and he relaive sock index price change. The model is esimaed wih he maximum likelihood algorihm according o Bernd e al. (974). A moving average residual erm is added o ake residual auocorrelaion ino accoun, and he residual variance is modeled using a GARCH(, ) formulaion. In addiion, he sandard errors are correced for heeroskedasiciy and auocorrelaion in he residuals (en lags) according o Whie (980) and Newey and Wes (987), respecively. According o he repored Ljung-Box (LB) saisics in Table 4, neiher absolue nor squared residuals show significan remaining auocorrelaion, rending an appropriae regression model. The esimaed locaion parameer equals Using he Wald es from Table 4, Hypohesis (ha β = ½) can be rejeced a he one percen significance level. Hence, since he locaion parameer is significanly larger han ½, he regression resuls provide evidence of an asymmeric fuures price, and ha he rue fuures value is closer o he bid quoe han o he corresponding ask quoe. In addiion, he coefficien β is very close o one. ) Thus, he opimal fuures hedge raio, which is obained as he inverse of β, is on average equal o one as well. ) A Wald es of he null hypohesis ha β is equal o one, resuls in a p-value equal o

18 Asymmeric Fuures Price Disribuion and Bid-Ask Quoes Table 4. Regression Resuls and Tes for Fuures Asymmery I Table 4 conains esimaion resuls from he model of relaive fuures ask quoe changes according o equaion (5). The coefficiens in each model are esimaed using daa from he sample period January, 00, hrough Sepember, 007, wih he maximum likelihood algorihm in Bernd e al. (974). In addiion, he sandard errors are correced for heeroskedasiciy and auocorrelaion in he residuals (0 lags) according o Whie (980), and Newey and Wes (987). The model equaions are: ΔA ΔSPD ΔS = β + β + β + ε, ε = e θe S S S f, f, 0 e I ~ N(0, h ) h = α + α e + α h 0 where Δ Af, / S denoes he relaive change in he fuures ask quoe, SPDf, / S is he fuures bid-ask spread, Δ S / S is he relaive sock index price change, I denoes he informaion se available a ime, N(0, h ) is he normal disribuion wih h as he condiional variance of fuures ask quoe changes a ime. Coefficien Esimae -value p-value β e β β θ α e α α Log Likelihood 6,93.5 LB Raw Residuals, 0 lags (p-value) 0.50 LB Squared Residuals, 0 lags (p-value) Wald es, H : β = ½ (p-value) Table 5 presens he resuls from he exended regression model in equaion (6), where Panel A conains he regression resuls and Panel B resuls from hypoheses ess. The exended model allows for differen locaion parameers in four disinc subsamples, defined by differen combinaions of shor- and long-erm mauriy fuures conracs and during Period I and Period II, i.e. before and afer he implemenaion of fuures syle selemen srucure respecively. Consequenly, β (, ) ( β 908 (, ) ) is an esimaor of he locaion parameer for long-erm fuures conracs during Period I (Period II), whereas β (, ) ( β (, ) ) corresponds o he locaion parameer for shor-erm conracs during Period I (Period II). From Panel B in Table 5,

19 Asia-Pacific Journal of Financial Sudies (009) v38 n6 Hypohesis in equaion (8) (ha all four locaion parameers are joinly equal o ½) can be rejeced a any reasonable significance level. Moreover, Hypohesis 3 (ha he locaion parameer is consan over he four differen subsamples) is rejeced a leas a he five percen level, indicaing significanly differen fuures asymmery levels during some of he subsamples. Table 5. Regression Resuls and Tes for Fuures Asymmery II Table 5 conains esimaion resuls from he model of relaive fuures ask quoe changes according o equaion (5). The coefficiens in each model are esimaed using daa from he sample period January, 00, hrough Sepember, 007, wih he maximum likelihood algorihm in Bernd e al. (974). In addiion, he sandard errors are correced for heeroskedasiciy and auocorrelaion in he residuals (0 lags) according o Whie (980), and Newey and Wes (987). The model equaions are: ΔA ΔSPD ΔS, ε = S S S e θ e, f, f, = β0 + β (, i j) Di, Qj, + β + ε i, j= e I ~ N(0, h ) h = α + α e + α h 0 where Δ Af, / S denoes he relaive change in he fuures ask quoe, SPDf, / S is he fuures bid-ask spread, Δ S / S is he relaive sock index price change, I denoes he informaion se available a ime, N(0, h ) is he normal disribuion wih h as he condiional variance of fuures ask quoe changes a ime, D, ( D, ) is a dummy variable defined o be equal o one if he fuures conrac mauriy > days ( days), and zero oherwise, and Q, ( Q, ) is a dummy variable equal o one during Period I (Period II), i.e. before (afer) April 5, 005. Panel A: Regression Resuls Coefficien Esimae -value p-value β e β (, ) β (, ) β (,) β (, ) β θ α 0.03e α α Log Likelihood 6,935.7 LB Raw Residuals, 0 lags (p-value) 0.40 LB Squared Residuals, 0 lags (p-value)

20 Asymmeric Fuures Price Disribuion and Bid-Ask Quoes Table 5. Regression Resuls and Tes for Fuures Asymmery II (coninue) Panel B: Wald Tess H : H : 3 H : 4 H : 5 Hypohesis χ -value p-value β (, ) = β (, ) = β (,) = β (, ) = ½ β (, ) = β (, ) = β (,) = β (, ) β (, ) = β (,), β (, ) = β (, ) β (, ) = β (, ), β (,) = β (, ) Figure 3. Esimaed Fuures Price asymmery Under Differen Fuures Marke Condiions Figure 3 displays he esimaed locaion parameer θ under differen fuures marke condiions; long-erm conracs in Period I (DQ), long-erm conracs in Period II (DQ), shorerm conracs in Period I (DQ), and shor-erm conracs in Period II (DQ). Long-erm (shor-erm) fuures conracs have a mauriy longer han (less han or equal o) days. Period I is beween January, 00, and April 5, 005, and Period II is beween April 8, 005, and Sepember, 007. Esimaed locaion parameer θ 0,5 0 DQ DQ DQ DQ Hypohesis 4 (ha here is no difference in fuures price asymmery for shor- and long-erm fuures conracs) canno be rejeced by a Wald es a any reasonable significance level. However, i is possible o rejec Hypohesis 5 (ha here is no difference in fuures price asymmery beween Period I and Period II) a he one percen significance level. The laer significan resul originaes from he large difference beween he wo coefficiens β (, ) and β (, ), i.e. he locaion parameer for longerm fuures conracs during Period I and Period II, which are esimaed a 0.8 and 0.44, respecively. Figure 3 illusraes he resul ha long-erm fuures conracs during Period I exhibi he highes level of price asymmery, wih rue fuures prices on average closer o bid quoes han o ask quoes. Thus, he resuls are consisen 90

21 Asia-Pacific Journal of Financial Sudies (009) v38 n6 wih he noion ha fuures price asymmery is relaed o fuures marke qualiy, since fuures conracs ha are long-erm raher han shor-erm, and are lised during Period I raher han during Period II, have relaively lower liquidiy, in erms of wider bid-ask spreads, and larger hedging errors according o he resuls in Table and Table Concluding Remarks A model is developed in order o esimae he asymmery of rue equilibrium fuures prices wih respec o corresponding observable fuures bid-ask spreads. Using daa from he acively raded OMXS 30 index fuures marke, clear evidence of asymmery in fuures prices is found; in general, fuures prices end o be closer o bid han o ask quoes. Evidence is also found in suppor of he hypohesis ha fuures price asymmery is more pronounced when he fuures marke exhibis low liquidiy. This sudy is he firs of is kind ha invesigaes sock index fuures price asymmery. While several oher sudies have examined how he size of he fuures bid-ask spread is affeced by various explanaory variables, his sudy conribues o previous research by focusing on he locaion of he rue equilibrium fuures price wihin he fuures spread. In addiion, in order o achieve he empirical esimaion of he degree of fuures price asymmery, his sudy develops a model wihin he classical fuures hedging framework. The model is furher exended o highligh he linkage beween fuures price asymmery and fuures marke liquidiy. The resuls of his sudy have several implicaions for fuure research. Firs, he resuls imply ha represening a fuures price wih he bid-ask midpoin resuls in a bias, paricularly in an illiquid fuures marke, overesimaing he rue fuures price. Hence, using he midpoin of he observed fuures bid-ask spread as a proxy for he unobserved rue equilibrium fuures price migh produce misleading conclusions in empirical fuures marke research if he rue fuures price is closer o he bid han o he ask quoe. Second, he model for esimaing he degree of fuures price asymmery in relaion o he fuures bid-ask spread provides a framework for esimaing he opimal hedge raio used for fuures hedging, while aking fuures price asymmery ino accoun. Accordingly, he model can be used o evaluae he imporance of fuures price asymmery for fuures hedging. This ask is, however, lef for fuure re- 9

22 Asymmeric Fuures Price Disribuion and Bid-Ask Quoes search. Finally, on a pracical noe, daily selemen in an exchange-raded fuures conrac is ofen based on he daily change in he midpoin of he closing fuures bid and ask quoes. Thus, using he midpoin fuures closing price migh no only inroduce a bias in daily mark-o-marke adjusmens due o asymmery, bu also an unwarraned dependence o fuures marke liquidiy and mauriy of he fuures conrac. 9

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