Management Science Letters

Transcription

1 Managemen Science Leers () 37 3 Conens liss available a GrowingScience Managemen Science Leers homepage: An EGARCH-BPNN sysem for esimaing and predicing sock marke volailiy in Morocco and Saudi Arabia: The effec of rading volume Salim Lahmiri * Deparmen of Compuer Science Universiy of Quebec a Monreal Monreal Canada A R T I C L E I N F O A B S T R A C T Aricle hisory: Received Ocober Received in Revised form November Acceped 3 January Available online February Keywords: EGARCH Volailiy Forecasing Arificial Neural Neworks In his sudy he backpropagaion neural nework (BPNN) is esed for he abiliy o forecas he daily volailiy of wo sock marke indices from he Middle Eas and Norh Africa (MENA) region using volume; namely Morocco and Saudi Arabia. Volailiy series were esimaed using he Exponenial Auo-Regressive Condiional Heeroskedasiciy (EGARCH) model. The simulaion resuls show ha rading volume helps improving he forecasing accuracy of BPNN in Morocco bu no in Saudi Arabia. As a resul volume represens valuable informaion flow o be used in he modeling and predicion of volailiy in Morocco. In addiion i is found ha BPNN overpredics volailiy during high volaile periods. This finding is imporan in financial applicaions such as asse allocaion and derivaives pricing. Growing Science Ld. All righs reserved.. Inroducion Volailiy plays a cenral role in derivaives pricing opimal porfolio selecion value-a-risk calculaions used in porfolio risk managemen and decision making in real-ime rading sysems. As a resul an exensive lieraure on volailiy forecasing is moivaed by hese applicaions. Indeed a number of saisical echniques have been developed o model and predic sock reurns such as GARCH family and sochasic volailiy models. Besides a srong link beween conemporaneous rading volume and condiional volailiy was found in he lieraure (Biswas & Gerard 7). Then one can use volume o forecas fuure volailiy. This relaionship was documened since early 97s. For insance Clark (973) inroduced he Mixure of Disribuion Hypohesis (MDH) o documen he conemporaneous relaionship beween sock reurns and volume. Clark (973) and Epps and Epps (976) suggesed ha he informaion arrival in he equiy marke can be represened by volume. In oher words volume is a proxy for he informaion flow. Since hen he MDH has received a large aenion in he academic lieraure. For example Lamoureux and Lasrapes (99) found evidence ha rading volume explains he variance of US sock reurns. Furhermore hey documened a subsanial reducion in volailiy persisence when volume is included in he variance * Corresponding auhor. Tel: addresses: lahmiri.salim@courrier.uqam.ca (S. Lahmiri) Growing Science Ld. All righs reserved. doi:.567/j.msl...7

2 38 equaion. Similarly Omran and McKenzie () repored ha rading volume explains he variance of UK sock reurns and he esimaes of he GARCH process are insignifican when volume is included in he condiional variance equaion. In a subsequen sudy Huang and Yang () concluded ha he volailiy persisence of Taiwanese sock marke declines significanly when volume is inroduced as explanaory variable. Thus hey rejec he MDH in he Taiwanese equiy marke. Lee and Rui () found a posiive relaionship beween volume and volailiy in he US UK and Japan sock marke. In anoher sudy Bohl and Henk (3) concluded ha he inclusion of rading volume in he condiional variance reduces volailiy persisence in mos of he Polish socks. Therefore hey concluded ha he MDH implicaions are no confirmed. Brooks (998) concluded ha augmening models of volailiy wih measures of lagged volume leads only o very modes improvemens in forecasing performance. Hu and Tsoukalas (999) examined he individual volailiy forecass produced by a number of GARCH models in he European Moneary Sysem (EMS). They concluded ha he neural nework combining model performed beer in he crisis period and i was proved superior o linear combining models. Tseng e al. (9) examined he variance forecasing abiliy of he radiional GARCH model and Grey-GARCH model using inernaional sock indices. They found ha he inegraed Grey-GARCH model provides a beer measure of he value of he forecass han he simple GARCH model. They concluded ha he inegraed model help enhance he one-period-ahead volailiy forecass. Oher sudies used arificial inelligence echniques o predic fuure sock marke volailiy because; unlike saisical models; hey are nonlinear models ha are capable o model noisy daa. For insance Hamid and Iqbal () used arificial neural neworks (ANN) for forecasing he volailiy of he S&P 5 Index fuures prices. They found ha forecass from neural neworks ouperformed implied volailiy forecass and ha hose forecass were no significanly differen from realized volailiy. Roh (7) found ha NN-EGARCH showed a good performance when compared wih he NN and NN-GARCH model o forecas he KOPSI volailiy. Bildirici and Ersin (9) combined ARCH/GARCH family models wih arificial neural neworks o predic daily volailiy of Isanbul Sock Exchange. Wang e al. () used backpropagaion neural nework o predic TXO (Taiwan exchange opions and fuures) price under differen volailiy models including hisorical volailiy implied volailiy deerminisic volailiy funcion GARCH and Grey model combined wih GARCH. They concluded ha in general he predicion accuracy depends on volailiy models and number of neurons in he hidden layer bu are no significanly relaed o acivaion funcions. Hung () used a fuzzy sysem o analyze clusering in generalized auoregressive condiional heeroskedasiciy (GARCH) models and geneic algorihms o esimae he parameers of he membership funcions and he GARCH models. Using daa from developed marke (Germany Canada Japan and USA) he simulaions showed ha he proposed mehod improved he forecasing accuracy in comparison wih convenional GARCH and EGARCH model. More recenly Hajizadeh e al. () proposed a hybrid sysem ha combines exponenial GARCH and arificial neural neworks o forecas he volailiy of S&P 5 index -days and 5-days ahead. The simulaions indicaed ha he proposed sysem ouperforms he convenional exponenial GARCH model alone. The main goal of his sudy is o forecas sock marke fuure volailiy using volume in wo emergen markes from he Middle Eas and Norh Africa (MENA) region while mos of previous sudies focused on developed and Asian markes (Omran & McKenzie ; Huang & Yang ; Lee & Rui ; Bohl & Henk 3; Tseng e al. 9). Two imporan sock markes in MENA region are considered: Morocco and Saudi Arabia. They are he mos acive marke in he Middle Eas and Norh Africa respecively. The EGARCH model is used o esimae and exrac sock marke volailiy. Then backpropagaion neural nework (BPNN) is employed o model he esimaed volailiy and perform forecass since hey are robus o noisy daa and capable o model nonlinear relaionships (Rumelhar e al. 986; Asalakis & Valavanis 9). Indeed volailiy and volume series are nonlinear and i is appropriae o approximae heir relaionship using nonlinear inelligen echniques such as arificial neural neworks. In addiion BPNN has proven is capabiliy o

3 S.Lahmiri / Managemen Science Leers () 39 ouperform radiional GARCH family models in he predicion of volailiy (Hamid & Iqbal Roh 7 Bildirici & Ersin 9; Hung Wang e al. Hajizadeh ). The reminder of he paper is organized as follows: Secion oulines he daa and volailiy esimaion. Secion 3 describes he forecasing echnique; namely he arificial neural neworks. Secion repors he resuls. Finally secion 5 concludes.. Daa and volailiy esimaion The sample consiss of daily index closing prices and rading volume. The sample covers observaions for he period January o June 6 for Morocco and Saudi Arabia. The series are aken from heir respecive sock marke official websies. The daily reurn daa are he firs difference of he log of sock prices. i.e.. R = *(log P log P - ) where P is he index price a ime. All reurns are in local currency. Similarly he change in volume a ime is defined by he variable V = *(log v log v - ); where v is he rading volume a ime. To esimae volailiy series a wo-sep process is followed. Firs he class of auoregressive moving average (ARMA) models (Davidson & MacKinnon 8) is used for condiioning on he pas of sock marke reurn series o model he reurn processes (see equaion ). This class allows capuring all linear dynamics in reurn series. Second he Exponenial GARCH (EGARCH) model (Nelson 99) is used o esimae sock reurns volailiy (see equaion 3). I is an exension of he GARCH model of Bollerslev (986) o accoun for asymmeric volailiy. The EGARCH model is esimaed using he maximum likelihood mehod (Davidson & MacKinnon 8). Hence he following model is esimaed for each marke index: p q () R = C+ ar + bε + ε i i j j i= j= where ε GED (h ) () and ε m n i log( h) = w+ Φ j / π + γ + β j log h j + η i= h i h i j= where w Φ j γ β j and ω are consan parameers. ε (3) The EGARCH model imposes no resricions on hese parameers. The variance componen h is defined as an asymmeric funcion of lagged disurbances ε -i. Hence unexpeced reurn variance ε is modeled as an EGARCH(mn). The selecion of he ARMA(pq) and EGARCH(mn) processes are based on he minimizaion of Akaike and Schwarz informaion crierions (Davidson & MacKinnon 8). Then he condiional variance series say volailiy series given by log(h ) - are exraced. Figure shows volailiy and changes in volume series for each marke. The laer are compued as log volume firs difference series as menioned before. Finally he esimaed condiional variance series from he EGARCH model and volume are normalized in he inerval [-] o obain accurae forecass. Then normalized daa will be fed o he backpropagaion neural neworks (BPNN). The normalizaion is performed as follows: x = ( x ( Max[ x] + Min[ x] )) Max[ x] Min[ x] () Generalized Error Disribuion (GED) was inroduced by Subboin (93). Nelson (99) used he GED o model he disribuion of sock marke reurns.

4 3 where x and x are respecively he original and he ransformed daa..35 Morocco 8 Morocco Saudi Arabia Saudi Arabia Fig.. Volailiy (lef column) and changes in volume series (righ column). 3. Arificial neural neworks The main objecive of his work is o predic he daily sock marke volailiy using backpropagaion neural nework (BPNN) (Rumelhar e al. 986). The inpu daa are he acual marke volailiy log(h ) and he acual change in marke volume V and he oupu is he fuure marke volailiy log(h + ). The opology of he nework is as follows: wo neurons in he firs layer neurons in he hidden layer and one neuron in he hird layer. The sigmoid ransfer funcion for example f(x) = /(+e -x ) is used in he hidden layer and a linear ransfer funcion in he oupu layer. The final oupu has he form: Ok s n f f xiwij w jk j= i= = where n is number of inpu unis s is number of neurons in hidden layer and m number of oupu neurons. And w ij denoes he connecion srengh beween unis i and j and k denoes he number of oupu unis. The objecive is o minimize he quadraic cos funcion E which is compued as follows: E = O y γ = k = where p is number of paerns in he raining se and O and y are he acual and he desired oupu respecively. The soluion of equaion 6 is found using he backpropagaion learning algorihm ha updaes he nework weighs and biases in he direcion in which he performance funcion decreases mos rapidly: he negaive of he gradien. In our sudy he nework is rained wih he Levenberg- Marquard (L-M) numerical algorihm (Nocedal & Wrigh ) where he weighs are adjused as follows: p m ( ) γ γ k k (5) (6)

5 S.Lahmiri / Managemen Science Leers () 3 T T k+ k μ Δ w= w w = J J + I J e where J is he Jacobian marix (firs derivaives) e is a vecor of nework errors (Eq. 6) and μ is a given parameer. The L-M algorihm approximaes he Hessian marix by compuing he Jacobian marix which is less complex han direcly compuing he Hessian marix. As a resul he L-M numerical algorihm is fas. The parameer μ is arbirarily se o. in our sudy. Finally in order o check he effeciveness of volume series in he predicion of fuure volailiy wo models are simulaed. The firs one includes pas volailiy and pas volume changes o predic fuure volailiy. For insance he following model is approximaed and simulaed: + ) = ( log( h V ) (8) log( h Ψ ) where log(h ) and V are respecively he esimaed volailiy and changes in log volume series as defined in previous secion. And ψ is he unknown nonlinear funcion o be approximaed wih arificial neural neworks. The second one includes only pas volailiy o predic fuure volailiy. For insance he following model is approximaed and simulaed: ( h ) ( V ) log + =Φ (9) where Φ is he unknown nonlinear funcion o be approximaed wih arificial neural neworks.. Resuls Training and es ses were chosen as follows: firs 8% observaions were used as he raining se and nex % observaions were used as he es se. The predicion performance is evaluaed using he roo mean of squared errors (RMSE) mean absolue error (MAE) and mean absolue deviaion (MAD). They are defined as follows: N R MSE = E P N = N = ( ) N M AE = E P MAD = N N = P P where E P and P are respecively he esimaed volailiy prediced volailiy and he average of prediced volailiy over he esing (ou-of-sample) period; for example = o N. Table shows he deails of he resuls obained for each marke. As shown in Table volume series improve he predicion accuracy in he case of he Moroccan sock marke. For insance he obained RMSE MAD and MAE when volume is included (excluded) as predicive variable are respecively.6 (.3).36 (.55) and.35 (.83). This finding is consisen wih Lamoureux and Lasrapes (99) and wih Omran and McKenzie () where GARCH models were used o esimae and predic fuure volailiy. In addiion he obained simulaion resuls for Morocco indicae ha volume decreases he predicion errors by.5.9 and.8 basis poins for RMSE MAD and MAE respecively. In oher words volume modesly improves he forecasing accuracy of he BPNN. This finding is consisen wih Brooks (998) who used GARCH process o esimae and predic fuure volailiy. Besides as shown in Table volume series do no improve he predicion accuracy in he case of he Saudi Arabian sock marke. The obained RMSE MAD and MAE when volume is included (7) () () ()

6 3 (excluded) as predicive variable are respecively.365 (.39).83 (.63) and.693 (.59). Thus volume increases he predicion errors by.33.6 and.7 basis poins for RMSE MAD and MAE respecively. The finding is consisen wih Huang and Yang () and Bohl and Henk (3) where GARCH models were used o esimae and predic fuure volailiy. Fig. compares he esimaed and he prediced volailiy for each marke. I clearly indicaes ha prediced volailiy is higher han he esimaed volailiy during volaile periods for boh markes and wheher volume is considered as predicive variable or no. This finding is more pronounced in he case of he Moroccan sock marke. In sum he simulaions resuls indicae ha rading volume helps improving he forecasing accuracy of BPNN in he predicion of sock marke volailiy in Morocco bu no in Saudi Arabia. In oher words rading volume conains valuable informaion o be used o predic fuure volailiy in Morocco. However he obained resuls sugges ha volume conains noisy informaion ha reduces he forecasing accuracy in Saudi Arabia. Table Performance measures Morocco Morocco Saudi Arabia Saudi Arabia Wih Volume No Volume Wih Volume No Volume RMSE MAD MAE Morocco esimaed prediced.5.5 Morocco esimaed prediced Saudi Arabia esimaed prediced 3.5 Saudi Arabia esimaed prediced Fig.. Simulaions wih volume (lef column) and when volume is excluded (righ column)

7 S.Lahmiri / Managemen Science Leers () Conclusion The predicabiliy of marke volailiy is imporan for praciioners o allocae asses and deermine expeced porfolio reurn. Moreover i is fundamenal o forecas closing prices of derivaives such as opions. In his sudy arificial neural neworks wih backpropagaion learning algorihm and Levenberg-Marquard (L-M) numerical raining algorihm were esed for heir abiliy o forecas he sock marke fuure volailiy in Morocco and Saudi Arabia. EGARCH model was used o esimae volailiy in each marke. Then neural neworks were used o simulae forecass. I is found ha rading volume improves he forecasing accuracy of arificial neural neworks in he predicion of sock marke volailiy in Morocco. Thus volume conains valuable informaion o be used o predic fuure volailiy in Morocco. However he obained resuls sugges ha rading volume conains noisy informaion ha reduces he forecasing accuracy in Saudi Arabia. In addiion i is found ha he prediced volailiy is larger han he esimaed volailiy during volaile periods for boh markes. As a resul high level of prediced volailiy should be aken wih cauion in pracice; namely in porfolio managemen and derivaives evaluaion. For fuure work pos 8 financial crisis period will be considered along wih oher Middle Eas sock markes. References Asalakis G.S. & Valavanis K.P. (9). Surveying sock marke forecasing echniques Par II: Sof compuing mehods. Exper Sysems wih Applicaions 36 (3) Bildirici M. & Özgür Ö.E. (9). Improving forecass of GARCH family models wih he arificial neural neworks: An applicaion o he daily reurns in Isanbul Sock Exchange. Exper Sysems wih Applicaions Bohl M.T. & Henke H. (3). Trading volume and sock marke volailiy: The Polish Case. Inernaional Review of Financial Analysis Bollerslev T. (986). Generalized Auoregressive Condiional Heroscedasiciy. Journal of Economerics Brooks C. (998). Predicing sock index volailiy: Can Marke Volume help? Journal of Forecasing Clark P.K. (973). A subordinaed sochasic process model wih finie variance for speculaive prices. Economerica Davidson R. & Mackinnon J.G. (8). Economeric Theory and Mehods. Oxford Universiy Press Inernaional Ediion. Epps T. & Epps M. (976). The sochasic dependence of sochasic price changes and ransacion volume: Implicaions for he mixure of disribuion hypohesis Economerica Girard E. & Biswas R. (7). Trading volume and marke volailiy: developed versus emerging sock markes. The Financial Review Hajizadeh E. Seifi A. Zarandi Fazel M.H. & Turksen I.B. (). A hybrid modeling approach for forecasing he volailiy of S&P 5 index reurn. Exper Sysems wih Applicaions Hamid S.A. & Zahid I. (). Using neural neworks for forecasing volailiy of S&P 5 index fuures prices. Journal of Business Research Hu M.Y. & Tsoukalas C. (999). Combining condiional volailiy forecass using neural neworks: An applicaion o he EMS exchange raes. Journal of Inernaional Financial Markes Insiuion and Money Huang B.-N. & Yang C.-W. (). An empirical invesigaion of rading volume and reurn volailiy of he Taiwan sock marke. Global Finance Journal Hung J.-C. (). Applying a combined fuzzy sysems and GARCH model o adapively forecas sock marke volailiy. Applied Sof Compuing Lamoureux C.G. & Lasrapes W.D. (99). Heeroskedasiciy in sock reurns daa: volume versus GARCH effecs. The Journal of Finance 5 () -9.

8 3 Lee B.-S. & Rui O.M. (). The dynamic of relaionship beween sock reurns and rading volume: domesic and cross-counry evidence. Journal of Banking and Finance McKenzie E. & Omran M.F. (). Heeroskedasiciy in sock reurns daa revisied: volume versus GARCH effecs. Applied Financial Economics Nelson D.B. (99). Condiional heeroskedasiciy in asse reurns: a new approach. Economerica Nocedal J. & Wrigh S.J. (). Numerical opimizaion. Springer. Roh T.H. (7). Forecasing he volailiy of sock price index. Exper Sysems wih Applicaions Rumelhar D.E. Hinon G.E. & Williams R.J. (986). Learning inernal represenaions by error propagaion. In Rumelhar D.E. and J.L. McClelland eds. Parallel disribued processing:exploraions in he microsrucure of cogniion. Cambridge MA MIT Press Subboin M.T. (93). On he law of frequency error. Maemaicheskii Sbornik Tseng C.-H. Cheng S.-T. & Wang Y.-H. (9). New Hybrid Mehodology for Sock Volailiy Predicion. Exper Sysems wih Applicaions 36 () Wang C.P. Lin S.H. Hung-Hsi Huang H.H. & Wu P.C. (). Using neural nework for forecasing TXO price under differen volailiy models. Exper Sysems wih Applicaions 39(5)