Liquidity and hedging effectiveness under futures mispricing: international evidence. A. Andani *, J.A. Lafuente **, A. Novales *** December 2008

Transcription

1 Liquidiy and hedging effeciveness under fuures mispricing: inernaional evidence A. Andani *, J.A. Lafuene **, A. Novales *** December 2008 Absrac: We analyze he hedging effeciveness of posiions ha replicae sock indexes using corresponding fuures conracs hrough he applicaion of a dynamic, sochasic hedging sraegy proposed by Lafuene and Novales (2003). Conclusive gains do no emerge in any of he markes analyzed over he period considered, relaive o he use of a consan uni hedge raio. These findings are consisen wih he rend observed in he IBEX 35 fuures marke sudy of Lafuene and Novales (2003). Our empirical evidence suggess ha, conrary o wha happens in less liquid markes, he discrepancy beween heoreical and quoed prices in index fuures conracs in fully developed markes does no represen a noise facor ha can be successfully exploied for hedging. Key words: fuures mispricing, hedging effeciveness, JEL classificaion: C22;F31;F47. Financial suppor from Spanish Minisry of Educaion hrough gran SEJ is graefully acknowledged. * Corporae Finance Planning Direcor. Marinsa-Fadesa ** Corresponding auhor: Deparameno de Finanzas y Conabilidad, Faculad de Ciencias Jurídicas y Económicas. Universia Jaume I, Casellón (Spain). E- mail: lafuen@cofin.uji.es. *** Deparameno de Economía Cuaniaiva. Faculad de Ciencias Económicas y Empresariales. Universidad Compluense de Madrid, Somosaguas (Spain).

2 1. INTRODUCTION Financial fuures are frequenly used in hedging operaions, in which he deerminaion of he hedge raio is he main issue. Several heoreical approaches have been proposed in he lieraure o design an opimal hedge wih fuures conracs (see Chen e al., 2003, for an excellen review ha considers minimum variance, meanvariance, expeced uiliy, mean exended-gini coefficien, and semivariance approaches). The usual approach akes ino accoun no only he dynamic naure of marke risk, bu also he fac ha he key idea of hedging is o combine spo and fuures rading o form a porfolio wih negligible flucuaions in is marke value. Under ha view, he decision is o choose he number of fuures conracs ha minimizes he condiional variance of he reurn on he hedged porfolio. The resuling opimal hedge raio is hen obained as he raio beween he condiional covariance of spo and fuures reurns and he condiional variance of fuures reurns. These condiional momens have usually been esimaed from a paricular specificaion of he GARCH family of models (see, for example, Lee and Yoder, 2007, Ku e al., 2007, Choudhry, 2003 and 2004, Park and Swizer, 1995 among many ohers). This sudy reviews he use of fuures conracs on a specific sock marke index as hedging insrumen for a porfolio ha replicaes he marke index. Afer showing ha he empirical evidence is consisen wih he absence of a common ARCH feaure beween he reurns from spo and fuures markes, we adop he heoreical raio proposed by Lafuene and Novales (2003), which is consisen wih he exisence of a noise specific o he fuure marke in addiion o a noise common o spo and fuures marke reurns. A bivariae model wih heeroskedasic disurbances is used o represen he dynamics of reurns in boh markes in order o esimae he minimum variance hedge raio. Afer esimaing wih daa for , empirical evidence obained from ouof-sample simulaions over 2006 for he Nikkei 225, S&P500, FTSE-100, DAX and IBEX 35 fuures markes shows no sysemaic improvemen in hedging effeciveness relaive o using a consan uni hedging raio, conrary o resuls in Lafuene and

3 Novales (2003) for he IBEX 35 index for We explore wheher his resul is consisen wih he rend poined ou by Lafuene and Novales (2003) in heir sochasic opimal hedge raio owards one over he period, wih a decreasing gain in hedging efficiency relaive o a uniary raio, which he auhors jusified on he basis of increased mauriy of a sill underdeveloped and illiquid marke. Our goal is o analyze wheher ha rend coninued afer 1996, as he Spanish marke increased liquidiy, as well as o examine he robusness of our empirical resuls by examining similar evidence in fully developed markes in he US, Japan and Germany. If confirmed, such a finding would sugges ha in maure index fuures markes wih high rading volume, he ime-varying noise ha characerizes basis risk canno be exploied o improve upon he hedging efficiency provided by a sysemaic uni raio. Our resuls are fully in line wih Roll e al. (2007), who presen empirical evidence suggesing ha liquidiy enhances he efficiency of he fuures-cash pricing sysem. The res of he paper is organized as follows. Secion 2 describes he daa used in he analysis and he resuls of esing for he presence of a common ARCH feaure in he spo and fuures markes reurns. Secion 3 presens he model used o deermine he opimal hedge raio and describes he esimaion of he relevan condiional momens. Secion 4 presens he empirical evidence on he evoluion of condiional momens over he analyzed period. A simulaion of hedging rading is performed o es he poenial implemenaion of he model, and secion 5 summarizes and makes concluding remarks. 2. STATISTICAL CHARACTERISTICS OF RETURNS We used daily closing daa for he IBEX35, FT100, NIKKEI225, DAX and SP500 indexes. We selec he rading day for he rollover of conracs according o he evoluion of he deph of fuures marke. Figure 1 shows he average relaive rading volume beween he neares o mauriy conrac and he nex o mauriy conrac. Wih he excepion of he S&P 500 fuures marke, he oher derivaives markes considered exhibi greaer rading volume for he nex o mauriy conrac all he way o expiraion. In he American marke, volumes raded reverse around five days before expiraion.

4 The ime period we consider, January 1997 o December 2006, is ineresing because of he occurrence of several evens: a) he financial crisis of 1998 ha significanly affeced he Unied Saes financial sysem; b) he echnology bubble burs in 2000; c) he subsequen deep generalized recession ha spread across markes and lased unil he beginning of 2003, and d) a subsequen period of sysemaic marke sabiliy, wih he excepion of isolaed crises due o geo-poliical ensions and inflaionary fears. The laer par of his period was characerized by abundan liquidiy in capial markes, wih low ineres raes. Table 1 presens he main saisics for he reurn series, compued as he firs differences of he logs of closing prices beween successive rading days. The sample mean daily reurn is negligible, as expeced from a sysemaically long and shor rading sraegy on consecuive rading days. Likewise, as is usually he case wih daily ime series, sock reurn disribuions show excess kurosis and some skewness, characerisics generally associaed wih condiional heeroskedasiciy. To assess he exisence of ARCH effecs in sock reurns, we perform Engle's Lagrange muliplier es. Empirical values of he es, no repored in he paper, sysemaically rejec he null, poining o he convenience of using some parameerizaion for second order momens of sock marke reurns in he family of GARCH models. In order o empirically jusify he use of our proposed model, which assumes he exisence of a noise common o spo and fuures reurns, ogeher wih a noise specific o he yield of he derivaive insrumen, we follow he approach of Engle and Kozicki (1993) o es he null hypohesis ha here is a linear combinaion of he reurns from he wo markes which is homoskedasic, i.e., ha he ARCH feaure is common o boh reurn series. The empirical values of he es saisic are presened in Table 2, sysemaically leading o rejecion of he hypohesis of a common ARCH feaure. This paern is consisen wih he proposed heoreical model.

5 3. OPTIMAL DYNAMIC HEDGING 3.1 The opimal hedge raio In accordance wih he empirical evidence above, we follow Lafuene and Novales (2003) consider ha he hedging problem can be specified: ds Min Var b h { h } s.. df T, ( ) Sd F, Td ds S d S dz S, S, 1, df F d F dz F dz T, f, T, S, T, 1, N, T, 2, where b denoes he spo posiion we wan o hedge, and h is he hedging fuures posiion, while S and F represen spo and fuures marke prices, respecively. We denoe he correlaion beween he wo Brownian processes: 12, Corr( dz1, dz2). denoes he size of he common noise shared by he wo markes. The discrepancy beween he price quoed in he fuures marke and he heoreical price according o he cos-of-carry valuaion model arises from a basis risk of size N,, ha we specifically aribue o he fuures marke. As shown in Lafuene and Novales (2003), he heoreical expression for he minimum variance (opimal) hedge raio solving he problem above is: h b 1 * 2 s, 12, s, N, 12, s, N, 2 12, s, N, , where N, represens he relaive imporance of he specific noise as S, compared o he common noise. Under he proposed model, he opimal hedge raio remains below one provided ha spo and fuures marke reurns do no share a single common noise. The opimal raio is posiive (implying a shor fuures posiion) when boh disurbances are posiively correlaed. In conras, if he correlaion beween he wo noises was negaive, he opimal hedge raio could lie eiher above or below 1.0.

6 3.2. Esimaing ime-varying variances for he heoreical noises Given he conclusive empirical evidence in he lieraure on he exisence of a coinegraion relaionship beween he logarihms of spo marke and fuures marke prices, our specificaion of he condiional mean for boh series of reurns incorporaes an error correcion erm. Lien (1996) shows ha disregarding he coinegraing relaionship could lead o a smaller han opimal fuures posiion and a relaively poor hedging performance. There is also abundan empirical evidence [see Lien and Yang (2006), among many ohers] supporing he hypohesis ha he coinegraion vecor is (1, -1) which, in urn, implies ha he empirical basis is saionary. Esimaed coinegraion vecors for he pair: [log(fuures price) log(spo price)] by Johansen s procedure, afer normalizing he firs enry o uniy are: S&P 500: [1.000, ], Nikkei 225: [1.000, ], FTSE100: [1.000, ], DAX: [1.000, -1,001], Ibex35: [1, -0,999]. In all cases, he null hypohesis of he coinegraion vecor being [1.000, ] is no rejeced a convenional significan levels. Hence, we define he error correcion erm as he spread beween he logarihm of he spo price and he fuure price. To capure he correlaions beween he reurn innovaions and esimae he condiional variance-covariance marix of spo and fuures markes reurns, we use he bivariae dynamic condiional correlaion (DCC) GARCH model proposed in Engle (2002). Mone Carlo experimens reveal no only ha he bivariae version of he DDC- MV-GARCH model provides a very good approximaion o a variey of ime-varying correlaion processes, bu also ha his model ofen compares favorably wih he simple mulivariae GARCH. The (DCC) GARCH specificaion combines he flexibiliy of univariae GARCH models wih a parsimonious parameric specificaion for he condiional correlaion. Furhermore, bearing in mind he objecives of he presen sudy, Ku e al (2007) compare he DCC-GARCH model proposed in Engle (2002) wih he consan correlaion specificaion, o find evidence of greaer hedging effeciveness from he model wih ime-varying correlaion. Hence, we represen he dynamics of spo and fuures markes reurns, r s, and r f,, hrough he error correcion model: rs rf,, n ( i) ( i) 11 ( i) ( i) 12 rs rf, i i , i s f s, ln S 1 ln F 1 f,

7 wih ~ 0, where 1 is he informaion se available a ime s, f, / 1 N, -1 and is he condiional variance-covariance marix of marke reurn innovaions 1. We represen he ime evoluion of he elemens in he condiional variancecovariance marix by a GARCH(p,q) specificaion wih possible asymmeric effecs: p q s, s Ai () 11 Ai () 12 s, i B( j) 11 B( j) 12 s, f, f i1 Ai () 21 Ai () 22 f, i j1 B( j) 21 B( j) 22 f, 1 2 D11 D12 s, 1 I s, 1, 2 D21 D 22 f, 1 I f, 1 I k, 1 1, si k, 1 0, k s, f 0, si k, 1 0, k s, f Wih regard o he condiional correlaion, he dynamics of he DCC model is: where: sf, ( ) 1 sf, 1 21 m,, k s, f h1 s, h f, h k, 1 k, k, m 2 m 2 1 s, h h h1 f, h Once he condiional momens have been esimaed, he condiional variance for fuures marke reurns, as well as heir condiional covariance and correlaion wih spo marke reurns can be recovered using he expressions in Lafuene and Novales (2003): ˆ f, ˆ s, ˆ N, 2 ˆ ˆ ˆ s, N, 12, ˆ ˆ ˆ ˆ ˆ 2 sf, s, s, N, 12, ˆ 12, ˆ s, ˆ ˆ 2 s, sf, ˆ 2 f, 2 s, 2 ˆ sf, 1 When he Normaliy assumpion was rejeced for he residuals, we esimaed he model using a -Suden condiional disribuion for he innovaions when evaluaing he log-likelihood funcion.

8 2 where, ˆ f, and ˆs, f, denoe he condiional variances of fuures and spo 2 ˆs, marke reurns and heir condiional covariance, as esimaed from he DCC-GARCH model. 4. EMPIRICAL EVIDENCE The sample informaion was divided ino wo sub-periods. The firs period runs from January 1997 o December 2005, which was used for iniial esimaion and specificaion esing. The second sub-period, from January 2006 o December 2006, was lef as an ou-of-sample window o es he effeciveness of simulaed hedging operaions The bivariae GARCH model Table 3 shows he parameers obained in he esimaion of he DCC- GARCH model. In all cases, we sough for he mos parsimonious specificaion possible 2. In he case of he S&P 500 and FTSE-100, a -Suden condiional disribuion was considered, while he Normal disribuion was used for IBEX 35, DAX and Nikkei225. In general, he esimaes show significan coefficiens for ARCH and GARCH effecs, suggesing volailiy clusering in boh marke reurns. Similarly, he parameers ha represen he cross effecs in mean and variance also reveal significan cross-marke ineracions. The speed of adjusmen o shor-run price deviaions from heir long-run equilibrium is also significan, hus evidencing ha he markes are arbiraged in such a way ha he empirical basis has a resriced evoluion over ime. Finally, he presence of significan asymmeric effecs should be noed for he SP500 as well as he Nikkei225. Figures 2a, 2b and 2c (see Appendix 2) show he evoluion over ime of he relaive imporance of he noise specific o he fuures marke, as compared o he common noise, ˆ ˆ N, / s,, in each of he sock markes considered. 2 To assess he abiliy of he esimaed model o capure he main saisical characerisics of marke reurns, a baery of sandard specificaion ess was employed, including he Ljung-Box Q-saisics on he sandardized residual and heir squared values. All series of residuals were found o be free of serial correlaion a he 5% significance level.

9 4.2. Hedging simulaions Having esimaed he model for he period , we incorporaed daa for he ou-of-sample period in 10-day windows. This is a compromise beween mainaining a consan hedge raio and changing he hedge oo ofen, which would imply unbearable ransacion coss. The model was re-esimaed every 10 days, obaining a each poin a hedge raio, before incorporaing addiional daa on a 10-day period for a new esimaion. Once he enire series of hedge raios had been obained for 2006, we implemened wo differen hedging sraegies by applying o each 10-day rading window (he ime inerval [+1, +10]), eiher he hedge raio esimaed he las day in each rolling sample (a ime ) or he average hedge raio compued over he las five rading days in each sample (from -4 o ). Thus, he 250 marke days in he year allowed for performing day hedging operaions wih each sraegy, excep in he case of he NIKKEI, for which only 24 were carried ou. 3 The volailiy of he series of reurns on he porfolio hedged wih he GARCH raio was hen obained under each of hese wo hedging sraegies, compuing he reducion in volailiy relaive o he spo posiion. The volailiy of he porfolio hedged wih he uniary raio was obained similarly, and he implied reducion in volailiy was also calculaed. Finally, we compared he reducion in volailiy obained by applicaion of each of he wo sraegies based on a GARCH raio and he sraegy based on imposing a consan uni raio: Volailiy( hedged posiion) Volailiy( Unhedged posiion) Hedging effeciveness 100 Volailiy( Unhedged posiion) where volailiy is measured by he sandard deviaion of reurns over he period chosen for comparison. We presen resuls obained hroughou he ou-of-sample period, as well as over each quarer. Tables 4 and 5 presen he resuls of applying he wo hedging sraegies described in he previous paragraph. The resuls obained do no exhibi a sysemaic advanage over he uni raio, which suggess ha he incorporaion of ransacion coss 3 Due o he availabiliy of a shorer number of marke days.

10 would make he applicaion of a dynamic hedging sraegy wih he GARCH raio even less ineresing. Finally, we now consider he gain or loss in erms of uiliy, aking ino accoun he ransacion coss from adjusing he posiion in he derivaives marke. To his end, we consider a specificaion of he expeced uiliy funcion: E U ( x) E ( x) 2 ( x) [as in Kroner and Sulan (1993), Lee e al. (2006) and Kofman and McGlenchy (2005), among ohers], where γ denoes he degree of risk aversion, wih he level of risk being measured by he condiional variance of reurns. Denoing ransacion coss by τ and assuming a zero expeced reurn, an invesor would have an expeced uiliy of 2 ** ( x ) if he hedge raio is updaed from * / h b o h ** / b, as agains an expeced uiliy equal o 2 * ( x ) if he hedge raio remains unchanged. Thus, an invesor whose uiliy is given by he specificaion considered will adjus he hedging posiion if and only if: ( 2 ( h / b) ( h / b) ) ( 2 ( h / b) ( h / b) ) 2 ** 2 ** 2 2 * 2 * 2 s, sf, f, s, sf, f, where ** ( h / ) b denoes he hedge raio applied as he resul of he las revision of he fuures posiion. To implemen his sraegy, we consider a risk aversion coefficien of 4 and average coss of % 4, and he opimal raio obained in he las rading day in each rolling sample,, is applied o he following 10 rading days (from +1 o +10). Thus, over he ou-of-sample period, we use he uiliy comparison rule every 10 rading days o decide on wheher o mainain he same hedge raio ha was applied previously, or o change i o he variance-minimizing raio calculaed in he immediaely preceding period. The resuls obained for each marke are presened in Table 6 in erms of aggregae uiliy for 2006, as well as in erms of he uiliy gain relaive o he non- 4 This corresponds o he MEFF Spanish commission of 1.3 Euros for he regular fuures conrac and he 2006 average value of he IBEX 35. As o he ransacion coss associaed o he bid-ask spread, we use he mean spread for he shor-erm index fuures conracs on FTSE-100 (1,4 ), as repored in Fahlenbrach and Sandas (2003). We applied he same commission o all indexes. Since he posiion does no change ofen, our resuls are robus o ransacion coss inside a (.0020%,.0060%) range.

11 hedged marke posiion. Managing he hedge raio according o he uiliy comparison rule ofen provides he highes uiliy gain, bu i is very similar o he one obained under he consan uni raio, as well as o he one emerging from applying he GARCH raio from he previous period. 5. CONCLUSIONS This paper analyzes he use of index fuures as a hedging insrumen for a porfolio ha replicaes he underlying asse for he fuures conrac. To his end, we have used he heoreical model proposed by Lafuene and Novales (2003), which includes a specific noise in he fuures price in addiion o he common noise ha i is assumed o share wih he spo marke price, according o he cos-of-carry valuaion model. We have analyzed daily closing daa on fuures and spo markes for he NIKKEI225, SP500, FT100, DAX and IBEX35 indexes over he period. The null hypohesis on he exisence of a common ARCH feaure [Engle and Kozicki (1993)] underlying he heeroskedasic behavior deeced in spo and fuures markes reurns is rejeced, validaing he exisence of a noise specific o he fuures marke, as included in our economeric model. We esimae an asymmeric bivariae errorcorrecion model wih a DCC-GARCH srucure o represen he condiional mean, variance and covariance of fuure and spo marke reurns, and we simulae ou-ofsample hedging sraegies ha apply a hedge raio calculaed from he esimaed economeric specificaion. The resuls show ha GARCH dynamic sraegies do no lead o a sysemaic improvemen in hedging effeciveness, as compared o he improvemen ha would be obained by applying a consan uni raio. These resuls are in sharp conras wih hose obained using inraday daa for he period by Lafuene and Novales (2003) for he Spanish marke. One reason migh be ha he presen sudy uses daily daa, which implies a loss of informaion on price flucuaions ha may bias upward he esimaion of co-movemen beween spo and fuures prices, moving opimal hedge raios closer o 1.

12 Bu we believe ha wha is really cenral o explain he differen resuls is he fac ha he Spanish marke was in 2006 a significanly more maure marke, wih a sufficienly high level of aciviy ha would quickly correc any arbirage opporuniy. Indeed, our resuls are consisen wih he rend deeced in Lafuene and Novales (2003) abou he opimal hedge raio for he Spanish marke gradually coming closer o 1 owards he end of he sample period, hereby limiing he poenial gain in hedging effeciveness obained from he dynamic GARCH raio. The similar conclusions we have reached for fully developed opion markes in he US, Japan and Germany reinforce ha inerpreaion. The empirical evidence for he Spanish fuures marke is also consisen wih he recen paper of McMillan and Quiroga (2008). These auhors show ha he equilibrium speed of adjusmen beween spo and fuures marke prices was reduced afer he inroducion of he mini-fuures conrac in he Spanish marke in November 2001, he effec being paricularly pronounced afer he second year, when mini-fuures conracs sared being more heavily raded. Even more significanly, he resul ha noisy deviaions from he no-arbirage relaionship in maure marke prices may be of no consequence for improving he efficiency of hedging a spo porfolio wih fuures conracs goes along he lines of Roll e al. (2007), who have shown evidence ha liquidiy enhances he efficiency of he fuures-cash pricing sysem for he S&P 500 sock index fuures marke.

13 REFERENCES Chen, S.-S., Lee, C.-F., Shresha, K., 2003, Fuures hedge raios: a review, The Quarerly Review of Economics and Finance 43, Choudhry, T., 2004, The hedging effeciveness of consan and ime-varying hedge raios using hree Pacific Basin sock fuures, Inernaional Review of Economics & Finance 4, Choudhry, T., 2003, Shor-run deviaions and opimal hedge raio: evidence from sock fuures, Journal of Mulinaional Financial Managemen 13, Engle, R.F., 2002, Dynamic condiional correlaion: A simple class of mulivariae generalized auoregressive condiional heeroskedasiciy models, Journal of Business and Economic Saisics 20, Engle,R.F., Kozicki, S., 1993, Tesing for common feaures, Journal of Business and Economic Saisics 11, Fahlenbrach, R., Sandas, P., 2003, Bid-Ask spreads and invenory risk: Evidence from he FTSE-100 Index Opions Marke. Universiy of Pennsylvania. Kofman P., McGlenchy, P., 2005, Srucurally sound dynamic index fuures hedging, Journal of Fuures Markes 25, Koumos, G., Tucker, M., 1996, Temporal relaionships and dynamics ineracions beween spo and fuures sock markes, Journal of Fuures Markes 16, Kroner, K.F, Sulan, J., 1993, Time-varying disribuions and dynamic hedging wih foreign currency fuures, Journal of Financial and Quaniaive Analysis 28, Ku, Y.H., Chen, H. Chen, K., 2007, On he applicaion of he dynamic condiional correlaion model in esimaing opimal ime-varying hedge raios, Applied Economics Leers 7, Lafuene, J.A., Novales, A., 2003, Opimal hedging under deparures from he cos-of-carry valuaion: Evidence from he Spanish sock index fuures marke, Journal of Banking & Finance 27, Lee, H., Yoder, J., Mielhamner, J., McCluskey, R., 2006, A random coefficien auoregressive Markov swiching model for dynamic fuures hedging, Journal of Fuures Markes 26, Lee, H., Yoder, J., 2007, Opimal hedging wih a regime swiching ime-varying correlaions GARCH model, Journal of Fuures Markes 27,

14 Lien, D. (1996), The effec of he coinegraion relaionship on fuures hedging: A noe, The Journal of Fuures Markes 16, Lien, D., Yang, Li., 2006, Spo-Fuures Spread, Time-Varying Correlaion, and hedging wih Currency Fuures, The Journal of Fuures Markes 26, McMillan D.G., Quiroga R., 2008, Efficiency of he IBEX spo-fuures basis: The impac of he mini-fuures, Journal of Fuures Markes 28, Park, T.H., Swizer, L.N., 1995, Bivariae Garch esimaion of he opimal hedge raios for sock index fuures: A noe, Journal of Fuures Markes 15, Roll, R., Schwarz, E., A. Subrahmanyam, 2007, Liquidiy and he law of one price: The case of he fuures-cash basis, The Journal of Finance, 52, 5,

15 Appendix 1. Tables Table 1 Descripive saisics of sock marke reurns NIKKEI225 SP500 FT100 DAX IBEX35 Spo Fuures Spo Fuures Spo Fuures Spo Fuures Spo Fuures Mean Sandard Dev Asymmery , Kurosis , Table 2 Tesing for common ARCH feaures K Min TR 2 NIKKEI SP FT DAX IBEX Criical values α= α= Noes: The firs panel shows he minimum T*R 2 in a se of regressions of (r s, -dr f, ) 2 on k lags of r 2 s,, r 2 f, and r s, r f,, over a grid of values for d, where T denoes he sample size. The las wo rows show criical values a he α- significance level.

16 Table 3 Maximum Likelihood esimaion of he parameers involved in he DCC-GARCH model NIKKEI225 SP500 FT100 DAX IBEX35 Spo mean equaion a ** ** ** ** a ** ** ** ** a(2) ** ** a(2) ** ** g s ** ** * ** Fuures mean equaion a ** ** ** a ** ** a(2) ** a(2) g f ** ** ** Spo Variance equaion w s ** ** A * ** ** A ** ** ** B ** ** ** * B ** ** D ** ** D ** ** Fuures Variance equaion w f * ** A ** ** A * B * ** ** B ** ** ** ** D ** ** D ** ** Correlaion dynamics k * ** ** * k ** ** ** ** ** * Significan a he 5% level ** Significan a he 1% level Noe: In he case of he S&P 500 and he FTSE-100, he condiional disribuion is a -Suden. Degrees of freedom were esimaed a 7.1 and 5.7 respecively.

17 Table 4 Ou-of-sample hedging effeciveness GARCH Hedging effeciveness Difference (%) Hedge Raio GARCH Uniary GARCH - Uni. NIKKEI225 January-March % % -1.07% April-June % % -0.22% July-Sepember % % -0.32% Sepember-December % % 0.42% Average % % -0.44% SP500 January-March % % -0.41% April-June % % -0.66% July-Sepember % % -0.49% Sepember-December % % -0.31% Average % % -0.50% FT100 January-March % % 0.03% April-June % % 0.20% July-Sepember % % 0.00% Sepember-December % % -0.28% Average % % 0.03% DAX January-March % % -1.52% April-June % % 0.00% July-Sepember % % -0.85% Sepember-December % % -0.14% Average % % -0.45% IBEX35 January-March % % -0.48% April-June % % -0.09% July-Sepember % % -1.72% Sepember-December % % -0.58% Average % % -0.63% Noe: The hedge raio obained for he las day in each rolling sample is applied o he following 10 rading days.

18 Table 5 Ou of-sample hedging effeciveness GARCH Hedging effeciveness Difference (%) Hedge Raio GARCH Uniary GARCH - Uni. NIKKEI225 January-March % % -1.60% April-June % % 0.00% July-Sepember % % -0.68% Sepember-December % % 0.20% Average % % -0.66% SP500 January-March % % -0.40% April-June % % -0.81% July-Sepember % % -0.39% Sepember-December % % -0.21% Average % % -0.51% FT100 January-March % % -0.20% April-June % % 0.03% July-Sepember % % -0.01% Sepember-December % % -0.57% Average % % -0.10% DAX January-March % % -1.20% April-June % % -0.01% July-Sepember % % -0.55% Sepember-December % % -0.13% Average % % -0.34% IBEX35 January-March % % -0.55% April-June % % -0.06% July-Sepember % % -1.57% Sepember-December % % -0.45% Average % % -0.58% Noe: The average hedge raio over he las five rading days in each rolling sample is applied o he following 10 rading days.

19 Table 6 Uiliy gains under differen hedging sraegies NIKKEI225 SP500 FT100 DAX IBEX35 Aggregae uiliy Spo posiion Uniary hedge raio GARCH hedge raio (*) GARCH hedge raio wih decision crierion (**) Uiliy gain on he spo posiion Uniary hedge raio 96.8% 94.0% 96.2% 91.0% 96.4% GARCH hedge raio (*) 96.3% 91.3% 94.4% 89.9% 94.9% GARCH hedge raio wih decision crierion (**) 97.0% 94.1% 96.3% 89.9% 96.5% (*) The hedge raio is changed every 10 days, applying he raio from he las rading day in each rolling sample. (**)The desirabiliy of applying a new raio was appraised every 10 days, he decision being made in accordance wih he expeced uiliy.

20 Appendix 2. Figures 2,50 Relaive average rading volume (Nex o mauriy / Neares o mauriy) 2,00 1,50 1,00 0,50 0, Time o mauriy (days) IBEX FT100 SP500 NIKKEI225 DAX Figure 1. Relaive volume raded in each sock marke: number of nex o mauriy conracs raded over number of Neares o mauriy fuures conracs raded, as a funcion of Time o mauriy.

21 Figure 2a. Raio of esimaed variances for specific and common noise componens: Nikkei225 and S&P500.

22 Figure 2b. Raio of esimaed variances for specific and common noise componens: FT100 and DAX.

23 Figure 2c. Raio of esimaed variances for specific and common noise componens: Ibex35.