Eleftheria Kostika HIGHER MOMENTS MODELING AND FORECASTING. Thesis by. A dissertation submitted for the degree of Doctor of Philosophy

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1 HIGHER MOMENTS MODELING AND FORECASTING Thesis by Elefheria Kosika A disseraion submied for he degree of Docor of Philosophy Ahens Universiy of Economics and Business Deparmen of Managemen Science and Technology December 007 1

2 ATHENS UNIVERSITY OF ECONOMICS & BUSINESS DEPARTMENT OF MANAGEMENT SCIENCE & TECHNOLOGY HIGHER MOMENTS MODELING AND FORECASTING Thesis by Elefheria Kosika A disseraion submied for he degree of Docor of Philosophy Supervisory Commiee: Professor Aposolos P. Refenes Associae Professor Dimiris A. Georgousos Assisan Professor Raphael N. Markellos

3 Acknowledgmens I would like o express my sincere appreciaion o my supervisor Professor Aposolos Refenes for his coninuous guidance and suppor hroughou he underaking of his Thesis. I am paricularly indebed o Dr. Raphael Markellos for he numerous discussions, ideas and suggesions he provided me hroughou my sudies. I would like o hank he members of my commiee, Professor Dimirios Georgousos, Professor Sofoklis Brissimis, Professor Kosas Syriopoulos, Dr. Chrisos Taranilis and Dr. Dimirios Kainourios, whose suggesions and advices made a subsanial impac on my developmen. Las bu no leas, I would like o hank my family for heir suppor. 3

4 Absrac This hesis exends he lieraure on porfolio selecion wih higher momens by invesigaing how non-normaliy of reurns and higher momens may affec hedging sraegies. In order o accoun for ime varying skewness and kurosis in opimal hedge raio esimaion, he ARCD model proposed by Hansen (1994) is employed, in which full condiional densiy is modelled allowing for condiional shape parameers. In a horserace of models, he dynamic hedging effeciveness of he ARCD is compared o ha obained by OLS, error-correcion, exponenial moving averages, and, univariae and mulivariae GARCH. Effeciveness is measured in-sample and ou-of-sample using he minimum variance mehod. Spo and fuures daily closing prices are used for sock indices from he US, UK and Germany for he period January 1999 o Sepember 004. The resuls sugges ha he hedging performance using he ARCD ouperforms ha obained by he compeing approaches. Also, in his hesis an alernaive, simplified mulivariae model is proposed, he simplified Mulivariae Auoregressive Condiional Densiy Model (S-ARCD) which is compaible wih he skewness and kurosis of he financial reurns and is easy o be implemened increasing he compuaional efficiency. I is based also, on he Auoregressive Condiional Densiy Model (ARCD) proposed by Hansen (1994) 4

5 and involves he esimaion only of he univariae specificaion of he above model. The condiional variances are calculaed by he simple univariae models, and he condiional covariance is hen impued from hese variance esimaes. The S-ARCD is illusraed o forecas he VaR of aggregae equiy porfolios for he US and UK and foreign exchange porfolio for EUR and GBP agains USD and is compared o he ad hoc mulivariae version of GARCH (Wang, Yao, 005) and BEKK models. The resuls, using boh saisical and economic crieria, sugges ha he simplified mulivariae version of ARCD performs a leas well as he oher wo models indicaing he higher momens imporance in volailiy forecasing and VaR calculaion. Finally, he hesis examines he Auoregressive Condiional Densiy (ARCD) applicaion in combinaion wih he framework of Hasbrouck (1995) in order o invesigae empirically he predicive abiliy of error-correcion models based on he coinegraion relaionship beween opion and spo prices. Alhough he coinegraion relaionship beween spo and opion prices has been sudied in he lieraure, he implicaions in erms of error-correcion modelling have no ye been empirically examined. Using daily index and opion closing prices from he US, France and Germany we idenify significan coinegraion relaionships in each marke and esimae nonlinear error-correcion models. Also, he role of higher momens in he coinegraion relaionship is no imporan. The error-correcion model resuls sugges ha boh opion and spo prices conain informaion abou opion price reurns. 5

6 Περίληψη H διατριβή µελετά τις στατιστικές ιδιότητες και τη µέτρηση κινδύνων λαµβάνοντας υπόψη την επίδραση της τρίτης και τέταρτης ροπής των συναλλαγµατικών ισοτιµιών και των παράγωγων χρηµατοοικονοµικών προϊόντων. Πιο συγκεκριµένα, ερευνάται πως η µη κανονικοποιηµένη κατανοµή των αποδόσεων των χρηµατοοικονοµικών προϊόντων µπορεί να αποφασίσει την λήψη αποφάσεων για την αντιστάθµιση των κινδύνων που ενέχουν. Πιο αναλυτικά, εστιάσαµε την έρευνα στον τύπο ετεροσκεδαστικότητας που εµφανίζεται στη χρονολογική σειρά των χρηµατοοικονοµικών προϊόντων καθώς και στις κατανοµές των προβλέψεων αυτών που εξαρτώνται από την εξαρτηµένη κατανοµή του κανονικοποιηµένου λάθους (he condiional disribuion for he normalized error). Στη διατριβή µας εξετάσαµε µια νέα τάξη παραµετρικών µοντέλων τα ονοµαζόµενα υποδείγµατα Αυτοπαλίνδροµης Υπό Συνθήκη Εκτίµησης Πυκνότητας (Auoregressive Condiional Densiy Esimaion Μοdels, ARCD) εναλλακτικά των υποδειγµάτων Αυτοπαλίνδροµης Υπό συνθήκη Ετεροσκεδαστικότητας (AuoRegressive Condiional Heeroscedasiciy, ARCH). Tα ARCD µοντέλα επιτρέπουν εκτός του µέσου και της διακύµανσης να µεταβάλλονται µε το χρόνο και άλλοι παράµετροι όπως η κυρτότητα και η συµµετρικότητα. Σκοπός της µελέτης αυτής είναι να καταλήξουµε αν οι παράµετροι που καθορίζουν το σχήµα της κατανοµής που ακολουθούν οι νοµισµατικές ισοτιµίες επηρεάζονται σηµαντικά από την πληροφόρηση που υπάρχει στην χρηµαταγορά. Στη συνέχεια µελετήθηκε η σκοπιµότητα µοντελοποίησης ανώτερων ροπών µέσω των οικονοµετρικών υποδειγµάτων ΑRCD στην αγορά των µετοχών, κατασκευάσθηκαν διάφορα επενδυτικά χαρτοφυλάκια µε υποκείµενους τίτλους µετοχές και παράγωγα προϊόντα, µετρήθηκε ο συνολικός κίνδυνος αυτών των χαρτοφυλακίων µε διάφορα εργαλεία αποτίµησης κινδύνου και προτάθηκαν 6

7 διάφορα µέτρα αντιστάθµισης κινδύνου (hedging measures) στο σύνολο διαφόρων χαρτοφυλακίων που περιέχουν µετοχές και συµβόλαια µελλοντικής εκπλήρωσης µε υποκείµενο τίτλο τους δείκτες των µετοχών. Το αποτέλεσµα της παραπάνω έρευνας, είναι η επιλογή της βέλτιστης αντιστάθµισης του κινδύνου (opimal choice of hedging insrumens beween spo and opion markes) των αγορών των µετοχών και των δικαιωµάτων προαίρεσης µε υποκείµενο τίτλο δείκτες των αντίστοιχων αγορών µετοχών. Τέλος, µια νέα µέθοδος µοντελοποίησης πολυµεταβλητών µοντέλων, το Simplified Mulivariae Auoregressive Condiional Densiy Model (S-ARCD) προτείνεται και εκτιµάται, το οποίο βασίζεται στην κατανοµή µας χρησιµοποιούµε την εξαρτηµένη συµµετρική suden s κατανοµή (condiional skewed suden s disribuion) η οποία επιτρέπει τόσο στη συµµετρικότητα και τη κύρτωση να εξαρτώνται από το χρόνο. Το παραπάνω µοντέλο εφαρµόζεται σε χαρτοφυλάκιο µετοχών και συναλλαγµατικών ισοτιµιών και γίνεται µέτρηση και πρόβλεψη του Value a Risk. Tα αποτελέσµατα του S-ARCD, χρησιµοποιώντας οικονοµικά και στατιστικά κριτήρια, συγκρίνονται µε τα αποτελέσµατα των οικονοµετρικών µοντέλων ad hoc GARCH(1,1) και ΒΕΚΚ και αποδεικνύεται ότι η σηµαντικότητα της κύρτωσης και συµµετρικότητας στην πρόβλεψη της µεταβλητότητας και του υπολογισµού του Value a Risk. Παράλληλα µε την εκτίµηση των οικονοµετρικών µοντέλων ΑRCD και S- ΑRCD, ολοκληρώθηκε η µελέτη σκοπιµότητας µοντελοποίησης ανώτερων ροπών µέσω υποδειγµάτων ARCD στο πλαίσιο µη γραµµικών υποδειγµάτων διόρθωσης σφάλµατος (error correcion) για την έρευνα κοινών στοχαστικών τάσεων µεταξύ της spo αγοράς των µετοχών και των τιµών των δικαιωµάτων προαίρεσης µε υποκείµενο τίτλο δείκτες των αντίστοιχων αγορών µετοχών (opions on indices) δηλαδή µελετήθηκε αν συνολοκληρώνονται και είναι co inegraed τα παραπάνω επενδυτικά προϊόντα. Επεκτείναµε τη µέθοδο του Hasbrouck (1995), και αποδεικνύουµε ότι τα µοντέλα αποτίµησης των δικαιωµάτων προαίρεσης στην παρουσία θορύβου εµπεριέχουν µια µη γραµµική συνολοκληρωµένη σχέση των 7

8 δικαιωµάτων προαίρεσης και των υποκείµενων τίτλων (nonlinear coinegraion relaionship beween spo and opion prices), ενώ ο ρόλος των ανώτερων ροπών δεν είναι σηµαντικός. Ερευνήσαµε αυτή τη σχέση µε το µοντέλο Black-Scholes χτίζοντας µοντέλα διόρθωσης λάθους δηλαδή error-correcion models, χρησιµοποιώντας καθηµερινές τιµές από τρεις αγορές µετοχών: Αµερική, Γερµανία και Γαλλία. Tα εµπειρικά αποτελέσµατα επιβεβαιώνουν τη σηµασία της µη γραµµικής συνολοκληρωµένης σχέσης των δικαιωµάτων προαίρεσης και των υποκείµενων τίτλων (nonlinear coinegraion relaionship beween spo and opion prices), για τον υπολογισµό των αναλογιών αντιστάθµισης κινδύνου (hedge raios esimaion) όσον αφορά τη µέθοδο της στρατηγικής αντιστάθµισης κινδύνου (hedging sraegies implemenaion). 8

9 Table of Conens Acknowledgmens... 3 Absrac... 4 Περίληψη... 6 Table of Conens... 9 Lis of Tables Lis of Figures Inroducion Εισαγωγή... Par A: Univariae and Mulivariae Higher Momens Modeling and Forecasing CHAPTER I ARCH-GARCH Processes: Review and Opimal Hedging I.1. The Processes I.1.1. Univariae ARCH-GARCH formulaions I.1.. Maximum Likelihood Esimaion I.. Daa Descripion I.3 The Economeric Mehodologies for opimal hedging I.3.1. Ordinary Leas Squares Model (OLS) I.3. Coinegraion and Error Correcion Modeling (ECM) Exponenially Weighed Moving Average Model (EWMA) I.3.4 Univariae GARCH(1,1) model I.4 Esimaion and Resuls of he Economeric Processes I.5. Summary and Discussion CHAPTER II Mulivariae GARCH Processes: Review and Opimal Hedging II.1. Mulivariae GARCH formulaions II.1.1 Bivariae BEKK(1,1,1) model II.1. The ad hoc mulivariae of GARCH(1,1) model... 6 II.. Daa Descripion II.3. Empirical Analysis II.3.1 Opimal Hedging Analysis for BEKK(1,1,1) model II.3. VaR Evaluaion for he mulivariae GARCH models

10 II.4. Summary and Discussion CHAPTER III Higher Momens Modeling: Univariae Approaches III.1. Mehodology: A model for condiional skewness and kurosis III.1.1. The generalized Suden disribuion III.. Daa Analysis and Empirical Applicaion III..1. Hedging Effeciveness... 9 III.3. Summary and Discussion CHAPTER IV Mulivariae Higher Momens Approaches IV.1. The Simplified Mulivariae Auoregressive Condiional Densiy Model 99 I.V.. Daa Descripion and Empirical Analysis IV.3. Evaluaion IV.4. Resuls IV.5. Summary and Discussion Par B: Coinegraion, Error Correcion and Higher Momens beween Spo and Opion Prices CHAPTER V The informaion Conen of Opion Prices V.1. Mehodology V.. Daa Descripion V.3. Empirical Applicaion V.4. Summary and Discussion Conclusions References Appendix A Appendix A1: Derivaion of he ARCD characerisic funcion wih mean zero and uni variance Appendix A: Derivaion of he hird and four momens of he ARCD and S- ARCD characerisic funcion Appendix B Criical Values for he independen Nyblom L c -Tes

11 Lis of Tables Table I. 1 Descripive Saisics of spo and fuures index reurns Table I. OLS Opimal Hedge Raio Esimaion Table I. 3 Error-Correcion Modelling Opimal Hedge Raio Esimaion Table I. 4 GARCH(1,1) esimaes for Spo and Fuures Index reurns Table I. 5 Comparison of hedging effeciveness beween models Table II. 1 Descripive Saisics of equiy indices and foreign exchanges reurns. 64 Table II. GARCH-BEKK(1,1) esimaes for Spo and Fuures Index reurns Table II. 3 Comparison of Hedging Effeciveness Table II. 4 Ad Hoc Mulivariae GARCH(1,1) esimaes for DJ, FTSE, (F+D), EUR, GBP, (E+G) Table II. 5 BEKK(1,1,1) esimaes for (FTSE,DJ), (EUR, GBP) Table II. 6 Ou of sample VaR evaluaion measures for he equiy indices porfolio a 95% confidence level Table II. 7 Ou of sample VaR evaluaion measures for he equiy indices porfolio a 99% confidence level Table II. 8 Ou of sample VaR evaluaion measures for he foreign exchange currencies porfolio a 95% confidence level Table II. 9 Ou of sample VaR evaluaion measures for he foreign exchange currencies porfolio a 99% confidence level Table III. 1 ARCD esimaes for Spo and Fuures Index reurns..89 Table III. Comparison of hedging effeciveness beween models Table IV. 1 S-ARCD esimaes for FTSE, DJ, (F+D), EUR, GBP,.(E+G) Table IV. Descripive Saisics of he esimaed covariance,, ˆ F D σ

12 σ,, Table IV. 3 Descripive Saisics of he esimaed covariance ρˆf Table IV. 4 Descripive Saisics of he esimaed correlaion, D, ρˆe Table IV. 5 Descripive Saisics of he esimaed correlaion, G, ρˆf Table IV. 6 Correlaion marix for he esimaed correlaion, D, ρˆe Table IV. 7 Correlaion marix for he esimaed correlaion, G, Table IV. 8 Uncondiional Unbiasedness esing Regressions for he equiy indices Table IV. 9 Uncondiional Unbiasedness esing Regressions for he currency series Table IV. 10. Ou of sample VaR evaluaion measures for he equiy indices. porfolio a 95% confidence level Table IV. 11 Ou of sample VaR evaluaion measures for he equiy indices porfolio a 99% confidence level Table IV. 1 Ou of sample VaR evaluaion measures for he foreign currencies porfolio a 95% confidence level Table IV. 13 Ou of sample VaR evaluaion measures for he foreign currencies porfolio a 99% confidence level Table V. 1 Coinegraion and Error-Correcion Analysis beween opion prices and Black-Scholes efficien prices calculaed using implied volailiies Table V. Coinegraion and Error-Correcion Analysis beween opion prices and Black-Scholes efficien prices calculaed using ARCD volailiy esimaors..13 Table V. 3 Daily Rae of Reurns for he porfolios including boh sraddle opions and long posiions on equiy indices..136 Table B. 1 Nyblom Tes Criical Values..164 ˆ E G 1

13 Lis of Figures Figure III. 1 Various Densiies for differen values of λ and η Figure III. Sandardized ARCD residual disribuions for Spo index reurns Figure IV.1a Time varying Condiional Covariances for S-ARCD model..107 Figure IV.1 b Time Varying Condiional Covariances for Ad Hoc GARCH(1,1) model Figure IV.1 c Time Varying Condiional Covariances for BEKK(1,1) model Figure V. 1 Scaerplo beween logarihmic observed call opion prices, Black- Scholes call opion prices and spo index prices..18 Figure V. Scaerplo beween logarihmic observed pu opion prices, Black- Scholes pu opion prices and spo index prices

14 Inroducion Undersanding and esimaing ime varying condiional variances and covariances is imporan for many issues in finance since here are many applicaions ha rely on mulivariae covariance models. I is essenial, for opimal hedging, asse allocaion, derivaives pricing and risk managemen, he accurae modelling and forecas of he asses reurns co-movemen. Bollerslev, Engle and Wooldridge (1988), Cecchei (1988), Myers and Thompson (1989), Baillie and Myers (1991), Kroner and Sulan (1993), Ng and Kroner (1998), argue ha financial prices are characerized by ime varying variances and covariances, presening a variey of mulivariae GARCH models. Bollerslev, Engle and Wooldridge (1988) suggesed he VEC model and he diagonal VEC (DVEC) model in which he variances depend only on heir pas squared errors and he covariances on heir pas cross-producs of errors. Given he excessively large parameers needed o esimae he VEC model and he necessiy o impose srong resricions on he parameers Engle and Kroner (1995) proposed he BEKK parameerizaion avoiding unrealisic assumpions such as ha he correlaion beween he condiional variances is consan (Consan correlaion model by Bollerslev 1990), and guaraneeing ha he ime varying covariance marix is posiive definie. Addiional models can be found in Engle, Ng and Rohschild (1990b) who proposed facor models (FGARCH), in Alexander and Chibumba (1997) who proposed he orhogonal GARCH models (O- GARCH), in Tse and Tsui (00) and of Engle (00) who suggesed he Dynamic Condiional Correlaion models (DCC). All he above models assume ha asse reurns are joinly normally disribued ignoring he fac of asymmery in volailiy 14

15 and covariance, fa ailness and skewness. However, asymmery and skewness in disribuion, is found in many financial asses since heir reurn disribuions depar far away from normaliy. For insance, French, Schwer and Sambaugh (1987) rejeced normaliy claiming significan condiional skewness in daily residuals of he SP500 reurns, Hong (1988) found abnormally high kurosis in daily NYSE sock reurns, Harvey (1995) observed deviaions form normaliy in emerging sock markes indices, Harvey and Siddique (1999) showed ha condiional skewness is imporan and consisen wih asymmeric variance in daily, weekly and monhly reurns of seleced markes. Since here is well esablished sylized evidence ha financial reurns exhibi fa ails and skewness, a lo of sudies focused on using of non normal disribuions o beer model his excess kurosis and skewness. More specifically, in he univariae framework, a large variey of condiional densiies has been employed o accommodae he asymmery and fa ailness. Hansen (1994) was he firs o propose a Skew-Suden disribuion which allows for condiional higher momens. Recenly, Harvey and Siddique (00a, 00b), Jondeau and Rockinger (006) and Yan (005), Brooks, Burke and Persand (005) among ohers, have discussed ways o joinly esimae ime varying condiional variance and skewness, bu heir resuling formulaion is difficul o be implemened, moreover in a mulivariae exension. More precisely, none of he popular mulivariae models are compaible wih he skewness and kurosis of asse reurns since hey assume mulivariae normaliy. A few sudies exis on he higher momens modelling in mulivariae approaches. Harvey, Ruiz and Shepard (1994) and Fiorenini, Senana and Galzolari (003) replace mulivariae Gaussian densiy wih suden densiy by leing condiional innovaions o follow a Suden- disribuion. Sahu, Dey, and Branco (001), and Bawens and Lauren (005) propose a mulivariae skew Suden densiy wih suppor on he full Euclidian space. Their main finding is ha his densiy improves he qualiy of ou of sample VaR forecass. More recenly, Hafner and Rombous (004) and Rombous and Verbeek (005) apply a mulivariae semi parameric 15

16 GARCH esimaion echnique o capure higher momens showing ha in wihin sample porfolios VaR he model s superioriy and robusness is confirmed. Azzallini (1996) and De Luca, Genon and Loperfido (006) propose he mulivariae Skew-GARCH model including a parameer o conrol skewness. Lee and Long (005) inroduce copula-based mulivariae GARCH, he C-MGARCH wih uncorrelaed dependen errors, arguing ha in erms of in sample model selecion and ou of sample mulivariae densiy forecas, he choice of copula funcions is more imporan han he volailiy models. The main drawback of he above models is ha are raher complex, and suffer from a large parameers esimaion and convergence problems. In he hesis, I propose for he firs ime, an alernaive, simplified mulivariae model, he simplified Mulivariae Auoregressive Condiional Densiy Model (S-ARCD) which is compaible wih he skewness and kurosis of he financial reurns and is easy o be implemened increasing he compuaional efficiency. I is based on he Auoregressive Condiional Densiy Model (ARCD) proposed by Hansen (1994) and involves he esimaion only of he univariae specificaion of he above model. The condiional variances are calculaed by he simple univariae models, and he condiional covariance is hen impued from hese variance esimaes. We illusrae he S-ARCD o forecas he VaR of aggregae equiy porfolios for he US and UK and foreign exchange porfolio for EUR and GBP agains USD and is compared o he ad hoc mulivariae version of GARCH (Wang, Yao, 005) and BEKK models. Our resuls, using boh saisical and economic crieria, sugges ha he simplified mulivariae version of ARCD performs a leas well as he oher wo models indicaing he higher momens imporance in volailiy forecasing and VaR calculaion. Also in he hesis, I exend he lieraure on porfolio selecion wih higher momens by invesigaing how non-normaliy of reurns and higher momens may affec hedging sraegies. Despie almos hree decades of research, he formulaion and implemenaion of an opimal fuures hedging sraegy remains a conroversial issue 16

17 in finance. Based on earlier research by Johnson (1960) and Sein (1961), Ederingon (1979) developed he firs and mos widely used OLS approach for esimaing he so-called minimum-variance saic hedge raio. Subsequen developmens in financial economerics wih respec o condiional heeroscedasiciy and coinegraion (for a comprehensive descripion and references see Mills and Markellos, 007), moivaed he developmen of more sophisicaed hedge raio esimaion approaches which can accoun for sylised facs of he daa. A variey of relevan echniques have evolved over recen years including, for example, error-correcion models, asymmeric ARCH models such as exponenial GARCH and Threshold ARCH models, univariae and mulivariae GARCH models, ec. (for a review see Chen e al, 003; Brooks and Chong, 001). Alhough more sophisicaed models are ofen found o be superior o OLS in erms of hedging performance, here is no consensus in he empirical lieraure wih respec o he mos appropriae approach. An issue ha has no received ye much aenion in he hedging lieraure concerns he imporance of condiional variaions in momens oher han he mean and variance. Higher momens, such as skewness and kurosis, can conrol he asymmeries and hick ails ha are ypically observed in he disribuions of financial reurns. Their role has been widely sudied wihin he conex of risk managemen and porfolio heory. A number of auhors, including Lai (1991), Prakash e al. (003), Sun and Yan (003), and, Jondeau and Rockinger (006), have provided evidence suggesing ha he incorporaion of higher momens in porfolio allocaion leads o superior approximaions of expeced uiliy and allows a very efficien way o compue opimal porfolios. Even if he firs wo momens are sufficien for asse pricing, higher momens are poenially imporan for hedging since variaions in he shape of he disribuion may affec he esimaion of he condiional mean and variance. Alhough models from he GARCH family are able under cerain assumpions and parameerisaions o produce hick-ailed and skewed disribuions, hey ypically assume ha he shape parameers are ime invarian. 17

18 Nelson (1996), applying EGARCH fi, suggesed ha since reurns disribuion deviae far away from normaliy, here is no reason o neglec higher momens modelling or assume hem as consans because of he resuling highly significan approximaion error. Various models have been developed in he lieraure in order o capure dependencies in higher momens (eg., see Hansen 1994; Harvey and Siddique, 1999, 00a, 00b; Jondeau and Rockinger, 006; Brooks, Burke and Persand, 005), ye heir usefulness for opimal hedge raion esimaion has no been ye invesigaed. In order o accoun for ime varying skewness and kurosis in opimal hedge raio esimaion, I examine and esimae for he firs ime he ARCD model proposed by Hansen (1994) is employed, in which full condiional densiy is modelled allowing for condiional shape parameers. The main reason for he above model selecion is is simpliciy in compuaion and evaluaion performance. In a horserace of models, he dynamic hedging effeciveness of he ARCD is compared o ha obained by OLS, error-correcion, exponenial moving averages, and, univariae and mulivariae GARCH. Effeciveness is measured in-sample and ou-of-sample using he minimum variance mehod. Spo and fuures daily closing prices are used for sock indices from he US, UK and Germany for he period January 1999 o Sepember 004. The resuls sugges ha he hedging performance using he ARCD ouperforms ha obained by he compeing approaches. Finally, in he lieraure of coinegraion an issue ha has no been formally examined ye is he coinegraion relaionship beween opion and spo prices for building error-correcion models of spo and opion reurns and exising resuls are limied. Mos of he empirical research in his area has applied some ype of Granger-ype causaliy es in invesigaing predicive relaionships beween spo and opion reurns. Hasbrouck (1995) proposed using a coinegraion approach whereby an equilibrium relaionship beween he nonsaionary spo and opion price levels is modelled. This approach assumes ha markes share an implici unobservable efficien price. The role of each marke in price discovery can be inferred by he 18

19 relaion of individual price series o his efficien price. Hasbrouck applied his concep in measuring price discovery for socks raded across differen exchanges. However, he noed ha coinegraion could also be applied o derivaive asse prices, opions or fuures, agains spo prices since hey are linked by arbirage condiions. Chakravary e al. (004) applied Hasbrouck s (1995) approach in exploring he conribuion of opion markes o price discovery. In he hesis, I use he framework of Hasbrouck (1995) in order o invesigae empirically he predicive abiliy of error-correcion models based on he coinegraion relaionship beween opion and spo prices. Using daily index closing prices from he US, France and Germany I firs calculae he implied volailiies and he condiional volailiies using he ARCD model for index opions. Then, I idenify coinegraion relaionships and esimae he corresponding error-correcion models. The error-correcion model resuls sugges ha boh opion and spo prices conain informaion abou opion price reurns. This is consisen wih Sephan and Whaley (1990) and implies ha spo markes are also imporan in he price formaion mechanism. We find ha he predicive abiliy of he error-correcion models is sronger for he German DAX, while he role of higher momens is insignifican for all he hree markes. The remainder of he hesis is organized as following: The firs par of he hesis consiss of hree chapers ha examine he impac of higher momens on porfolio selecion, risk managemen and opimal invesmen rules. The second par of he hesis consiss of wo chapers ha examine he informaion conen of opion prices and he conribuion of coinegraion o price discovery. In paricular: In Chaper I, a review on he univariae ARCH-GARCH processes is presened, heir esimaion is discussed and heir Maximum Likelihood Esimaion is implemened. Then, he following economeric mehodologies for opimal hedging are described: he Ordinary Leas Squares Model (OLS), he Error Correcion Modelling (ECM), he Exponenially Weighed Moving Average Model (EWMA) 19

20 and he Univariae GARCH(1,1) model. Finally, he esimaion and resuls of he economeric processes are presened. Chaper II presens he Mulivariae GARCH economeric models which allow for ime varying condiional variance covariance marix of he financial asses reurns. Then, he Bivariae BEKK(1,1,1) model and he ad hoc mulivariae of GARCH(1,1) model are exensively described. The opimal hedging analysis for BEKK(1,1,1) model is compared o he oher economeric mehods presened in he previous chaper. Finally, he VaR of differen porfolios is calculaed using he mulivariae GARCH models and applying differen VaR measures such as he condiional and uncondiional coverage, he Roo Mean Square Error and he Capial Employed, he VaR Evaluaion of he porfolios is discussed. Chaper III presens he higher order economeric models which allow for ime varying condiional skewness and kurosis. Hansen (1994) and Harvey and Siddique (1999) mehodologies are developed. The maximum likelihood under non normaliy and he quasi-maximum likelihood are esimaed. Then, he daa and he mehodology for he opimal hedge raio esimaion are illusraed. Finally, a comparison of he hedging effeciveness of he alernaive models concludes he chaper. Chaper IV inroduces he simplified mulivariae ARCD model (S-ARCD) and briefly describes BEKK and he ad hoc mulivariae version of GARCH (Wang, Yao, 005) models. The empirical resuls of he above saisical mehods on he VaR esimaion and forecasing are presened and analysed for he equiy indices and he exchange rae series. A comparison of he VaR performance of he alernaive mulivariae models is developed, implemening boh saisical and economic evaluaion. Chaper V examines he framework of Hasbrouck (1995) in order o invesigae empirically he predicive abiliy of error-correcion models based on he coinegraion relaionship beween opion and spo prices. Firsly, he implied volailiies and he condiional volailiies using ARCD model are calculaed for boh 0

21 call and pu opion prices, in order o consider for ime variaion in higher momens. Then, he coinegraion relaionships and esimaions of he corresponding errorcorrecion models are calculaed for he daily index closing opion prices from he US, France and Germany. Finally, a simple rading sraegy is applied so as o idenify he predicive abiliy of he error correcion models. The hesis ends up wih he conclusions, giving a summary, he main resuls and suggesions for furher research. 1

22 Εισαγωγή Βασική επιδίωξη των διαφόρων χρηµατοοικονοµικών οργανισµών (τράπεζες, εταιρείες διαχείρισης χαρτοφυλακίου, αµοιβαίων κεφαλαίων, κ.λπ.) είναι η µεγιστοποίηση της απόδοσης των υπό διαχείριση κεφαλαίων και, κατά συνέπεια, του κέρδους. Για την επίτευξη του προαναφερόµενου στόχου, προβαίνουν στην επένδυση των κεφαλαίων στις παγκόσµιες χρηµαταγορές αναλαµβάνοντας ποικιλόµορφους κινδύνους όπως επιτοκιακούς, συναλλαγµατικούς, πιστωτικούς λειτουργικούς και κινδύνους ρευστότητας. Ως αποτέλεσµα, είναι απαραίτητη η ποσοτικοποίηση µε αξιοπιστία αυτών των κινδύνων καθώς και η εκτίµηση των επιπτώσεων που µπορεί να έχει µια σηµαντική απόκλιση από τη µέση πρόβλεψη ενός αποτελέσµατος. Τα τελευταία χρόνια παρατηρείται µια ραγδαία εξέλιξη στο χώρο διαχείρισης κινδύνου και ιδιαίτερα σε τεχνικό επίπεδο (στατιστικά µοντέλα και τεχνικές). Ορισµένοι από τους παράγοντες που έκαναν επιτακτική την ανάγκη για µέτρηση και διαχείριση του κινδύνου είναι: α) η απελευθέρωση (deregulaion) των διεθνών χρηµαταγορών β) η αύξηση της µεταβλητότητας (volailiy) των αγορών γ) αύξηση του όγκου συναλλαγών των χρεογράφων και των παραγώγων δ) αύξηση του ενδιαφέροντος για τη σχέση κινδύνου / απόδοσης (risk / reurn) ε) αύξηση των απαιτήσεων και των κανονισµών από τις εποπτικές αρχές πάνω σε θέµατα διαχείρισης κινδύνου. H παρούσα διατριβή µελετά τη στατιστική και οικονοµική σηµασία της µεταβλητής µε το χρόνο συνδιακύµανσης των διάφορων χρηµατοοικονοµικών προϊόντων. Οι εφαρµογές περιλαµβάνουν παράγωγα χρηµατοοικονοµικά προϊόντα,

23 καθώς και βέλτιστους επενδυτικούς κανόνες οι οποίοι προκύπτουν τόσο από την οικονοµετρική θεωρία όσο και τη θεωρία των πραγµατικών δικαιωµάτων. Είναι κοινά αποδεκτό στην ακαδηµαϊκή βιβλιογραφία ότι η σωστή και ακριβής µοντελοποίηση καθώς και η πρόβλεψη της κατανοµής που ακολουθούν οι αποδόσεις των διάφορων χρηµατοοικονοµικών προϊόντων είναι απαραίτητη όσον αφορά τη βέλτιστη αντιστάθµιση των κινδύνων αγοράς, τη βέλτιστη σύνθεση χαρτοφυλακίων καθώς την τιµολόγηση πραγµατικών δικαιωµάτων. Οι Bollerslev, Engle και Wooldridge (1988), Cecchei (1988), Myers and Thompson (1989), Baillie and Myers (1991), Kroner and Sulan (1993), Ng and Kroner (1998), υποστηρίζουν ότι οι τιµές των χρηµατοοικονοµικών προϊόντων χαρακτηρίζονται από µεταβαλλόµενες µε το χρόνο διακυµάνσεις και συνδιακυµάνσεις προτείνοντας µια οικογένεια πολυµεταβλητών υποδειγµάτων Γενικευµένης Αυτοπαλίνδροµης Υπό συνθήκη Ετεροσκεδαστικότητας (Mulivariae Generalized AuoRegressive Condiional Heeroscedasiciy, GARCH). Οι Bollerslev, Engle και Wooldridge (1988) παρουσίασαν το διανυσµατικό υπόδειγµα σε όρους διόρθωσης των σφαλµάτων (Vecor Error Correcion model, VECM) και το διαγώνιο διανυσµατικό υπόδειγµα σε όρους διόρθωσης των σφαλµάτων (Diagonal Vecor Error Correcion model, DVECM) των οποίων οι διακυµάνσεις εξαρτώνται µόνο από τα τετράγωνα των σφαλµάτων των προηγούµενων περιόδων και οι συνδιακυµάνσεις εξαρτώνται από τα χιαστί γινόµενα. των σφαλµάτων των προηγούµενων περιόδων. εδοµένου τόσο του µεγάλου αριθµού των παραµέτρων όσο και της ανάγκης για αυστηρούς κανονισµούς και συνθήκες που χρειάζονται για να εκτιµηθεί το VECM οικονοµετρικό µοντέλο, οι Engle και Kroner (1995) πρότειναν το διπαραµετρικό ΒΕΚΚ µοντέλο αποφεύγοντας µη ρεαλιστικές υποθέσεις όπως το ότι η συσχέτιση µεταξύ των υπό συνθήκη διακυµάνσεων είναι σταθερή (O Bollerslev, 1990, είχε προτείνει το Consan correlaion model) και εξασφαλίζοντας το γεγονός ότι ο πίνακας των µεταβαλλόµενων µε το χρόνο συνδιακυµάνσεων είναι θετικά ορισµένος. Επιπρόσθετα µοντέλα έχουν προταθεί από τους Engle, Ng, και Rohschild (1990b) τα ονοµαζόµενα υποδείγµατα Παραγόντων Γενικευµένης 3

24 Αυτοπαλίνδροµης Υπό συνθήκη Ετεροσκεδαστικότητας (Facor GARCH (FGARCH)), τους Alexander και Chibumba (1997) οι οποίοι ανέπτυξαν το κανονικοποιηµένο υπόδειγµα Γενικευµένης Αυτοπαλίνδροµης Υπό συνθήκη Ετεροσκεδαστικότητας (Οrhogonal GARCH (O-GARCH)), τους Tse και Tsui (00) οι οποίοι πρότειναν τα δυναµικά υπό συνθήκη µοντέλα συσχέτισης (Dynamic Condiional Correlaion models (DCC)). Όλα τα προαναφερόµενα οικονοµετρικά υποδείγµατα θεωρούν ότι οι αποδόσεις των χρηµατοοικονοµικών χρεογράφων ακολουθούν την κανονική κατανοµή αγνοώντας την ύπαρξη ασυµµετρίας, κύρτωσης και λεπτοκύρτωσης στη διακύµανση και τη συνδιακύµανση. Εντούτοις, η ασυµµετρία και η κύρτωση επηρεάζουν σηµαντικά τα χρηµατοοικονοµικά προϊόντα αφού οι κατανοµές των αποδόσεών τους δεν ακολουθούν τους κανόνες της κανονικής κατανοµής. Παραδείγµατος χάρη οι French, Schwer και Sambaugh (1987) απέδειξαν ότι υπάρχει σηµαντική ασυµµετρία και κύρτωση στα κατάλοιπα των αποδόσεων του δείκτη S&P, ο Hong (1995) παρατήρησε ότι υπάρχει ασυνήθιστα µεγάλη κύρτωση στις καθηµερινές αποδόσεις του δείκτη NYSE, ο Harvey (1995) βρήκε µεγάλες αποκλίσεις από την κανονική κατανοµή στις αναδυόµενες αγορές µετοχών ενώ οι Harvey και Siddique (1999) απέδειξαν ότι η υπό συνθήκη ασυµµετρία είναι σηµαντική και ακόλουθη µε την διακύµανση των καθηµερινών, εβδοµαδιαίων και µηνιαίων αποδόσεων συγκεκριµένων χρηµαταγορών. Εφόσον είναι εξακριβωµένο γεγονός ότι οι αποδόσεις των διάφορων χρηµατοοικονοµικών προϊόντων παρουσιάζουν λεπτοκύρτωση και ασυµµετρία, πολλές ακαδηµαϊκές µελέτες επικεντρώθηκαν στην εφαρµογή µη κανονικοποιηµένων κατανοµών για τη σωστή µοντελοποίηση της υπερβάλλουσας ασυµµετρίας και κύρτωσης. Πιο συγκεκριµένα, στην κατηγορία των µοντέλων µίας µεταβλητής, υπάρχουν πολλές κατανοµές που λαµβάνουν υπόψη την κύρτωση και ασυµµετρία. Ο Hansen (1994) ήταν ο πρώτος που πρότεινε την Skew-Suden κατανοµή στην οποία τόσο η ασυµµετρία όσο και η κύρτωση είναι µεταβαλλόµενες σε σχέση µε το χρόνο. Πρόσφατα, οι Harvey και Siddique (00a, 00b), οι 4

25 Jondeau και Rockinger (006) καθώς και οι Yan (005), Brooks, Burke, και Persand (005) µεταξύ των άλλων ανέπτυξαν οικονοµετρικά υποδείγµατα της από κοινού εκτίµησης των συµµεταβαλλόµενων µε το χρόνο διακύµανσης και κύρτωσης, αλλά η µοντελοποίηση τους είναι δύσκολο να υλοποιηθεί τόσο για µία όσο και για περισσότερες µεταβλητές. Συµπερασµατικά, κανένα από τα πιο γνωστά πολυµεταβλητά οικονοµετρικά µοντέλα, δεν είναι συµβατά µε την κύρτωση και την ασυµµετρία των αποδόσεων των χρεογράφων αφού υποθέτουν ότι ισχύει η κανονικότητα σε πολυµεταβλητά συστήµατα. Πολύ λίγες σε αριθµό µελέτες υπάρχουν που εξετάζουν τη µοντελοποίηση των παραµέτρων της κύρτωσης και της ασυµµετρίας σε πολυµεταβλητές ανελίξεις. Οι Harvey, Ruiz και Shepard (1994), καθώς και οι Fiorenini, Senana και Galzolari (003) αντικαθιστούν την πολυµεταβλητή Gaussian κατανοµή µε τη Suden κατανοµή επιτρέποντας τις υπό συνθήκη µεταβολές να ακολουθούν τη Suden κατανοµή. Οι Sahu, Dey, και Βranco (001) καθώς και οι Bawens και Lauren (005 ) πρότειναν την πολυµεταβλητή Skew Suden κατανοµή η οποία υποστηρίζεται πλήρως από την Ευκλείδεια γεωµετρία και απέδειξαν ότι οι προβλέψεις του Value a risk είναι πιο ακριβείς από τις αντίστοιχες που υπολογίζονται µε βάσει τη Gaussian κατανοµή. Πρόσφατα, οι Hafner και Rombous (004) και οι Rombous και Verbeek (005) εφάρµοσαν µια πολυµεταβλητή ηµιπαραµετρική GARCH µέθοδο η οποία λαµβάνει υπόψη την κύρτωση και ασυµµετρία αποδεικνύοντας ότι το Value a Risk διάφορων χαρτοφυλακίων υπολογίζεται µε µεγαλύτερη ακρίβεια και συνεπώς µε µικρότερο σφάλµα. Οι Azzallini (1996) και De Luca καθώς και οι Genon και Loperfido (006) πρότειναν το πολυµεταβλητό Ασύµµετρο υπόδειγµα Γενικευµένης Αυτοπαλίνδροµης Υπό συνθήκη Ετεροσκεδαστικότητας (Mulivariae Skew- GARCH) το οποίο περιλαµβάνει µια επιπλέον παράµετρο που υπολογίζει τη µεταβολή της ασυµµετρίας σε σχέση µε το χρόνο. Οι Lee και Long (005) εισήγαγαν στην προϋπάρχουσα βιβλιογραφία την έννοια των Copula συναρτήσεων στα πολυµεταβλητά υποδείγµατα Γενικευµένης Αυτοπαλίνδροµης Υπό συνθήκη 5

26 Ετεροσκεδαστικότητας (C-GARCH) στα οποία τα εξαρτηµένα λάθη είναι µη συσχετισµένα υποστηρίζοντας ότι στην τόσο εντός όσο και εκτός δείγµατος πολυµεταβλητή συνάρτηση πυκνότητας η επιλογή των Copula συναρτήσεων είναι πιο σηµαντική από την επιλογή των µοντέλων αποτίµησης της διακύµανσης του ε λόγω δείγµατος. Το κύριο µειονέκτηµα όλων των προαναφερόµενων µοντέλων είναι η πολυπλοκότητά τους καθώς και ότι είναι απαραίτητη η εκτίµηση ενός µεγάλου αριθµού παραµέτρων µε αποτέλεσµα να δηµιουργούνται προβλήµατα σύγκλισης. Στη διατριβή παρουσιάζεται για πρώτη φορά στη βιβλιογραφία ένα εναλλακτικό πολυµεταβλητό µοντέλο το απλοποιηµένο πολυµεταβλητό υπόδειγµα Αυτοπαλίνδροµης Υπό συνθήκη Πυκνότητας (S-ARCD), το οποίο λαµβάνει υπόψη την ασυµµετρία και κύρτωση των αποδόσεων των χρεογράφων ενώ είναι πολύ εύκολο να εφαρµοστεί µειώνοντας το χρόνο εκτίµησης του και αυξάνοντας την υπολογιστική απόδοση του. Είναι βασισµένο στο υπόδειγµα Αυτοπαλίνδροµης Υπό συνθήκη Πυκνότητας του Hansen (ARCD) και περιλαµβάνει την εκτίµηση µόνο των µονοµεταβλητών µορφών του προαναφερόµενου υποδείγµατος. Οι υπό συνθήκη διακυµάνσεις υπολογίζονται από τα απλά µονοµεταβλητά υποδείγµατα ARCD, ενώ η υπό συνθήκη συνδιακύµανση συνεπάγεται από τις προηγούµενες εκτιµήσεις των διακυµάνσεων. Τα δεδοµένα που χρησιµοποιούνται στην ανάλυση περιλαµβάνουν πληθώρα δεικτών µετοχών από Ευρωπαϊκά και Αµερικάνικα χρηµατιστήρια, καθώς και συναλλαγµατικές ισοτιµίες του ευρώ και της λίρας σε σχέση µε το δολάριο. Η στατιστική ανάλυση αναδεικνύει την κεντρική σηµασία του S-ARCD για την πρόβλεψη του Value a Risk συγκρίνοντας τη µε τη στατιστική ανάλυση δύο άλλων υποδειγµάτων των ad hoc GARCH και του ΒΕΚΚ. Τα αποτελέσµατα επιβεβαιώνονται τόσο από οικονοµικά όσο και στατιστικά κριτήρια καταλήγοντας ότι το S-ARCD υπόδειγµα αποτίµησης µε ασυµµετρία και κύρτωση παρουσιάζει το µικρότερο σφάλµα τιµολόγησης, και αποδεικνύοντας ότι οι µεταβαλλόµενες µε το χρόνο ασυµµετρία και κύρτωση είναι πολύ σηµαντικές για τον υπολογισµό και την πρόβλεψη του Value a Risk διαφόρων χαρτοφυλακίων. 6

27 Επίσης, στη διατριβή αναπτύσσονται βέλτιστοι κανόνες επιλογής χαρτοφυλακίου λαµβάνοντας υπόψη την ασυµµετρία και κύρτωση, µελετώντας τον τρόπο επίδρασης της µη κανονικότητας των αποδόσεων των χρεογράφων στην στρατηγική αντιστάθµισης κινδύνου. Παρόλο τη συνεχή έρευνα τριών σχεδόν δεκαετιών, η διατύπωση και εφαρµογή βέλτιστων στρατηγικών αντιστάθµισης κινδύνου µε τα προθεσµιακά συµβόλαια µελλοντικής εκπλήρωσης παραµένουν επίµαχα σηµεία στη χρηµατοοικονοµική ακαδηµαϊκή βιβλιογραφία. Στη βάση της έρευνας των Johnson (1960) και Sein (1961) o Ederingon (1979) πρώτος ανέπτυξε την ευρέως γνωστή µέθοδο των ελαχίστων τετραγώνων (OLS) για την αποτίµηση της ελάχιστης σε διακύµανση στατικής αναλογίας αντιστάθµισης κινδύνου. Οι συνεχείς εξελίξεις στην ακαδηµαϊκή βιβλιογραφία της οικονοµετρίας σε σχέση µε την υπό συνθήκη ετεροσκεδαστικότητα και τη συνολοκλήρωση (οι Mills και Markellos (007) κάνουν µια ολοκληρωµένη αναφορά στα παραπάνω θέµατα) οδήγησαν στην εφαρµογή πιο τεκµηριωµένων και πολύπλοκων µεθόδων υπολογισµού της αναλογίας αντιστάθµισης κινδύνου οι οποίες λαµβάνουν υπόψη τις ιδιαιτερότητες των δεδοµένων. Οι πιο πρόσφατες τεχνικές περιλαµβάνουν για παράδειγµα υποδείγµατα σε όρους διόρθωσης των σφαλµάτων (Error Correcion Models (ECM)), ασυµµετρικά υποδείγµατα της υπό συνθήκης ετεροσκεδαστικότητας όπως εκθετικά υποδείγµατα της γενικευµένης αυτοπαλίνδροµης υπό συνθήκης ετεροσκεδαστικότητας (ΕGARCH) και τµηµατικά µοντέλα ή µοντέλα «κατωφλιού» (hreshold models), µονοµεταβλητά και πολυµεταβλητά υποδείγµατα της γενικευµένης αυτοπαλίνδροµης υπό συνθήκης ετεροσκεδαστικότητας (Mulivariae GARCH) κ.λπ.. Αν και τα πιο εξελιγµένα οικονοµετρικά µοντέλα είναι πιο συχνά ποιοτικά ανώτερα της απλής µεθόδου ελαχίστων τετράγωνων (ΟLS) όσον αφορά την απόδοση στη µέτρηση της αναλογίας αντιστάθµισης των κινδύνων αγοράς, η ακαδηµαϊκή έρευνα δεν έχει καταλήξει στο ερώτηµα ποια είναι η πιο ασφαλής και ακριβής µέθοδος αποτίµησης της προαναφερόµενης αναλογίας. 7

28 Στη βιβλιογραφία της αντιστάθµισης κινδύνων, ένα ζήτηµα που δεν έχει εξεταστεί ακόµη αφορά στη σηµασία των υπό συνθήκη µεταβολών των ροπών πέραν της διακύµανσης και του µέσου. Οι τρίτες και τέταρτες ροπές όπως η ασυµµετρία και η κύρτωση ελέγχουν τη λεπτοκύρτωση, το δείκτη λοξότητας και το δείκτη κυρτότητας που παρατηρούνται στις αποδόσεις των χρεογράφων. Η επίδρασή τους στους κανόνες επιλογής βέλτιστων επενδυτικών αποφάσεων για τη σύνθεση και µέτρηση του κινδύνου χαρτοφυλακίων έχει µελετηθεί εκτενέστερα. Αρκετοί ακαδηµαϊκοί, συµπεριλαµβανοµένου των Lai (1991), Prakash (003), Sun και Yan (003), καθώς και οι, Jondeau and Rockinger (006), παρέχουν αριθµητικά παραδείγµατα αποδεικνύοντας ότι η ενσωµάτωση των τρίτων και τέταρτων ροπών στη σύνθεση χαρτοφυλακίων οδηγεί σε υπέρτερες προσεγγίσεις της προσδοκώµενης συνάρτησης χρησιµότητας και συνεισφέρει αποτελεσµατικά στον υπολογισµό βέλτιστων χαρτοφυλακίων. Ακόµη και αν οι δύο πρώτες ροπές είναι επαρκείς για την τιµολόγηση των χρεογράφων, οι τρίτες και τέταρτες ροπές είναι εξίσου σηµαντικές για την αντιστάθµιση των κινδύνων αγοράς αφού οι µεταβολές στο σχήµα της κατανοµής των αποδόσεων των χρεογράφων επηρεάζει την εκτίµηση των υπό συνθήκη µέσου και διακύµανσης. Αν και τα υποδείγµατα από την οικογένεια των υποδειγµάτων της γενικευµένης αυτοπαλίνδροµης υπό συνθήκη ετεροσκεδαστικότητας (GARCH) είναι ικανά κάτω από ορισµένες προϋποθέσεις και παραµέτρους να παράγουν λοξότητες και κυρτώσεις στις κατανοµές αποδόσεων, κατά κανόνα υποθέτουν ότι οι παράµετροι που καθορίζουν το σχήµα της κατανοµής είναι σταθερές και αµετάβλητες µε το χρόνο. Ο Nelson (1996), εφάρµοσε το εκθετικό υπόδειγµα Γενικευµένης Αυτοπαλίνδροµης υπό Συνθήκη Ετεροσκεδαστικότητας (GARCH) υποδεικνύοντας ότι δεδοµένου ότι οι αποδόσεις αποκλίνουν αρκετά από την κανονική κατανοµή δεν υπάρχει λόγος να αγνοούµε τη µοντελοποίηση των τρίτων και τέταρτων ροπών τους ή να υποθέτουµε ότι είναι σταθερές, εφόσον τα αποτελέσµατα που προκύπτουν εµπεριέχουν µεγάλο σφάλµα εκτίµησης. ιαφορετικά υποδείγµατα έχουν προταθεί στην ακαδηµαϊκή βιβλιογραφία έτσι ώστε να καταγραφεί η ύπαρξη της κύρτωσης και της ασυµµετρίας 8

29 στην κατανοµή των αποδόσεων (π.χ. Hansen 1994; Harvey και Siddique, 1999, 00a, 00b; Jondeau και Rockinger, 006; Brooks, Burke και Persand, 005), όµως η επίδρασή τους στην επιλογή της βέλτιστης αναλογίας αντιστάθµισης των κινδύνων αγοράς δεν έχει ακόµη εξεταστεί. Η διατριβή συνεισφέρει στη βιβλιογραφία εφαρµόζοντας για πρώτη φορά το υπόδειγµα Αυτοπαλίνδροµης Υπό συνθήκη Πυκνότητας του Hansen (ARCD) το οποίο εκτιµά πλήρως την υπό συνθήκη συνάρτηση πυκνότητας επιτρέποντας στις παραµέτρους που καθορίζουν το σχήµα της εν λόγω συνάρτησης να µεταβάλλονται σε σχέση µε το χρόνο. Το παραπάνω υπόδειγµα χρησιµοποιείται για πρώτη φορά για τον υπολογισµό της βέλτιστης αναλογίας αντιστάθµισης των κινδύνων αγοράς και επιλέχθηκε λόγω της απλότητάς του τόσο στην υπολογιστική υλοποίησή του όσο και στην αξιολόγηση της απόδοσής του. Μέσω συγκριτικής ανάλυσης, µε τη µέθοδο των ελαχίστων τετραγώνων (OLS), το υπόδειγµα σε όρους διόρθωσης των σφαλµάτων (Error Correcion Models (ECM)), το εκθετικό υπόδειγµα της γενικευµένης αυτοπαλίνδροµης υπό συνθήκης ετεροσκεδαστικότητας (ΕGARCH) και τα µονοµεταβλητά και πολυµεταβλητά υποδείγµατα της γενικευµένης αυτοπαλίνδροµης υπό συνθήκης ετεροσκεδαστικότητας (Mulivariae GARCH) προκύπτει ότι τόσο εντός όσο και εκτός δείγµατος η αποτελεσµατικότητα στη µέτρηση της αντιστάθµισης των κινδύνων αγοράς µε βάση το υπόδειγµα Αυτοπαλίνδροµης υπό συνθήκη Πυκνότητας του Hansen (ARCD) βελτιώνεται σηµαντικά. Τέλος, όσον αφορά τη θεωρία συνολοκλήρωσης, το ζήτηµα της συνολοκληρωµένης σχέσης µεταξύ των δικαιωµάτων προαίρεσης και των υποκείµενων τίτλων στην τιµολόγηση των υποδειγµάτων σε όρους διόρθωσης των σφαλµάτων (error correcion model, ECM) δεν έχει εξεταστεί ενώ τα αποτελέσµατα είναι περιορισµένα. Το µεγαλύτερο µέρος της εµπειρικής µελέτης στον τοµέα της συνολοκλήρωσης εφαρµόζει τη θεωρία και τον έλεγχο Granger Causaliy για να διερευνήσει τις πιθανές σχέσεις µεταξύ των αποδόσεων των δικαιωµάτων προαίρεσης και των υποκείµενων τίτλων. Ο Hasbrouck (1995) πρότεινε, 9

30 χρησιµοποιώντας τη µέθοδο συνολοκλήρωσης, τη µοντελοποίηση µιας σχέσης ισορροπίας µεταξύ των αποδόσεων των δικαιωµάτων προαίρεσης και των υποκείµενων τίτλων όπου αυτό είναι εφικτό. Η παραπάνω µέθοδος υποθέτει ότι οι χρηµαταγορές µοιράζονται µια εσωτερική, µη παρατηρήσιµη από όλους, σωστή αποτίµηση. Ο ρόλος κάθε χρηµαταγοράς στην κοινοποίηση της εσωτερικής πληροφόρησης για την πορεία των τιµών της είναι δυνατό να συνάγεται από τη σχέση µεταξύ των ανεξάρτητων τιµών της και των αντίστοιχων προαναφεροµένων εσωτερικών τιµών. Ο Hasbrouck εφάρµοσε την παραπάνω θεωρία για τη µέτρηση της πληροφόρησης των τιµών µετοχών οι οποίες είναι διαπραγµατεύσιµες σε δύο και πάνω διαφορετικές χρηµαταγορές. Εντούτοις, παρατήρησε ότι η συνολοκλήρωση µπορούσε να εφαρµοσθεί µεταξύ παράγωγων προϊόντων και των υποκείµενων τίτλων τους αφού συνδέονται µε κοινούς κερδοσκοπικούς κανόνες (arbirage condiions). Στην παρούσα διατριβή, χρησιµοποιώ τη µέθοδο του Hasbrouck (1995) µε σκοπό την εµπειρική διερεύνηση της ικανότητας πρόβλεψης των υποδειγµάτων σε όρους διόρθωσης των σφαλµάτων (error correcion model, ECM) η οποία βασίζεται στη συνολοκληρωµένη σχέση µεταξύ των δικαιωµάτων προαίρεσης και των υποκείµενων τίτλων τους. Βάσει των ιστορικών καθηµερινών δεδοµένων από τις ΗΠΑ, Γαλλία και Γερµανία, στην αρχή υπολογίσαµε τις τεκµαρτές διακυµάνσεις των δικαιωµάτων προαίρεσης αγοράς και πώλησης µε υποκείµενους τίτλους τους χρηµατιστηριακούς δείκτες καθώς και τις υπό συνθήκη διακυµάνσεις µε βάση το υπόδειγµα Αυτοπαλίνδροµης Υπό συνθήκη Πυκνότητας του Hansen (ARCD), το οποίο λαµβάνει υπόψη τη µεταβολή των ανώτερων ροπών, (συµµετρία και κυρτότητα) σε σχέση µε το χρόνο. Στη συνέχεια εξακριβώσαµε τις συνολοκληρωµένες σχέσεις και εκτιµήσαµε τα υποδείγµατα σε όρους διόρθωσης των σφαλµάτων (error correcion models, ECM). Από την ανάλυση, τα αποτελέσµατα δείχνουν ότι τόσο τα δικαιώµατα προαίρεσης όσο και οι υποκείµενοι τίτλοι περιέχουν πληροφόρηση σχετικά µε την πορεία εξέλιξης των αποδόσεων τους, ενώ για τον δείκτη µετοχών DAX η ικανότητα πρόβλεψης του αντίστοιχου 30

31 υποδείγµατος σε όρους διόρθωσης των σφαλµάτων (error correcion model, ECM) είναι µεγαλύτερη. Το παραπάνω συµπέρασµα είναι σε συµφωνία µε τους Sephen και Whaley (1990) υποδηλώνοντας ότι οι spo αγορές παίζουν σηµαντικό ρόλο στο µηχανισµό αποτίµησης της πληροφόρησης, ενώ ο ρόλος των ανώτερων ροπών δεν είναι σηµαντικός. H διατριβή αποτελείται από δύο µέρη. Το πρώτο µέρος αποτελείται από τρία κεφάλαια, τα οποία εξετάζουν την επίδραση των τρίτων και τέταρτων ροπών στη σύνθεση βέλτιστων χαρτοφυλακίων, στη διαχείριση των κινδύνων αγοράς και στην απόφαση βέλτιστων επενδυτικών κανόνων. Το δεύτερο µέρος αποτελείται από δύο κεφάλαια που εξετάζουν την εσωτερική πληροφόρηση των δικαιωµάτων προαίρεσης και το ρόλο της συνολοκλήρωσης στην εξέλιξη των τιµών των χρεογράφων. Πιο συγκεκριµένα: Στο πρώτο κεφάλαιο παρουσιάζεται ανασκόπηση των µονοµεταβλητών υποδειγµάτων της Αυτοπαλίνδροµης υπό συνθήκη Ετεροσκεδαστικότητας (ARCH) και της Γενικευµένης Αυτοπαλίνδροµης υπό συνθήκη Ετεροσκεδαστικότητας (GARCH). Η εκτίµησή των παραµέτρων τους και ο τρόπος υπολογισµού της µέγιστης πιθανοφάνειας τους συζητείται. Στη συνέχεια, οι παρακάτω οικονοµετρικές µέθοδοι περιγράφονται: η µέθοδος Ελαχίστων Τετραγώνων (Ordinary Leas Squares Model, OLS), το υπόδειγµα σε όρους διόρθωσης των σφαλµάτων (he Error Correcion Modelling, ECM), το εκθετικό οικονοµετρικό µοντέλο (he Exponenially Weighed Moving Average Model, EWMA) και το οικονοµετρικό µοντέλο της Γενικευµένης Αυτοπαλίνδροµης υπό συνθήκη Ετεροσκεδαστικότητας (GARCH). Τέλος, η υλοποίησή τους και τα πιθανά προβλήµατα εκτίµησής τους σε διάφορες χρηµατοοικονοµικές εφαρµογές σχολιάζονται. Στο δεύτερο κεφάλαιο παρουσιάζονται τα Πολυµεταβλητά Οικονοµετρικά µοντέλα της Γενικευµένης Αυτοπαλίνδροµης υπό συνθήκη Ετεροσκεδαστικότητας (Mulivariae GARCH), τα οποία επιτρέπουν τη µεταβολή του πίνακα συνδιακύµανσης των αποδόσεων των χρηµατοοικονοµικών προϊόντων σε σχέση µε 31

32 το χρόνο. Στη συνέχεια, το διπαραµετρικό µοντέλο της Γενικευµένης Αυτοπαλίνδροµης υπό συνθήκη Ετεροσκεδαστικότητας BEKK(1,1,1) και το απλοποιηµένο πολυµεταβλητό µοντέλο της Γενικευµένης Αυτοπαλίνδροµης υπό συνθήκη Ετεροσκεδαστικότητας (Ad hoc GARCH(1,1) συζητούνται µε λεπτοµέρεια καθώς και πως εκτιµούνται οι παράµετροι τους. Αναλύεται η ικανότητα αντιστάθµισης κινδύνου διάφορων χρηµατοοικονοµικών προϊόντων όταν εφαρµόζεται το διπαραµετρικό µοντέλο της Γενικευµένης Αυτοπαλίνδροµης υπό συνθήκη Ετεροσκεδαστικότητας BEKK(1,1,1) και συγκρίνουµε τα αποτελέσµατα µε άλλες οικονοµετρικές µεθόδους που αναλύθηκαν στο προηγούµενο κεφάλαιο. Τέλος, υπολογίζεται ο κίνδυνος διάφορων χαρτοφυλακίων χρησιµοποιώντας τα προαναφερόµενα πολυµεταβλητά GARCH µοντέλα και εφαρµόζονται οι παρακάτω δείκτες εκτίµησης του κινδύνου VaR: το condiional and uncondiional coverage, το Roo Mean Square Error and το Capial Employed, έτσι ώστε να εκτιµηθεί η αποδοτικότητα και ικανότητα πρόβλεψης της κάθε πολυµεταβλητής οικονοµετρικής µεθόδου όσον αφορά τον υπολογισµό της διακύµανσης και συνεπώς του κίνδυνου των χαρτοφυλακίων. Στο τρίτο κεφάλαιο περιγράφονται τα οικονοµετρικά µοντέλα που επιτρέπουν στις τρίτες και τέταρτες ροπές να µεταβάλλονται µε το χρόνο. Αρχικά τα υποδείγµατα του Hansen (1994) και των Harvey και Siddique (1996) αναπτύσσονται καθώς και ο τρόπος υπολογισµού της µέγιστης πιθανοφάνειας των παραπάνω µοντέλων. Στη συνέχεια, περιγράφονται τα δεδοµένα τα οποία χρησιµοποιούνται στην ανάλυση και η οικονοµετρική µεθοδολογία που χρησιµοποιείται για την εκτίµηση της βέλτιστης αναλογίας αντιστάθµισης των κινδύνων αγοράς. Σε κάθε περίπτωση, τα αποτελέσµατα συγκρίνονται µε προηγούµενες µελέτες και παρέχονται αριθµητικά παραδείγµατα. Στο τέταρτο κεφάλαιο παρουσιάζεται το απλοποιηµένο πολυµεταβλητό υπόδειγµα Αυτοπαλίνδροµης Υπό συνθήκη Πυκνότητας (S-ARCD), ενώ παράλληλα γίνεται σύντοµη περιγραφή των ad hoc GARCH και ΒΕΚΚ πολυµεταβλητών οικονοµετρικών µοντέλων. Πρώτα παρουσιάζονται οι τρεις στατιστικές µέθοδοι που 3

33 χρησιµοποιούνται στην ανάλυση. Στη συνέχεια, οι τρεις µέθοδοι συγκρίνονται µε τη χρήση δεδοµένων από δείκτες µετοχών και συναλλαγµατικών ισοτιµιών για τον υπολογισµό και την πρόβλεψη του Value a Risk και ακολούθως σχολιάζονται τα αποτελέσµατα. Στο πέµπτο κεφάλαιο εξετάζεται το θεωρητικό πλαίσιο του Hasbrouck (1995) σε συνδυασµό µε την εφαρµογή του υποδείγµατος ΑRCD για τον υπολογισµό των διακυµάνσεων των χρηµατιστηριακών δεικτών έτσι ώστε να ερευνηθεί εµπειρικά η ικανότητα πρόβλεψης των υποδειγµάτων σε όρους διόρθωσης των σφαλµάτων (error correcion model, ECM) που βασίζονται στη συνολοκληρωµένη σχέση µεταξύ των δικαιωµάτων προαίρεσης και των υποκείµενων τίτλων τους. Πρώτα, οι συνολοκληρωµένες σχέσεις και οι εκτιµήσεις των υποδειγµάτων σε όρους διόρθωσης των σφαλµάτων (error correcion model, ECM) υπολογίζονται για τις καθηµερινές τιµές των δικαιωµάτων προαίρεσης των δεικτών των µετοχών των χωρών των Η.Π.Α, Γαλλίας και Γερµανίας. Σε κάθε περίπτωση εφαρµόζεται µια απλή στρατηγική διαχείρισης χαρτοφυλακίων έτσι ώστε να εξακριβωθεί η ικανότητα πρόβλεψης των υποδειγµάτων σε όρους διόρθωσης των σφαλµάτων. Στο τελευταίο κεφάλαιο παρουσιάζονται τα σηµαντικότερα συµπεράσµατα της διατριβής και προτείνονται ιδέες για µελλοντική έρευνα. 33

34 Par A: Univariae and Mulivariae Higher Momens Modeling and Forecasing 34

35 Chaper I. ARCH-GARCH Processes: Review and Opimal Hedging CHAPTER I ARCH-GARCH Processes: Review and Opimal Hedging The saisical properies of asse reurns (equiies, bonds, foreign exchange currencies) are subjec o coninual ineres and research, mainly because he knowledge of hem has imporan economic implicaions for he money marke equilibrium, and he inernaional rade including he assessmen of open posiions in reserves, paymens of underlying asses in impors and expors area, and money lending and borrowing. Also, he wide flucuaions in sock and foreign currencies marke exchanges are of coninual concern o invesors and o heir managemen decisions because hese changes in volailiy may have adverse effec on risk managemen, hedging sraegies and heir invesmen decisions. Especially i is crucial o idenify properly he ype of heeroscedasiciy in he daa-generaion process and he predicive disribuions, which depend on he condiional disribuion for he normalized error. Accurae specificaion plays an imporan role in inernaional finance, paricularly in he area of he pricing of foreign currency opions and fuures and he selecion of mean-variance efficien porfolios of inernaional asses. An exensive lieraure documens he behaviour of he asse reurns disribuion. Tradiional ARMA/ARIMA models have concenraed on modelling 35

36 Chaper I. ARCH-GARCH Processes: Review and Opimal Hedging and forecasing he condiional mean (a locaion parameer) assuming variance and all higher order momens o be consan. A naural exension has been o model he condiional variance (a scale parameer), so as o capure he mos known characerisic of asse reurns is he volailiy clusering firs noed by Mandelbro (1963): Large changes end o be followed by large changes, of eiher sign, and small changes end o be followed by small changes. Fama (1970) also observed he alernaion beween periods of high and low volailiy: Large price changes are followed by large price changes, bu of unpredicable sign. This exension was moivaed by he empirical fac ha volailiy in many economic ime series and paricularly in financial ime series did no remain consan over ime. Engle (198) proposed o model ime varying condiional volailiy wih he Auoregressive Condiional Heeroscedasiciy (ARCH) processes. Bollerslev (1986) showed ha high ARCH order has o be seleced in order o cach he dynamic of condiional variance inroducing he Generalised ARCH (GARCH) models. Since hen, a wide range of GARCH formulaions appeared in he lieraure such as IGARCH, Nelson (1990), EGARCH, Nelson (1991), APARCH of Ding, Granger and Engle (1993) e..c.. Several simplified versions of hese models have been applied o differen asse daa series (e.g. Milhoj 1987, Bollerslev 1987, Hsieh 1988 e..c.). All he above models are univariae and ry o esimae he properies of financial asses wihou considering heir co-movemen. However, undersanding and esimaing ime varying condiional variances and covariances is imporan for many issues in finance since here are many applicaions ha rely on mulivariae covariance models. I is essenial, for opimal hedging, asse allocaion, derivaives pricing and risk managemen, he accurae modelling and forecas of he asses reurns co-movemen. As a resul, Bollerslev, Engle and Wooldridge (1988), Cecchei (1988), Myers and Thompson (1989), Baillie and Myers (1991), Kroner and Sulan (1993), Ng and Kroner (1998), argue ha financial prices are characerized by ime varying variances and covariances, presening a variey of mulivariae GARCH models. Bollerslev, Engle and Wooldridge (1988) suggesed 36

37 Chaper I. ARCH-GARCH Processes: Review and Opimal Hedging he VEC model and he diagonal VEC (DVEC) model in which he variances depend only on heir pas squared errors and he covariances on heir pas crossproducs of errors. Given he excessively large parameers needed o esimae he VEC model and he necessiy o impose srong resricions on he parameers Engle and Kroner (1995) proposed he BEKK paramerizaion avoiding unrealisic assumpions such as ha he correlaion beween he condiional variances is consan (Consan correlaion model by Bollerslev 1990), and guaraneeing ha he ime varying covariance marix is posiive definie. Addiional models can be found in Engle, Ng and Rohschild (1990b) who proposed facor models (FGARCH), in Alexander and Chibumba (1997) who proposed he orhogonal GARCH models (O- GARCH), in Tse and Tsui (00) and of Engle (00) who suggesed he Dynamic Condiional Correlaion models (DCC). Despie almos hree decades of research, he formulaion and implemenaion of an opimal fuures hedging sraegy remains a conroversial issue in finance. Based on earlier research by Johnson (1960) and Sein (1961), Ederingon (1979) developed he firs and mos widely used OLS approach for esimaing he so-called minimum-variance saic hedge raio. Subsequen developmens in financial economerics wih respec o condiional heeroscedasiciy and coinegraion (for a comprehensive descripion and references see Mills and Markellos, 007), moivaed he developmen of more sophisicaed hedge raio esimaion approaches which can accoun for sylised facs of he daa. A variey of relevan echniques have evolved over recen years including, for example, error-correcion models, exponenial moving averages, univariae and mulivariae GARCH models, ec. (for a review see Chen e al, 003; Brooks and Chong, 001). Alhough more sophisicaed models are ofen found o be superior o OLS in erms of hedging performance, here is no consensus in he empirical lieraure wih respec o he mos appropriae approach. In his chaper, he univariae processes of ARCH-GARCH models are discussed and heir economeric mehodology is oulined. Nex, he maximum 37

38 Chaper I. ARCH-GARCH Processes: Review and Opimal Hedging likelihood esimaion of boh univariae and mulivariae ARCH-GARCH models is defined. In he hird secion, he differen economeric models employed for he hedging effeciveness are described. Nex, he esimaion and he resuls of he above economeric processes are presened. The las Secion concludes and presens he poenial problems and implicaions in he esimaion of he above processes. I.1. The Processes I.1.1. Univariae ARCH-GARCH formulaions Much recen ineres in economerics and empirical finance has cenered on modelling he emporal variaion in he condiional second order momens of asse reurns changes. Parallel o he success of he sandard linear ime series models, which use he condiional han he uncondiional mean, he key insigh offered by ARCH formulaions lies in he disincion beween he condiional and he uncondiional second order momens. The general form for ARCH model posulaes he condiional variance o be a non-rivial funcion of he curren informaion se. The ARCH(q) 1 formulaion in which σ depends linearly on q lagged values of he squared innovaions ε is described by he following equaion: q h = ω + αε (1) i= 1 i -1 where σ is he condiional variance and ε is he disurbance erm (or unpredicable par). Of course, for his model o be well defined and he condiional variance o be 1 For a more deailed discussion of ARCH(q) he reader is referred o he original paper by Engle (198). 38

39 Chaper I. ARCH-GARCH Processes: Review and Opimal Hedging posiive, almos surely he parameers mus saisfy ω>0 and a 1 0,.,a q 0. Anoher formulaion of he ARCH(q) process is he following: h = ω+ ( α L+ α L α L ) ε () q 1 q Defining as A L α L α L α L q ( ) = q and v = ε h he equaion () can be rewrien as: ε = ω + A(L) ε + v (3) As a resul, he ARCH(q) model is an auoregressive process of he squared error erms and is covariance saionary if he sum of he posiive auoregressive parameers is less han one, q i= 1 i α L p1. If he ARCH(q) process is covariance i saionary, he respecive uncondiional variance is given by he following ype: σ = 1 ω q i= 1 α i (4) In he majoriy of he empirical applicaions, o be correcly specified an ARCH model, i is decen o impose a relaively long lag srucure of he squared innovaions in he condiional variance equaion, in order o guaranee a posiively definie variance. Unforunaely, economic heory offers lile guidance as o which variables should be imporan in deermining he observed ime variaion in he condiional variances. In his ligh i seems of immediae pracical ineres o exend he ARCH class of models o allow for boh a longer memory and a more flexible lag srucure. Numerous ad hoc parameric formulaions have been suggesed in he 39

40 Chaper I. ARCH-GARCH Processes: Review and Opimal Hedging lieraure. One of he mos successful has been he generalized ARCH, GARCH model inroduced by Bollerslev (1986). GARCH(p,q). The Generalized ARCH (GARCH) model can be expressed as: q p = ω + αiε-i + β j - j = ω + ( ) ε + ( ) i= 1 j= 1, p (5) h h A L B L h where ω > 0, α i 0 and β i 0 for all i in order o ensure he posiiviy of h. Provided ha he sum of he posiive parameers β j is less han one, q j β il p 1, and he j= 1 polynomials 1-B(L), A(L) have no common roos, he above posiiviy consrain is saisfied if all coefficiens in he infinie power expansion of he polynomial B( L) are non negaive. Rearranging he parameers of equaion (5), he 1 B( L) GARCH(p,q) processes can be rewrien as: q p p = + i -i + j -j bjv j + v i= 1 j= 1 j= 1 (6) ε ω αε β ε which is an auoregressive moving average process wih squared innovaions of orders max(p,q) and p, he ARMA(max(p,q),p). The above process is second order saionary if he sum of q α + p i i= 1 j= 1 β <1 is smaller han j one. The uncondiional variance of he GARCH(p,q) process is equal o: q α + In case he esimae of he sum i j is very close or equal o uniy hen he i= 1 j= 1 formulaion of he GARCH(p,q) process is called he inegraed IGARCH(p,q) model developed by Engle and Bollerslev (1986). p β 40

41 Chaper I. ARCH-GARCH Processes: Review and Opimal Hedging σ = 1 q ω p α β j i i= 1 j= 1. (7) I.1.. Maximum Likelihood Esimaion The maximum likelihood esimaion (MLE) of he ARCH processes is easy o implemen once he densiy funcion of he sandardized errors z (θ) = ε (θ) /σ (θ) is specified, where θ is he vecor of he unknown parameers o be esimaed for he condiional mean, variance and densiy funcion. The log likelihood funcion is generally deermined as: 1 l( y, θ) = log( f( z( θ); θ)) log( h ( θ)) (8) where y is he se of T observaions saisfying he regression equaion: y = xβ + ε (9) and x is he vecor of predeermined explanaory variables which could include lagged values of y. The full sample log-likelihood is simply: T L( θ) = l ( y θ). (10) = 1, 41

42 Chaper I. ARCH-GARCH Processes: Review and Opimal Hedging The maximum likelihood esimaor (MLE) is found by maximizing equaion (10) or by aking he derivaive of he log likelihood of he h observaion wih T l( y, θ) respec o he parameer θ equal o zero: = 0. θ = 1 Case I: Normal disribued sandardized innovaions z (θ) When he condiional disribuion of y is Gaussian wih mean xβ and variance h : 1 ( y x β) f( y x, y 1) = exp( ), he se of parameers o be πh h esimaed are only for he condiional mean and variance since he densiy funcion is uniquely deermined by he firs wo momens. As a resul, he log-likelihood funcion is esimaed by he following equaion: ( y x β) l ( y ) = log( ) log( ) (11) 1 1, θ h π h and he full sample log likelihood is hen: T 1 1 L( θ) = log f( y, θ) = ( )log( π) log( h ( y x β) T T T ) = 1 = 1 = 1 h (1) In order o calculae he firs and second derivaives of he log of he condiional likelihood of he h observaion wih respec o he parameer vecor θ, he condiional variance is rewrien as: h = ω s, where ω = (a 0, a 1,., a q,β 1,.,β p ), and s = (1,ε -1,.,ε -q, h -1,.,h -p ). Then, he firs and second derivaives of he log likelihood wih respec o he condiional mean are: l ( y ; θ) ε x 1 h ε h ( ) = + h β h β h (13) 4

43 Chaper I. ARCH-GARCH Processes: Review and Opimal Hedging l ( y ; θ) x x 1 h h ε ε x h ε h 1 h ( ) ( ) ( ) β β β β β β β = + + h h h h h h (14) where q h = a x ε + β h p j i i i β j (15) i= 1 j= 1 β The firs and second derivaives of he log likelihood wih respec o he condiional variance are: l( y; θ) 1 h ε h = ( ) ω h ω h (16) l ( y ; θ) ε 1 1 ( h h h h ε ) ( ) ( ) = ω ω h ω h ω h ω ω h (17) where h = s + ω h p i βi (18) i= 1 ω Generally, in order o maximize he likelihood funcion he mehod of scoring as in Engle (198) or he Bernd, Hall, Hall and Hausman (BHHH, 1974) algorihm. Case II: Non-Normal disribued sandardized innovaions z (θ) Since here is well esablished sylized evidence ha financial reurns exhibi faer ails han he normal disribuion, Bollerslev (1987) proposed ha he innovaions z (θ) migh be drawn from a sandardized disribuion wih η degrees of freedom o be esimaed by maximizing he log likelihood funcion. The above menioned densiy funcion is given by he following ype: 43

44 Chaper I. ARCH-GARCH Processes: Review and Opimal Hedging -( η+ 1) Γ (( η+ 1)/) z f( z η) = (1 + ) (19) Γ ( η/) π( η ) η where Γ( ) is he gamma funcion. When η=, The sample log likelihood condiional on he firs T observaions along wih he specificaion of he condiional mean and variance given by equaions (9) and () is specified as follows: T T Γ ( η+ 1)/ 1 log f( y θ) = Tlog log( h) π( η ) Γ( η/) = 1 = 1 T ( η+ 1) ( y x β) log 1 + h( η ) = 1 (0) The log likelihood is maximized numerically wih respec o η, β and h subjec o he consrain η>. Several approaches have been proposed for he condiional disribuion of he innovaions such as he power exponenial disribuion by Baillie and Bollerslev (1989), he generalized disribuion by Bollerslev (1994), he generalized exponenial disribuion by Nelson (1991), and he Gram Charlier ype disribuion by Jondeau and Rockinger (001). I.. Daa Descripion The daa used in he presen sudy includes daily spo prices and respecive fuure conrac prices for he UK Financial Times Sock Exchange Index (FTSE), he US Dow Jοnes Index (DJ) and he German Deuscher Akien-Index (DAX). The sample covers he period from 4 January 1999 o 0 Sepember 004 for all hree 44

45 Chaper I. ARCH-GARCH Processes: Review and Opimal Hedging markes. Non-rading days were removed and rading daes beween spo and fuures prices were mached leaving a oal of 1443, 1435 and 1450 observaions for he FTSE, DJ and DAX, respecively. The las 60 observaions for each series were lef for ou-of-sample hedge raio performance comparison. There are four delivery monhs for all hree markes: March, June, Sepember and December. Due o he size of he markes considered, a leas wo conracs were raded a any ime, hus faciliaing conrac rollover. Descripive saisics of logarihmic reurns for he spo and fuures series under sudy are provided in Table 1. There is srong evidence ha all series are non-normally disribued wih high peaks and fa ails. For all bu one series, here is negaive asymmery in he disribuion. Obs. Mean S. Dev. Skewness Κurosis JB Spo FTSE 1, DJ 1, DAX 1, Fuures FTSE 1, DJ 1, DAX 1, Table I. 1 Descripive Saisics of spo and fuures index reurns. Jarque-Bera (JB) is asympoically disribued as a Chi-squared wih degrees of freedom under he null hypohesis of normaliy. 45

46 Chaper I. ARCH-GARCH Processes: Review and Opimal Hedging I.3 The Economeric Mehodologies for opimal hedging I examine four univariae economeric approaches for hedge raio esimaion, namely: OLS, error-correcion, EWMA, univariae GARCH. These models are presened briefly, since hey have been exensively described in he lieraure. I.3.1. Ordinary Leas Squares Model (OLS) As suggesed by Ederingon (1979), he opimal hedge raio can be esimaed simply via OLS by regressing spo reurns ( S ) on fuures reurns ( F ): S = c + β F + ε (1) where εis he regression error erm. The one-period minimum variance consan OLS hedge raio is simply he slope coefficien β (denoed hereafer ashr ). I.3. Coinegraion and Error Correcion Modeling (ECM) Ghosh (1993), Kroner and Sulan (1993) and Lien (1996), among ohers, have shown ha a regression such as (1) will be misspecified if he fuures and spo price are coinegraed. Naurally, ignoring he resuling error-correcion relaionship may lead o a biased hedge raio. The opimal hedge raio, accouning for he coinegraing relaionship beween spo and fuure markes, is he coefficien β ( HR ECM ) esimaed from he following error correcion model esed by Myers and Thompson (1989): 46

47 Chaper I. ARCH-GARCH Processes: Review and Opimal Hedging m n S = α u + β F + ξ F + φ S + ε () -1 i -1 j -1 i= 1 j= 1 where u -1 are lagged residuals from he coinegraing regression represening he error correcion erm Exponenially Weighed Moving Average Model (EWMA) The exponenially weighed moving average model, popularized by Riskmerics TM (1996), is a special case of he ARCH( ) model of Engle (198) which allows capuring some of he persisence in volailiy. The one-period ahead EWMA variance forecas and covariance can be calculaed as, respecively: n i 1 λ i i= 1 h FF, = (1 λ ) F (3a) n i 1 SF, = λ λ i i i= 1 h (1 ) S F (3b) The persisence in volailiy is conrolled by he decay parameer λ ( 0<λ<1) which is ofen se o The opimal hedge raio can be esimaed as: HR EWMA = h h SF, FF, (3c) 47

48 Chaper I. ARCH-GARCH Processes: Review and Opimal Hedging I.3.4 Univariae GARCH(1,1) model Cecchei and Figlewski (1988) firs proposed using an ARCH model for opimal hedge raio esimaion in order o allow he reurn covariance marix o vary over ime. Several exensions of his basic model have been developed ever since. For example, using he popular GARCH(1,1) specificaion, he condiional variances of he spo ( h SS, ) and fuures ( h FF, ) reurns can be modeled respecively as: h = c + aε + β h (4a) SS, SS 1 S, -1 1 S, -1 h = c + a ε + β h (4b) FF, FF 1 F, -1 1 F, -1 where ε and S, 1 ε F, 1are he lagged squared residuals from he condiional mean equaions for he spo and fuures reurns respecively. The covariance of reurns can hen be esimaed by: h = ρ h h (4c) SF, SS, FF, where ρ is he ime invarian correlaion beween spo and fuures reurns. Finally, he opimal hedge raio can be calculaed as: HR GARCH = h h SF, FF, (4d) 48

49 Chaper I. ARCH-GARCH Processes: Review and Opimal Hedging I.4 Esimaion and Resuls of he Economeric Processes In his Secion, firs he parameers of he proposed four economeric processes are esimaed and he resuls for he opimal hedge raios are discussed. Then, he robusness of he resuls is checked, applying he Akaike Informaion Crieria (AIC) and he Bayesian Informaion Crierion or Schwarz Informaion Crierion (BIC) crieria. Table provides regression summary saisics and residual diagnosic es resuls for he OLS approach. The esimaed hedge raios are less han uniy for all hree markes. The inercep was found o be saisically insignifican and was omied. The residual ess saisics are all highly significan a he 1% level and sugges serious violaions of he OLS assumpions for all hree regressions esimaed. More specifically, he Jarque-Berra es saisics imply srong nonnormaliy in he residuals while he Durbin-Wason (DW) and he Breusch- Godfrey (BG) es resuls sugges serial correlaion in he residuals. Finally, Engle s (198) ARCH es deecs srong heeroscedasiciy in he residuals. All hese resuls jusify he implemenaion of he alernaive esimaion echniques discussed in he mehodology secion. HR OLS s.e. R LL JB DW BG ARCH FTSE DJ DAX Table I. OLS Opimal Hedge Raio Esimaion. LL denoes he log likelihood of he model esimaed. Jarque-Bera (JB) is asympoically disribued as a Chi-squared wih degrees of freedom under he null hypohesis of normaliy. DW is he Durbin-Wason es saisic of firs order auocorrelaion in he residuals. The Breusch-Godfrey (BG) 49

50 Chaper I. ARCH-GARCH Processes: Review and Opimal Hedging Lagrange Muliplier ess he null of zero auocorrelaion in he residuals of order p. The ARCH ess he null hypohesis of homoskedasiciy in he residuals of order p. The BG and ARCH es saisics are asympoically disribued as χ wih p degrees of freedom and are implemened for p=0 lags. For he coinegraion and error-correcion modelling approach, he order of inegraion for he variables involved mus firs be esablished. The presence of a uni roo is a sylized fac for boh spo and fuures prices, somehing ha was confirmed by a variey of saionariy ess (resuls are available upon reques from he auhors). Coinegraion beween spo and fuures prices was esed for using he Johansen and Juselius (1990) approach. The null hypohesis of non-coinegraion was rejeced for all hree markes. More specifically, he relevan maximum eigenvalue (race) es saisics assumed values of (1.9), 41.9 (48.8) and 39.1 (33.) for he FTSE, DJ and DAX, respecively. All hese are significan a he 5% level. On he basis of he Granger Represenaion heorem (Engle and Granger, 1987), coinegraion implies error-correcion models linking spo and fuures reurns and hese were esimaed as following for each marke (sandard errors appear in brackes): ( S ) = u ( F ) ( F ) FTSE, -1 FTSE, FTSE, 1 (0.08) (0.0046) (0.0046) ( F ) (0.071) FTSE, ( S )= u (F ) (F ) DJ, -1 DJ, DJ, 1 (0.0490) (0.0049) (0.0048) (F ) ( S ) ( S ) DJ, DJ,-1 DJ,- (0.0051) (0.0496) (0.031) 50

51 Chaper I. ARCH-GARCH Processes: Review and Opimal Hedging ( S ) = u ( F ) ( F ) DAX, -1 DAX, DAX, 1 (0.0347) (0.0059) (0.0059) ( S ) ( S ) DAX, 1 DAX, (0.0358) (0.079) HR ECM s.e. R LL JB DW BG ARCH FTSE DJ DAX Table I. 3 Error-Correcion Modelling Opimal Hedge Raio Esimaion. LL denoes he log likelihood of he model esimaed. Jarque-Bera (JB) is asympoically disribued as a Chi-squared wih degrees of freedom under he null hypohesis of normaliy. DW is he Durbin-Wason es saisic of firs order auocorrelaion in he residuals. The Breusch-Godfrey (BG) Lagrange Muliplier ess he null of zero auocorrelaion in he residuals of order p. The ARCH ess he null hypohesis of homoskedasiciy in he residuals of order p. The BG and ARCH es saisics are asympoically disribued as χ wih p degrees of freedom and are implemened for p=0 lags. The error-correcion modeling resuls show ha all hedge raios and errorcorrecion erm coefficiens in he hedge equaions are significanly differen han zero a he 1% level. Lagged fuures and spo reurns ener significanly in hese regressions. The resuling hedge raios are reproduced along wih regression diagnosics in Table 3. The hedge raio values are comparable in magniude o hose obained by OLS. Alhough he likelihood of he hedge regressions has now increased compared o he OLS, he residuals remain srongly non-iid. More specifically, as suggesed by he ess, nonnormaliy, auocorrelaion and heeroscedasiciy sill prevail in he residuals. Nex, I urn o he esimaion of he hedge raios using he models which allow for variaions in he condiional volailiy. Using he EWMA approach, he average values of he dynamic hedge raios were esimaed a , and for he FTSE, DJ and DAX, respecively. These values are somewha lower 51

52 Chaper I. ARCH-GARCH Processes: Review and Opimal Hedging han hose obained previously. The GARCH(1,1) model esimaion resuls are presened in Table 4. All coefficiens are posiive and saisically significan. The near-uniy sum of he coefficiens suggess very high persisence in he condiional variances. The average values of he GARCH-esimaed hedge raios are , and for he FTSE, DJ and DAX, respecively. c SS, FF, ε S, -1 c F, -1 hs, -1 F, -1 FTSE DJ DAX Spo Fuures Spo Fuures Spo Fuures (0.0084) ε (0.0149) h (0.0171) 0.03 (0.0076) (0.0134) (0.0155) (0.0070) (0.0100) (0.0115) (0.0068) (0.0099) (0.0117) (0.0151) (0.016) (0.0147) (0.015) (0.0108) (0.013) LL Nyblom Join Tes , AIC BIC Table I. 4 GARCH(1,1) esimaes for Spo and Fuures Index reurns. Sandard errors appear in brackes. Also, I evaluae and compare he hedging performances of he models considered. We follow he procedure proposed by Kroner and Sulan (1993) and calculae he porfolio variance as following: var( R HR R ) (5) i, F, 5

53 Chaper I. ARCH-GARCH Processes: Review and Opimal Hedging where HR are he hedge raios esimaed using each one of he four models: OLS, ECM, EWMA, GARCH. Obviously, for he OLS and ECM saic hedge raios are esimaed and he ime subscrip is no applicable. The opimal hedging sraegy is he one which achieves he maximum variance porfolio reducion. Unhedged Naive OLS EWMA ECM GARCH In-sample FTSE DJ DAX Ou-of-sample FTSE DJ DAX Table I. 5 Comparison of hedging effeciveness beween models Table 5 displays he in-sample and ou-of-sample hedging performance of each approach. We also include resuls from an unhedged posiion along wih hose from a naïve hedging approach which assumes a uniary hedge raio over he whole sample. A firs observaion is ha all he models are able o offer a significan reducion in he porfolio variance in comparison o he unhedged porfolio and he naïve mehod. In in-sample daa, he GARCH model consisenly ouperforms is compeiors for all markes sudied. The EWMA performed wors in-sample among he six modeling approaches for all hree markes. For he ou-of-sample daa, he resuls are somewha mixed. On average, for he hree markes sudied, he increase 53

54 Chaper I. ARCH-GARCH Processes: Review and Opimal Hedging in porfolio variance for each one of he models considered was: OLS (7.4%), EWMA (9.3%), ECM (7.1%), and GARCH (8.4%). I.5. Summary and Discussion The accurae modelling of he condiional is imporan for financial asses pricing and risk managemen. Turning o he relaive lieraure, he opic has received limied aenion apar from he ARCH-GARCH processes. The objecive of his chaper was o idenify he process ha evaluaes efficienly he hedge raio effeciveness of differen financial asses porfolios and also he forecasing hedging abiliy. To his end, I have examined alernaive affine specificaions of univariae processes and heir esimaion resuls have been compared. More precisely, he dynamic hedging effeciveness of he OLS, errorcorrecion, exponenial moving averages, and, univariae GARCH processes is examined. Effeciveness is measured in-sample and ou-of-sample using he minimum variance mehod. Spo and fuures daily closing prices are used for sock indices from he US, UK and Germany for he period January 1999 o Sepember 004. I found, ha under boh in-sample and ou-of-sample, using he minimum variance mehod, he radiional consan hedge raios indicae poor hedging performance. Since, asses and markes affec each oher no only in erms of expeced reurns bu also in erms of volailiy, o improve he effeciveness of he opimal hedging, in he nex chaper, I sudy he relaion beween he volailiies and co-volailiies of he spo and fuures markes so as o compue ime varying hedge raios. More precisely, a bivariae MGARCH model is applied for he condiional variance covariance specificaion and he hedge raios are compued as a by-produc of esimaion and updaed by insering new observaions. 54

55 Chaper II. Mulivariae GARCH: Review and Opimal hedging CHAPTER II Mulivariae GARCH Processes: Review and Opimal Hedging 3 Undersanding and esimaing ime varying condiional variances and covariances is imporan for many issues in finance since here are many applicaions ha rely on mulivariae covariance models. I is essenial, for opimal hedging, asse allocaion, derivaives pricing and risk managemen, he accurae modelling and forecas of he asses reurns co-movemen. Bollerslev, Engle and Wooldridge (1988), Cecchei (1988), Myers and Thompson (1989), Baillie and Myers (1991), Kroner and Sulan (1993), Ng and Kroner (1998), argue ha financial prices are characerized by ime varying variances and covariances, presening a variey of mulivariae GARCH models. Bollerslev, Engle and Wooldridge (1988) suggesed he VEC model and he diagonal VEC (DVEC) model in which he variances depend only on heir pas squared errors and he covariances on heir pas cross-producs of errors. Given he excessively large parameers needed o esimae he VEC model and he necessiy o impose srong resricions on he parameers 3 Par of his work has been acceped for presenaion in he 17 h European Financial Managemen Associaion Conference o be held in Ahens, Greece, on June 5-8,

56 Chaper II. Mulivariae GARCH: Review and Opimal hedging Engle and Kroner (1995) proposed he BEKK parameerizaion avoiding unrealisic assumpions such as ha he correlaion beween he condiional variances is consan (Consan correlaion model by Bollerslev 1990), and guaraneeing ha he ime varying covariance marix is posiive definie. Addiional models can be found in Engle, Ng and Rohschild (1990b) who proposed facor models (FGARCH), in Alexander and Chibumba (1997) who proposed he orhogonal GARCH models (O- GARCH), in Tse and Tsui (00) and of Engle (00) who suggesed he Dynamic Condiional Correlaion models (DCC). In his chaper, he ARCH processes are exended in a mulidimensional framework, discussing he consrains o be imposed so as o reduce he number of parameers o be esimaed and o achieve a more parsimonious specificaion. The remainder of he paper is organized as follows: he nex secion inroduces he mulivariae ARCH models. Nex, BEKK and he ad hoc mulivariae version of GARCH (Wang, Yao, 005) models are examined in deails. The hird secion describes he daa and he empirical resuls on he VaR esimaion and opimal hedging, and he final secion concludes. II.1. Mulivariae GARCH formulaions The accurae esimaion of ime varying covariances beween asse reurns is crucial for asse pricing, forecasing and risk managemen. Consider an n- dimensional vecor sochasic process (Y ). If θ is he finie vecor of parameers hen: y = µ ( θ) + ε (6) where µ ( θ) is he condiional mean vecor and ε, he innovaion process is equal o ε 1/ = H ( θ) z whereh 1/ ( θ) is an N N posiive definie marix. Furhermore, if z has zero mean and variance equals o he ideniy marix of order N, hen he condiional variance marix of yis calculaed by he following equaion: 56

57 Chaper II. Mulivariae GARCH: Review and Opimal hedging Var( y I ) = Var ( y ) = Var ( ) = H Var ( z )( H ) = H (7) 1/ 1/ ε 1 For a sysem of n regression equaions he mulivariae framework could be presened as: y = Bx + ε ε I 1 f 0, H (8) H = g( H H..., ε ε ) 1,, 1,,... where B is a n k marix of unknown parameers, x is a 1 k vecor of explanaory variables and εis he 1 n vecor of whie noise residuals. The f( ) funcion represens he condiional mulivariae densiy funcion of he residuals and g( ) is a funcion of he lagged condiional covariance marix and he residuals. Engle and Kroner (1995) proposed he following vecor generalizaion of he mulivariae GARCH(p,q) model specificaion: H = CC + Aε A + Aε A Aε A + BH B + B H B B H B ' ' ' ' ' ' ' q q q q q q q p ' ' ' ' iε iε i i j j j i= 1 j= 1 = CC + ( A A) + ( B H B ) (9) where C, Α, Β, are n n marices of parameers and H is guaraneed o be posiive definie as long as CC' is posiive definie which can be ensured numerically when C is a lower riangular marix. The main drawback of he unconsrained specificaion is he large number of parameers o be esimaed since he required parameers are equal o [(n(n+1)/) + n (p+q)]. For insance, if he model is applied o a porfolio including n=10 risky asses, i ends up wih 3,080 parameers o be esimaed, which is no feasible. In pracice, i is necessary o impose consrains on he parameers of he volailiy equaion o reduce he dimension of he parameer space and simplify he dynamics. 57

58 Chaper II. Mulivariae GARCH: Review and Opimal hedging A number of models in he economeric area have been proposed by imposing differen consrains in mulivariae GARCH in order o reduce he esimaed parameers and also o achieve a more parsimonious specificaion wih a posiive definie condiional variance covariance marix. Bollerslev (1990) proposed he diagonal mulivariae GARCH (p,q), named DVEC model in which each elemen of H depends on is own lag and he previous value of ε i ε ij and is defined as: H C A vech ε ε (30) o o ' o = + i ( i i) + Bi vech( H i) where vech( ) denoes he operaor ha sacks he lower riangular porion of a n n marix as a [(n(n+1)/)] 1parameer vecor. A and B are diagonal marices, and o o C, A, B o are symmeric n n marices, implied by he relaions: C=vech(C o ), A= diag[vech( A o )] and B=diag[vech(B o )]. Necessary and sufficien condiions on he parameers o ensure ha he condiional variance marices are posiive definie are more easily derived by expressing he model in erms of Hadamard producs (denoed as ). From he above formulaion, H is posiive o o definie for all provided ha C, A, B o and he iniial variance marix H 0 are posiive definie, and he number of parameers o be esimaed is reduced o (n(n+1))/)(1+p+q). So for example, for n=10, he diagonal GARCH(1,1) requires he esimaion of 165 parameers, quie fewer han he unconsrained mulivariae GARCH referred previously. The above formulaion, even under diagonal assumpion, requires large scale sysems which are sill highly parameerized and difficul o esimae in pracice. The main disadvanage is he difficuly o guaranee he posiiviy of H wihou imposing srong resricions on he parameers 4. Anoher drawback of he diagonal model is is srucure since i is no invarian wih respec o linear combinaions. I is noed ha he invariance requiremen is imporan for a 4 Gourieroux (1997) gives sufficien condiions for he posiiviy of H. These condiions are obained by wriing he model for he marix H iself raher han for is vecorized version. 58

59 Chaper II. Mulivariae GARCH: Review and Opimal hedging number of financial applicaions as asses aggregaion and opimal hedging of muual funds. Engle and Kroner (1995), based on an earlier approach adaped by Baba e al. (1987), proposed a new parameerizaion of he H, he so called BEKK model (Baba-Engle-Kraf-Kroner), which is based on he specral decomposiion of he condiional variance covariance marix in order o build a mulivariae GARCH model. The BEKK(1,1,K) model is defined as: q p ' ' ' ' = + iε iε i i + j j j i= 1 j= 1 (31) H CC ( A A) ( B H B ) where ' CC is symmeric and posiive definie, he second and hird erm of he equaion (31) q p ' ' ' ( A ε ε i i iai), ( BjH jbj) i= 1 j= 1 are expressed in quadraic forms, in order o ensure ha he variance covariance marix H is posiive definie, and he summaion limi K deermines he generaliy of he process. For K=1, he number of parameers o be esimaed in a BEKK(1,1,1) model is n(5n+1)/. However, i is invarian wih respec o linear combinaions. To reduce his number, and consequenly o reduce he generaliy, one can impose a diagonal BEKK model, for example he A i and B j marices o be diagonal. This model is like a diagonal VEC model (DVEC) bu i is less general, alhough i is guaraneed o be posiive definie while he DVEC is no. Also, he parameers o be esimaed by a BEKK model are less han hese by a DVEC model because he parameers governing he dynamics of he covariance equaion in he BEKK model are he producs of he corresponding parameers of he wo variance equaions in he same model. For example, in a bivariae model: he DVEC model conains 9 parameers while he BEKK model conains only 7 parameers. Similar o he univariae GARCH models, he sandard esimaion mehod for he BEKK model is he quasi-maximum likelihood esimaion (qmle) faciliaed by assuming y F 1 N(0, H ), since he qmle is a soluion of a 59

60 Chaper II. Mulivariae GARCH: Review and Opimal hedging high dimensional nonlinear opimizaion problem. The consisency and asympoic normaliy of he qmle have been esablished by Come and Lieberman (003). The main drawback when esimaing he mulivariae GARCH models is he excessively large parameers needed o esimae, even afer imposing several resricions. As a resul, he menioned models are rarely used when he number of series is larger han 3 or 4. To overcome he above problem, differen models, more simplified were proposed by imposing a common dynamic srucure on all he elemens of H, which resul in less parameerized models, such as ha he correlaion beween he condiional variances is consan (Consan correlaion model by Bollerslev 1990), and guaraneeing ha he ime varying covariance marix is posiive definie. Addiional models can be found in Engle, Ng and Rohschild (1990b) who proposed facor models (FGARCH), in Alexander and Chibumba (1997) who proposed he orhogonal GARCH models (O-GARCH), in Tse and Tsui (00) and of Engle (00) who suggesed he Dynamic Condiional Correlaion models (DCC). Finally, Wang and Yao (005) and Harris, Soja and Tucker (007) proposed an ad hoc mulivariae mehod and a simplified process respecively, using univariae GARCH models in order o allow he reurn covariance marix o vary over ime, which are easy o be implemened increasing he compuaional efficiency. In he nex secion, I analyze furher BEKK and he ad hoc mulivariae version of GARCH (Wang, Yao, 005) models. II.1.1 Bivariae BEKK(1,1,1) model following: Using a BEKK represenaion, he condiional variance marix is he H = CC' + Aε ε ' A + BH B (3)

61 Chaper II. Mulivariae GARCH: Review and Opimal hedging where C, Α, Β are marices, wih C being upper riangular, symmeric and posiive definie. The condiional variance marix is posiive definie since he second and hird erms in he above equaion (5) are expressed in quadraic forms. This means ha no oher consrains for he marices Α and Β are necessary. For he case of he bivariae GARCH(1,1), he BEKK model is esimaed in a resriced form wih C as a lower riangular marix, and, Α, Β being diagonal marices. This can be expressed by he following equaions: h11, h1, c11 0 c11 c a1 0 ε 1, 1 ε1, 1ε, 1 a1 0 = h1, h + +, c c1 0 c1 0 a ε1, 1ε, 1 ε, 1 0 a β1 0 h11, 1 h1, 1 β β h 1, 1 h, 1 0 β (33) or h = c + a ε + β h 11, , , 1 h = c + c + a ε + β h, 1, 1, 1 h = c c + aa ε ε + β β h 1, , 1, 1 1 1, 1 (34) where c 11, c, c 1 are consans and ε 1, 1 ε, 1 are he lagged residuals from he condiional mean equaion. In he above bivariae model he variables used are spo prices (y 1 ) and fuures prices (y ). For our empirical analysis we evaluae: a) The opimal hedge raio which is esimaed by: HR BEKK h = h 1,, (35) 61

62 Chaper II. Mulivariae GARCH: Review and Opimal hedging and b) he Value a risk (VaR) of a porfolio using he variance covariance approach (VCV). II.1. The ad hoc mulivariae of GARCH(1,1) model Wang and Yao (005) firs proposed an ad hoc mulivariae mehod using univariae GARCH models in order o allow he reurn covariance marix o vary over ime. More precisely, using he popular GARCH(1,1) specificaion, he condiional variances of wo reurn series y 1,, y,, can be modeled respecively as: h = c + aε + β h (36) 1, , , -1 h = c + aε + β h (37), 1, -1, -1 where 1, 1 ε and ε, 1 are he lagged squared residuals from he condiional mean equaions for he spo and fuures reurns respecively. Then, o model he condiional covariance, he following seps are implemened. Firsly, a new series is consruced as: x ( y + y ) 1,, 1, =. Secondly, he condiional variance of he new series h 1, is esimaed from anoher univariae GARCH(1,1) as: h = c + a ε + β h (38) 1, 1 1 1, , -1 equaion: σ Finally, he ime varying condiional covariance of reurns is given by he ( h + h ) = h (39) 1,, 1, 1,. 6

63 Chaper II. Mulivariae GARCH: Review and Opimal hedging I is clear ha he above proposal overcomes he difficulies due o overparamerizaion, and can be implemened in compuaionally efficien manner since all he componens of H are pracically fied separaely. However he simpliciy in boh he srucure and he feasibiliy does come wih a price unforunaely. The main drawback of he above approach is ha he implied esimaor of he condiional variance covariance marix is no guaraneed ha i is posiive definie. Also, he condiional covariance esimaor may no follow he model specificaion of he univariae models used for he variances esimaion. II.. Daa Descripion The daa used in he presen sudy includes he same sample as in he fis chaper, i.e. daily spo prices and respecive fuure conrac prices for he UK Financial Times Sock Exchange Index (FTSE), he US Dow Jοnes Index (DJ) and he German Deuscher Akien-Index (DAX). The sample covers he period from 4 January 1999 o 0 Sepember 004 for all hree markes. Non-rading days were removed and rading daes beween spo and fuures prices were mached leaving a oal of 1443, 1435 and 1450 observaions for he FTSE, DJ and DAX, respecively. The las 60 observaions for each series were lef for ou-of-sample hedge raio performance comparison. There are four delivery monhs for all hree markes: March, June, Sepember and December. Due o he size of he markes considered, a leas wo conracs were raded a any ime, hus faciliaing conrac rollover. This sample is used for he hedge raio esimaion by he BEKK model. The second sample, used in his sudy, includes daily closing prices for he UK Financial Times Sock Exchange Index (FTSE), and he US Dow Jοnes Index (DJ) and daily spo prices of wo exchange series, he EUR and he GBP agains USD. The sample covers he period from 3 January 000 unil 30 June 007 for all four daa series, a oal of 195 observaions for boh he exchange raes and 1861 for he FTSE and Dow Jones indices respecively. The las 100 observaions for each series are lef for 63

64 Chaper II. Mulivariae GARCH: Review and Opimal hedging ex ane (ou of sample) porfolio VaR esimaion, applying boh BEKK(1,1) and he ad hoc GARCH(1,1) models. Descripive saisics of logarihmic reurns of all series daa for he second sample under sudy are provided in Table, since he firs sample was analysed in he firs chaper. There is srong evidence ha all series are non-normally disribued wih high peaks and fa ails. For all series, here is negaive asymmery in he disribuion. The Ljung-Box pormaneau es for all series excep FTSE shows no significan auocorrelaion while he ARCH-LM es for serial correlaion in squared reurns reveals volailiy clusering in all series and more significanly in equiy indices. Obs. Mean S. Dev. Skewness Κurosis JB Q 1 (10) ARCH(4) Equiy Indices FTSE 1, DJ 1, FX Spo agains USD EUR 1, GBP 1, Table II. 1 Descripive Saisics of equiy indices and foreign exchanges reurns. Jarque-Bera (JB) is asympoically disribued as a Chi-squared wih degrees of freedom under he null hypohesis of normaliy. The Q 1 (k) represens he Ljung-Box pormaneau es of he reurn series. ARCH(4) saisic ess he null hypohesis ha he firs four parial auocorrelaion of squared reurns are zero. 64

65 Chaper II. Mulivariae GARCH: Review and Opimal hedging II.3. Empirical Analysis II.3.1 Opimal Hedging Analysis for BEKK(1,1,1) model The resuls of he Bivariae GARCH(1,1) model are esimaed and presened in Table. All coefficiens are saisically significan and he average hedge raios are esimaed a , and for he FTSE, DJ and DAX, respecively. FTSE DJ DAX c SS (0.073) (0.0180) (0.079) β (0.0143) a (0.016) c SF (0.0107) c FF (0.050) β (0.0119) a (0.0168) Table II. GARCH-BEKK(1,1) esimaes for Spo and Fuures Index reurns. Sandard errors in brackes (0.003) (0.0095) (0.0034) (0.0151) (0.006) (0.0093) (0.0068) (0.0135) (0.05) (0.031) (0.0079) (0.0131) LL AIC BIC

66 Chaper II. Mulivariae GARCH: Review and Opimal hedging Unhedged Naive OLS EWMA ECM GARCH BEKK(1,1) In-sample FTSE DJ DAX Ou-of-sample FTSE DJ DAX Table II. 3 Comparison of Hedging Effeciveness Table 3 displays he in-sample and ou-of-sample hedging performance of each approach. We also include resuls from an unhedged posiion along wih hose from a naïve hedging approach which assumes a uniary hedge raio over he whole sample. The in sample hedge raios are 0.854, and , and he ou of sample hedge raios are , and for he FTSE, DJ and DAX, respecively. In comparison o he unhedged porfolio and he naïve mehod, he reducion in he porfolio variance, for he in sample, flucuaes wih a facor ranging beween 3.53 and For he ou of sample hedging performance, he above facor flucuaes beween 1.97 and 8.00 compared o he unhedged and he naïve porfolio mehod. In boh in-sample and ou-of-sample daa, he BEKK(1,1,1) model consisenly underperforms slighly GARCH(1,1) bu ouperforms all he oher compeiors for all markes sudied. Is closes compeior for he in-sample and ou of sample daa is he GARCH(1,1). The EWMA performed wors in-sample among he five modeling approaches for all hree markes. 66

67 Chaper II. Mulivariae GARCH: Review and Opimal hedging II.3. VaR Evaluaion for he mulivariae GARCH models For he empirical implemenaion of he ad hoc Mulivariae GARCH(1,1) model, (Wang, Yao, 005), he condiional variances are esimaed using he simple version of GARCH(1,1) model applying equaions 36 and 37 for each daa series separaely. We hen consruc he new series for he equiy indices FTSE and DJ so as o esimae he condiional covariance by equaion 38. The condiional variance of he new series is esimaed applying anoher univariae version of GARCH(1,1). The above procedure is implemened, also, for he foreign exchange series currencies EUR and GBP. The new series, from he currencies EUR and GBP agains USD, is consruced and used for he condiional covariance calculaion. Also, he BEKK(1,1,1) model is esimaed for all daa. Tables 4 and 5 presen he parameers for he ad hoc mulivariae version of GARCH and BEKK models respecively. For boh GARCH(1,1) and Bivariae GARCH(1,1) models, all coefficiens are posiive and saisically significan. Especially, for he simple univariae GARCH(1,1), he near-uniy sum of he coefficiens suggess very high persisence in he condiional variances. Ad Hoc GARCH(1,1) FTSE DJ (F+D) EUR GBP (E+G) c 11,, c, c (0.0038) ε, ε, ε , -1, -1 h1, -1, -1 1, -1 (0.013) h, h , -1 (0.0138) (0.001) (0.0087) (0.0103) (0.001) (0.0096) (0.0110) (0.0005) (0.0043) (0.0045) (0.0187) (0.004) (0.0440) (0.0008) (0.0058) (0.0073) LL Nyblom Join Tes

68 Chaper II. Mulivariae GARCH: Review and Opimal hedging Table II. 4 Ad Hoc Mulivariae GARCH(1,1) esimaes for DJ, FTSE, (F+D), EUR, GBP, (E+G). Sandard errors appear in brackes. For he Nyblom Join es, he 1% criical value is equal o.8. BEKK(1,1) (FTSE, DJ) (EUR,GBP) c (0.0194) (0.010) β (0.0030) (0.0017) a (0.0117) (0.0103) c (0.0188) (0.0105) c (0.0151) (0.010) β (0.006) (0.009) a (0.0177) (0.0117) LL Table II. 5 BEKK(1,1,1) esimaes for (FTSE,DJ), (EUR, GBP). Sandard errors appear in brackes. For he empirical analysis, we compue ou of sample one day period VaR forecass using he variance covariance approach 5 (VCV) for boh equiies and foreign exchange porfolios, since he VCV approach considers and reveals direcly he volailiy and correlaion effec in he Value a risk (VaR) esimaion. The volailiy is updaed as in Hull and Whie (1987) procedure in order o capure he volailiy clusering. The more accurae and efficien variance covariance esimaion (VCV) is he one which gives he lower level of capial o cover agains unexpeced porfolio s losses and also he smaller average deviaion beween he esimaed VaR 5 We resric our aenion o he variance covariance approach bu also and oher mehodologies such as hisorical simulaion or parameric Riskmerics approach could be applied as well. 68

69 Chaper II. Mulivariae GARCH: Review and Opimal hedging and he acual reurn. Brooks and Persand (003) showed ha he forecased porfolio s VaR based on he VCV approach is calculaed as: a VaR T a = F h 100 i i 1 i, 1(, ) p + + 1, T p, + 1, T (40) where i=, BEKK(1,1,1), ad-hoc GARCH(1,1), T is he forecas horizon, here equal o 1 day period, α is he desired confidence level, i F + 1, T 1 is he inverse of he cumulaive disribuion funcion and is he porfolio s forecased condiional variance which is given by he following ype: h = a h + b h + abσ (41) p, + 1 1, + 1, + 1 1, + 1 where h 1, + 1, h, + 1, are he forecased condiional variances esimaed from he wo models for he indices and foreign currencies respecively andσ 1, + 1 is he forecased condiional covariance of he wo indices or he currencies esimaed from he respecive model, wih a and b being he proporion invesed in each asse. In his sudy he weighs a and b are each equal o 0.5, while he cumulaive disribuion funcion is he normal disribuion and he significance confidence level is chosen as 5% and 1% which corresponds o a value of equal o 1.65 and.33 for he normal disribuion respecively. In order o compare he VaR forecass accuracy esimaed by he wo models he following measures for VaR evaluaion are performed: Uncondiional Coverage Kupiec (1995) proposed an uncondiional es (LRun) so as o es he proporion of imes VaR is exceeded in a given sample and under he correc VaR model wih he null hypohesis ha expeced violaion frequency is equal o he desired significance level. The LRun follows an asympoic chi-square disribuion 69

70 Chaper II. Mulivariae GARCH: Review and Opimal hedging wih one degree of freedom χ (1) and compues he appropriae likelihood raio saisic as: un T N N T N N LR = ln (1 p) p + ln (1 N / T) ( N / T) (4) where T is he sample size, N is he number of failures or violaions, and p is he desired significance. Condiional Coverage Chrisofferson (1998) developed a es saisic (LR ind ) o accoun for uncondiional coverage and also for serial independence of VaR esimaes. This is very useful since we can conclude if a model rejecion is due he uncondiional coverage failure or clusering of he excepions or boh. For esing he independence of he VaR violaions, he saisic is asympoically χ disribued wih one degree of freedom and is derived as: [ π π π π π π π π ] ind LR = v ln( /(1 )) + v ln((1 )/ ) + v ln( /(1 )) + v ln(1 )/ ) (43) where v ij is he number of observaions of I wih value i followed by j, π 00 = v00 /( v00 + v01), 10 v10 v10 v11 π = /( + ), π = ( v01 + v11)/( ν00+ ν01+ ν10+ ν11). 1, if exceedence occurs Τhe indicaor I is consruced as: I = 0, if no exceedence occurs. The join es for condiional coverage capuring boh uncondiional coverage and he independence is simply given by he sum of he above individual ess and follows a χ disribuion wih wo degrees of freedom: cc un ind LR = LR + LR (44) 70

71 Chaper II. Mulivariae GARCH: Review and Opimal hedging Roo Mean Square Error (RMSE) In VaR models evaluaion, he roo mean square error is a frequenly used measure of he difference beween he VaR esimaed values and he acually observed porfolios reurns. The model wih he smaller RMSE is considered as he mos accurae VaR forecasing model. I is defined as: 1 RMSE = r VaR T T i ( + 1 p, + 1) (45) = 0 Sandard Deviaion of Capial Employed The Economic Capial or Capial Employed is considered as he amoun o be se aside in order o cover mos of he poenial losses a a predeermined level. Is sandard deviaion is calculaed as: 1 SD( VaR) = ( VaR VaR ) T T i i p, + 1 p (46) = 0 where VaR i p i VaRp is he average esimaed porfolio VaR given by he following ype: 1 T i = VaRp, + 1. The lower he sandard deviaion of he capial employed, he T = 0 mos accurae is he model used for he VaR calculaion since he uncerainy of he compulsory capial used o cover he unexpeced porfolio s losses is reduced. A rolling window of 100 observaions is used for he ou of sample esimaion of each model. The model which has he greaer percenage of VaR exceedences in he ou of sample period he highes RMSE and he highes sandard deviaion of he capial employed is ranked as he wors model, while he mos accurae is he one wih he lower percenage of exceedences and also wih he lowes RMSE and sandard deviaion of he capial employed. Tables 6 and 7 and ables 8 and 9 repor 71

72 Chaper II. Mulivariae GARCH: Review and Opimal hedging he esimaed five measures for he ou of sample VaR evaluaion of boh equiy and currency porfolios a he 95% and 99% levels respecively. FTSE-DJ Porfolio 95% conf. level un LR ind LR cc LR RMSE SD(VaR) Ad-Hoc GARCH(1,1) BEKK(1,1) Table II. 6 Ou of sample VaR evaluaion measures for he equiy indices porfolio a 95% confidence level. The un LR saisic ess he null hypohesis ha he proporion of VaR exceedences is equal o he nominal significance level and has a chi squared disribuion wih criical value 3.84 a he 5% significance level. The ind LR ess he null hypohesis ha he VaR exceedences are serially uncorrelaed and has a chi squared disribuion wih criical value 3.84 a he 5% significance level. The cc LR ess he null hypohesis of boh uncondiional coverage and ha he VaR exceedences are serially uncorrelaed and has a chi squared disribuion wih criical value 5.99 a he 5%. FTSE-DJ Porfolio 99% conf. level un LR ind LR cc LR RMSE SD(VaR) Ad-Hoc GARCH(1,1) BEKK(1,1) Table II. 7 Ou of sample VaR evaluaion measures for he equiy indices porfolio a 99% confidence level. The un LR saisic ess he null hypohesis ha he proporion of VaR exceedences is equal o he nominal significance level and has a chi squared disribuion wih criical value 6.63 a he 1% significance level. The ind LR ess he null hypohesis ha he VaR exceedences are serially uncorrelaed and has a chi squared disribuion wih criical value 6.63 a he 1% significance level. The cc LR ess he null hypohesis of boh uncondiional coverage and ha he VaR exceedences are serially uncorrelaed and has a chi squared disribuion wih criical value 9.1 a he 1%. 7

73 Chaper II. Mulivariae GARCH: Review and Opimal hedging un ind cc EUR-GBP Porfolio LR LR LR RMSE SD(VaR) 95% conf. level Ad-Hoc GARCH(1,1) E BEKK(1,1) Table II. 8 Ou of sample VaR evaluaion measures for he foreign exchange currencies porfolio a 95% confidence level. The un LR saisic ess he null hypohesis ha he proporion of VaR exceedences is equal o he nominal significance level and has a chi squared disribuion wih criical value 3.84 a he 5% significance level. The ind LR ess he null hypohesis ha he VaR exceedences are serially uncorrelaed and has a chi squared disribuion wih criical value 3.84 a he 5% significance level. The cc LR ess he null hypohesis of boh uncondiional coverage and ha he VaR exceedences are serially uncorrelaed and has a chi squared disribuion wih criical value 5.99 a he 5%. EUR-GBP Porfolio 99% conf. level un LR ind LR cc LR RMSE SD(VaR) Ad-Hoc GARCH(1,1) E BEKK(1,1) E Table II. 9 Ou of sample VaR evaluaion measures for he foreign exchange currencies porfolio a 99% confidence level. The un LR saisic ess he null hypohesis ha he proporion of VaR exceedences is equal o he nominal significance level and has a chi squared disribuion wih criical value 6.63 a he 1% significance level. The ind LR ess he null hypohesis ha he VaR exceedences are serially uncorrelaed and has a chi squared disribuion wih criical value 6.63 a he 1% significance level. The cc LR ess he null hypohesis of boh uncondiional coverage and ha he VaR exceedences are serially uncorrelaed and has a chi squared disribuion wih criical value 9.1 a he 1%. 73

74 Chaper II. Mulivariae GARCH: Review and Opimal hedging Boh models provide correc uncondiional and condiional coverage close o he significance levels since heir un LR, ind LR, and cc LR values do no violae boh he 5% and 1% olerance levels, and herefore heir VaR forecass are adequae. Summarizing he resuls for he uncondiional and condiional coverage, he VaR predicions obained from boh models are all wihin he percenage exceedences hreshold. Since, boh models have a good uncondiional and condiional coverage performance, he examinaion of he roo mean square deviaion and he sandard deviaion of he capial employed is required. For boh equiy and foreign currency porfolios, he BEKK(1,1,1) model provides he lower RMSE a 5% and 1% levels compared o he ad hoc mulivariae GARCH(1,1) model. Considering he capial employed, he bivariae BEKK(1,1,1) model produces again he lowes sandard deviaion of he capial employed allowing he uncerainy, over he required capial reserved o cover agains unexpeced adverse price movemens, o be highly reduced. II.4. Summary and Discussion A number of recen sudies sugges he use of mulivariae models for he accurae esimaion of he co-movemen of financial asses. In response o his alarming empirical evidence, his chaper examined he esimaion and forecasing abiliy of he mulivariae BEKK and ad-hoc GARCH models. I found, using he equiy indices FTSE, DJ and he foreign currencies EUR and GBP agains USD ha he BEKK(1,1,1) provides more accurae VaR esimaion boh in and ou of sample, compared o he ad-hoc GARCH(1,1) model. The main disadvanages are he compuaional ime consuming in order o esimae is parameers and he complexiy of is mulivariae version. 74

75 Chaper II. Mulivariae GARCH: Review and Opimal hedging In he nex chaper, I exend he lieraure on porfolio selecion wih higher momens by invesigaing how non-normaliy of reurns and higher momens may affec hedging sraegies. 75

76 Chaper III. Higher Momens Modeling: Univariae Approaches CHAPTER III Higher Momens Modeling: Univariae Approaches 6 An issue ha has no received ye much aenion in he hedging lieraure concerns he imporance of condiional variaions in momens oher han he mean and variance. Higher momens, such as skewness and kurosis, can conrol he asymmeries and hick ails ha are ypically observed in he disribuions of financial reurns. Their role has been widely sudied wihin he conex of risk managemen and porfolio heory. A number of auhors, including Lai (1991), Parkas e al. (003), Sun and Yam (003), and, Jondeau and Rocking (006), have provided evidence suggesing ha he incorporaion of higher momens in porfolio allocaion leads o superior approximaions of expeced uiliy and allows a very efficien way o compue opimal porfolios. Even if he firs wo momens are sufficien for asse pricing, higher momens are poenially imporan for hedging since variaions in he shape of he disribuion may affec he esimaion of he condiional mean and variance. Alhough models from he GARCH family are able under cerain assumpions and parameerisaions o produce hick-ailed and skewed disribuions, hey ypically assume ha he shape parameers are ime invarian. 6 Par of his chaper has been published o he proceedings of he Sixeenh Hellenic-Saisical Conference under he ile Modeling he sochasic behaviour of exchange raes based on ARCD. 76

77 Chaper III. Higher Momens Modeling: Univariae Approaches In he economeric lieraure, here are many sudies which have aemped o sudy uncondiional skewness. Since (1965), when Fama poined o he fac ha sock reurns were no normally disribued, a lo of research presened evidence which enhances he imporance of skewness in financial heory and is applicaions in asse managemen. Ardii (1967), firsly, found ha second and hird momens were significan measures and mos of he informaion abou any probabiliy disribuion is conained in is firs hree momens. Exending he above resuls, Jean (1971, 1973) presened a formulaion which akes ino accoun he hird momen in he Capial Asse pricing Model (CAPM). Cooley (1977) found ha invesors prefer asses which follow posiively skewed disribuions since hey are less risky even if hey have lower reurns han asses wih higher risk quanile. Beedles and Simkowiz (1978) and Sears (1983) noiced a negaive relaionship beween opimum diversificaion and skewness. Also, Beedles and Simkowiz (1980) found evidence for persisence in skewness and kurosis using cross-secional daa. Chaerjee (1988) concluded ha financial asses reurns disribuion is very complicaed and he idea of modeling and forecasing he enire densiy is required. Alles and Kling (1994) showed ha he skewness of he socks and bonds vary boh in magniude and sign, and i is more negaive during economic upurns and less negaive during downurns. More recenly, Chen e al (000) poined ou ha small firm s socks end o possess posiive skewness in conras o he observed negaive skewness for large firm s socks. This exisence of skewness was explained by he leverage effec and heerogeneiy among invesors. Since here is well esablished sylized evidence ha financial reurns exhibi fa ails and skewness, various models have been developed in he lieraure in order o capure dependencies in higher momens (eg., see Hansen 1994; Harvey and Siddique, 1999, 00a, 00b; Jondeau and Rockinger, 006; Brooks, Burke and Persand, 005). Focusing on he lieraure on condiional densiy, he only noable research for he higher momens modeling are Gallan, Hsieh, Tauchen (1991), Hansen (1994) and Harvey and Siddique (1999). Gallan e al proposed a semi-non 77

78 Chaper III. Higher Momens Modeling: Univariae Approaches parameric mehod based on a series expansion of he Gaussian densiy in order o accommodae he complex condiional dependency of highly condiionally heerogeneous financial ime series. Their idea was o conver he densiy o a polynomial of he pas hisory reurns. Hansen (1994) argued ha Gallan s approach was no parsimonious while for he semi non parameric parameers esimaion, a large daa se is required in order o have accurae and raional resuls. Alernaively, Hansen (1994) proposed a generalized ARCH formulaion, he Auoregressive Condiional Densiy model (ARCD) by allowing he shape parameers beyond he variance and he mean o depend upon condiional informaion and using an exension of he condiional funcion, named as condiional skewed disribuion. Harvey and Siddique (1999) presened a GARCH-ype model allowing for imevarying volailiy, skewness and kurosis using a raher complicaed funcion, he condiional non-cenral disribuion and indicae a significan presence of condiional skewness and kurosis. In addiion, he condiional ime varying skewness is imporan and consisen wih asymmeric variance, since heir specificaions allowing for ime-varying skewness and kurosis ouperform specificaions wih consan hird and fourh momens. Recenly, a number of auhors, including Lai (1991), Prakash e al. (003), Sun and Yan (003), and, Jondeau and Rockinger (006), have provided evidence suggesing ha he incorporaion of higher momens in porfolio allocaion leads o superior approximaions of expeced uiliy and allows a very efficien way o compue opimal porfolios. More analyically, Prakash e al applied polynomial goal programming, in which invesor preferences for skewness can be incorporaed, in order o deermine he opimal porfolio from Lain American, US and European capial markes. The empirical findings sugges ha he incorporaion of skewness ino an invesor s porfolio decision causes a major change in he resulan opimal porfolio. Even if he firs wo momens are sufficien for asse pricing, higher momens are poenially imporan for hedging since variaions in he shape of he disribuion may affec he esimaion of he condiional mean and variance. Similar 78

79 Chaper III. Higher Momens Modeling: Univariae Approaches resuls have been found in Sun and Yan (003), using socks from boh Japanese and US markes and a boosrap mehod, creaing porfolios opimally formed and applying he polynomial goal programming mehod, which considers preference for skewness, grealy enhances skewness persisence over ime. Jondeau and Rockinger (006), builds a framework on a GARCH model wih a condiional generalized disribuion for residuals showing ha for exchange rae and sock marke daa cross secionally and a daily frequency, here is srong evidence for co-variabiliy of momens beyond volailiy. In risk managemen, he research ineres has been concenraed in he esimaion of he financial reurns disribuion ails. Taking ino accoun ha ARCH/GARCH models or he sochasic simulaion mehods forecasing abiliy is relaively poor in he condiional volailiy predicion, i is mandaory o implemen new mehods in order o capure he higher momens effec. More recenly, a few researchers implemened he quanile regression mehods for densiy forecasing. Among ohers, Granger and Sin (000) sudy applies auoregressive quanile regressions and finds evidence ha perform beer han he asymmeric power ARCH models, and Taylor (000) approach which uses neural nework o esimae poenially non-linear models and concludes in favor of he quanile regressions compared o Gaussian and empirical GARCH(1,1) formulaions. Finally, he parameric approaches used exensively owards densiy forecasing. Weigend and Shi (000) implemened he Markov processes framework for predicing condiional probabiliy disribuions of ime series fuure values and hey found ha heir mehod performs beer compared o GARCH model. Addiionally, Schienkopfs e al (000) implemened a neural framework ha uses a mixure of Gaussian Disribuions in order o esimae he condiional disribuion semi-nonparamerically, allowing for non-lineariy in condiional mean and variance and esimae a condiional densiy wih ime varying higher momens. Their sudy found only a marginal improvemen over GARCH and GARCH- formulaions. 79

80 Chaper III. Higher Momens Modeling: Univariae Approaches Since various approaches have been proposed in he lieraure in order o capure dependencies in higher momens (eg. see Hansen 1994; Harvey and Siddique, 1999, 00a, 00b; Jondeau and Rockinger, 006; Brooks, Burke and Persand, 005), ye heir usefulness for opimal hedge raion esimaion has no been ye invesigaed. The presen paper exends he lieraure on porfolio selecion wih higher momens by invesigaing how non-normaliy of reurns and higher momens may affec hedging sraegies. In order o accoun for ime varying skewness and kurosis in opimal hedge raio esimaion, he ARCD model proposed by Hansen (1994) is employed, in which full condiional densiy is modelled allowing for condiional shape parameers. The main reason for he above model selecion is is simpliciy in compuaion and evaluaion performance. In a horserace of models, he dynamic hedging effeciveness of he ARCD is compared o ha obained by OLS, error-correcion, exponenial moving averages, and, univariae and mulivariae GARCH, approaches which have been sudied exensively in he preceding chapers. Effeciveness is measured in-sample and ou-of-sample using he minimum variance mehod. Spo and fuures daily closing prices are used for sock indices from he US, UK and Germany for he period January 1999 o Sepember 004. The resuls sugges ha he hedging performance using he ARCD ouperforms ha obained by he compeing approaches. The remainder of he chaper is organized as follows: he nex secion illusraes he mehodology for he ARCD approach. The following secion describes he daa and he empirical resuls on he esimaion of he hedge raios. The hird secion compares he hedging effeciveness of he alernaive models described previously while he final secion concludes he paper. 80

81 Chaper III. Higher Momens Modeling: Univariae Approaches III.1. Mehodology: A model for condiional skewness and kurosis III.1.1. The generalized Suden disribuion Auoregressive Condiional densiy Model, proposed by Hansen (1994), is based on he selecion of a disribuion wih parsimonious parameerizaion. The parameric vecor of his disribuion is srucured o be ime varying and o depend on he condiioning informaion. This disribuion funcion has a densiy of he following general form: d f( y α( x, θ )) = P( y y x) (47) dy where (y, x : = 1,., n) is he observed sample, θ is a parameer vecor having finie dimensions and α can be defined as: α = a( x, θ) is a low dimensional and ime varying parameer inerpreing he influence of x on he condiional disribuion. The primary objec is o parameerize he funcion f( y a ) in order o have as a resul: α = ( µ, σ, η) where µ and σ are he condiional mean and variance respecively, and ηcomprises he remaining parameers influence in he condiional disribuion. The above menioned parameers are defined as follows: µ = µ ( θ, x ) = E( y x ) (48) σ = σ ( θ, x ) = E(( y µ ) x ) (49) η = η( θ, x ) (50) 81

82 Chaper III. Higher Momens Modeling: Univariae Approaches The above formulaions of he condiional mean and variance allow normalizing he variable y and defining he normalized variable z as follows: y µ ( θ, x) z( θ) = (51) σ( θ, x ) Conrariwise y, he condiional densiy of he normalized variable z depends only on he parameer η, and is defined as: d g( zη) = P( z p zη) (5) dz The relaionship ha links he condiional densiies of he variables y and z is given by he following ype: f( y µ, σ, η ) = ( η ) (53) 1 g z σ Mos ARCH-ype formulaions use probabiliy models wih ime invarian η. Hansen (1994) achieved o build a model ha allows for condiional higher momens. Using he sandardized residuals he inroduces a disribuion where asymmeries can occur, and models he remaining dynamics of y using he appropriae disribuion while mainaining he assumpion of a zero mean and uni variance. By assuming ha parameers are dependen on pas realizaions, he shows ha parameers may be ime varying and also he shape momens may be ime varying. The goal is o selec a densiy funcion g( zη ) which generalises he sandard normal, is sufficienly flexible o generae he range of shapes such as heavy ails, skewness or bi-modaliy, is sufficienly parsimonious ha η can be adequaely modelled using ime series echniques, and also is available in closed 8

83 Chaper III. Higher Momens Modeling: Univariae Approaches form. Building on he sandard seup of Hansen (1994), a generalizaion of he Suden- disribuion wih zero mean and uni variance can be used which allows for asymmery and fa-ailedness o vary over ime. Alhough alernaive skewed Suden- disribuions have been considered in he lieraure (eg., Jondeau and Rockinger, 003, 006; Harvey and Siddique, 1999, 00a, 00b), we seleced his specificaion since i offers compuaional advanages (see Harvey and Siddique, 1999; Brooks and Persand, 00) and he variaion in he shape parameers may be smaller and easier o manage numerically. The disribuion has he following densiy funcion: -( η+ 1) 1 bz+ k g( zη, λ) = bd (1 + ( ( ) ) for η- 1-λ z - < k b (54a) and, -( η+ 1) 1 bz+ k g( zη, λ) = bd (1 + ( ( ) ) for η- 1+ λ z - k b (54b) where z are he normalized reurns of he spo and fuures reurns, < η <, -1< λ < 1, η are he degrees of freedom, λ is he skewness, while k,b,d are consan parameers defined by he following equaions: η - k = 4 λd ( ), η -1 b = 1+ 3λ - k, η + 1) Γ( ) d = η (55) π( η -) Γ( ) The above densiy funcion encompasses a large se of convenional densiies. For insance, when λ se is equal o zero, he above disribuion reduces o he radiional Suden- disribuion. If in addiion η =, he Suden- disribuion collapses o a normal densiy. I is well known ha he radiional 83

84 Chaper III. Higher Momens Modeling: Univariae Approaches Suden- disribuion wih η degrees of freedom allows for he exisence of all momens up o η h. The Figure 1, below, displays various densiies for differen values of skewness and degrees of freedom, noicing ha λ conrols skewness and he righ of lef ailness. Figure III. 1 Various Densiies for differen values of λ and η. Therefore, given η f, Hansen s skewed- disribuion is well defined. For he skewness and he degrees of freedom, he following general specificaion is proposed: η = η( e 1, e,..., e 1, ) (56) More precisely, he degrees of freedom η and skewness λ are specified respecively as following: 84

85 Chaper III. Higher Momens Modeling: Univariae Approaches η = c + cε + c ε (57) λ = p + pε + p ε (58) where c0, c1, c3, p1, p, p 3 are consans. Hansen proposes a logisic ransformaion in order o impose he above consrains on skewness and degrees of freedom parameers. Suppose ha η is real valued and is relaionship o λ be as following: ( U L) η = L+ 1 + exp( λ ) (59) Implemening he above equaion, η is consrained o lie in he region [L, U] wih lower bound L and upper bound U, while λ is allowed o vary over he enire line beween -1 and 1. In combinaion wih a law of moion for η such as he afore menioned equaion (58) we obain he relaionship η = η( e ) = c + cε + c ε, which is quie flexible even if i is consrained o he region [L, U]. Jondeau and Rockinger (003) derive he exac formulas for he calculaion of he skewness and kurosis respecively: 3 3 ( m3 3αm + a ) E Z = (60) 3 b and 4 4 ( m4 4αm3 + 6a m 3 a ) E Z = (61) 4 b where = + λ and m 1 3 m 3 = λ + λ 16 c (1 ) ( η ) ( η 1)( η 3) if η > 3 (6a) 85

86 Chaper III. Higher Momens Modeling: Univariae Approaches m 4 = + λ + λ 4 ( ) 3( η ) ( η 4) if η > 4 (6b) Since Z variable is normalized wih zero mean and uni variance 7, he skewness is equal o he hird momen (equaion 60) and he kurosis equals he 4 fourh momen (equaion 61). The excess kurosis is defined as XKu= E Z 3. Taking ino accoun ha he generalised Suden- disribuion is meaningful for η f and-1< λ < 1, Jondeau and Rockinger (003) underline ha skewness obained from he above equaion (60) is well defined when η f3and kurosis calculaed from he afore menioned equaion (61) exiss if η f 4. The condiional log-likelihood of he full ARCD model is calculaed as: T 1 LL = ln g( z; η)- lnh = max( p, q) + 1 (63) The maximum likelihood esimae (MLE) of he model is he value ˆ θ which maximizes he above condiional log likelihood using he Bernd-Hall-Hall- Hausman (BHHH) opimizaion algorihm. The ARCD shape parameers η can be esimaed via sandard ieraive mehods by assuming an arbirary reasonable iniial value η 1. I is advisable o compue robus sandard errors since hese generae asympoically valid confidence inervals for he pseudo rue parameer values which minimize he informaion disance beween he rue probabiliy and he quasi-likelihood measure. In his manner, we can achieve he maximum possible accuracy in our resuls. Finally, he Nyblom L-saisic, for esing he consancy of he esimaed parameers, akes he form: 7 In Appendix A, we show ha z variable has zero mean and uni variance. 86

87 Chaper III. Higher Momens Modeling: Univariae Approaches L i = 1 n Gi n V (64) = 1 ii where G i are he cumulaive scores and V ii is he i-h diagonal elemen of he esimaed variance. The L-saisic is used o es he saionariy of he parameers of he disribuion funcion and can be considered as an LM es of he null hypohesis ha he parameers are sable. Asympoic criical values for he Nyblom es along wih an exensive analysis can be found in Hansen (1990). The above model is applied o calculae he condiional variances for he spo reurns and he fuures reurns, respecively: h = c + a ( ε - h ) + β ( h ) (65a) SS, SS, 1 S, -1 S, -1 1 S, -1 h = c + a ( ε - h ) + β ( h ) (66b) FF, FF, 1 F, -1 F, -1 1 F, -1 While i is sraighforward o model he variances of he reurns, he covariance mus be deal wih in a somewha differen way since he esimaed covariance marix should be posiive definie for each period. In order o assure he posiive definieness, we consrain he spo reurns and he fuures reurns o have consan correlaion hroughou he sample, so changes in he covariance are only due he variances which are always posiive. Given ha he correlaion esimae beween spo and fuures prices is ypically in he range 90% - 99% and ha he underlying series are highly volaile, he consancy assumpion of correlaion is unlikely o inroduce noiceable errors (a similar argumen is made by Cecchei e al., 1988). The covariance of reurns is given by he following equaion: h = ρ h h (67) SF, SS, FF, Finally, he opimal hedge raio can be calculaed as: 87

88 Chaper III. Higher Momens Modeling: Univariae Approaches HR ARCD h SF, =. (68) hff, III.. Daa Analysis and Empirical Applicaion The daa used in he presen sudy includes daily spo prices and respecive fuure conrac prices for he UK Financial Times Sock Exchange Index (FTSE), he US Dow Jοnes Index (DJ) and he German Deuscher Akien-Index (DAX). The sample covers he period from 4 January 1999 o 0 Sepember 004 for all hree markes, he same sample which has been used in Chaper 1. Non-rading days were removed and rading daes beween spo and fuures prices were mached leaving a oal of 1443, 1435 and 1450 observaions for he FTSE, DJ and DAX, respecively. The las 60 observaions for each series were lef for ou-of-sample hedge raio performance comparison. There are four delivery monhs for all hree markes: March, June, Sepember and December. Due o he size of he markes considered, a leas wo conracs were raded a any ime, hus faciliaing conrac rollover. Descripive saisics of logarihmic reurns for he spo and fuures series under sudy are provided in Table I.1, page 45. The ARCD model esimaion resuls are presened in Table 6. We repor he condiional variance, degrees of freedom, skewness and Nyblom es values. ARCD FTSE DJ DAX Spo Fuures Spo Fuures Spo Fuures Condiional Variance c SS, FF, ε S, -1 c (0.0087) ε 0.13 F, 1 (0.00) hs, -1 F, -1 h (0.0097) (0.008) (0.001) (0.0098) (0.0080) (0.0140) (0.0075) (0.0084) (0.0168) (0.008) (0.0061) (0.000) (0.0081) (0.0169) (0.0169) (0.0088) 88

89 Chaper III. Higher Momens Modeling: Univariae Approaches Nyblom L σ es c c SS, FF, ε -h S,- 1 S,-1 hs, -1 F, -1 ε -h F,- 1 F,-1 h Condiional Degrees of Freedom c (1.6170) ε -1 (c 1 ) (1.796) (1.5687) (1.191) (0.4013) (0.5634) (0.3976) (0.565) (1.5539) (0.56) (0.011) (0.344) ε -1 (c ) (0.1677) (0.65) (0.016) (0.015) (0.048) (0.0531) Nyblom L σ es c ε -1 (c 1 ) ε -1 (c ) Condiional Skewness p (0.0939) (0.0977) (0.0837) (0.0416) (0.0999) (0.1066) ε -1 (p 1 ) (0.0039) (0.0070) (0.0654) (0.0781) (0.043) (0.0103) ε -1 (p ) (0.003) (0.003) (0.0347) (0.0371) (0.013) (0.003) Nyblom L σ es p ε -1 (p 1 ) ε -1 (p ) Log Likelihood LR es Nyblom Join Tes AIC Crieria BIC Crieria Table III. 1 ARCD esimaes for Spo and Fuures Index reurns. Numbers in brackes under he parameer esimaes give he sandard errors values. Nyblom L saisic has been inroduced by Nyblom (1989) and modified by Hansen (1990) for esing he consancy of he esimaed parameers. I akes he form: Li =1/n*(Σ(Gi /Ṽii) where Gi are he 89

90 Chaper III. Higher Momens Modeling: Univariae Approaches cumulaive scores, Ṽii is he ih diagonal elemen of he esimae variance Ṽ and can be considered as he LM es of he null hypohesis ha all parameers are sable. The asympoic criical values for he Nyblom es have been presened in Hansen (1990). For he Nyblom es, he 1% criical value is equal o 0.75, and for he Nyblom Join es, he 1% criical value is equal o.8. LR saisic is he likelihood raio es of he null hypohesis of normal disribuion agains he alernaive ha he daa reurns follow a skewed suden disribuion. The LR saisic is asympoically disribued as a Chisquared wih degrees of freedom and is criical value a 1% significance level is 9.1. In general, all coefficiens of he condiional variance, degrees of freedom and skewness are saisically highly significan and mus no be ignored. As shown in Figure 1, he kernel densiy approximaion of he sandardized residuals closely resembles a Suden- for all he hree indices. Similar resuls are obained for he fuures reurns. FTSE 90

91 Chaper III. Higher Momens Modeling: Univariae Approaches DJ DAX Figure III. Sandardized ARCD residual disribuions for Spo index reurns For he condiional variances, he condiional degrees of freedom and he condiional skewness, he Nyblom L es indicaes ha he parameers are all sable since he es saisic is less ha he 1% level criical value of 0.75, implying ha he ARCD model is well specified and fis he daa capuring he higher momens ime variaion. A log likelihood raio (LR) saisic is applied o es he null hypohesis ha he series follow he normal disribuion agains he alernaive of ime varying higher momens. Since he normal disribuion (of GARCH model) is nesed o he skewed Suden- disribuion of ARCD model, he LR saisic is calculaed by he following formula: LR= -[ LGARCH ( LARCD where LGARCH and LARCD 91