From Discrete to Continuous: Modeling Volatility of the Istanbul Stock Exchange Market with GARCH and COGARCH

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1 MPRA Munich Personal RePEc Archive From Discree o Coninuous: Modeling Volailiy of he Isanbul Sock Exchange Marke wih GARCH and COGARCH Yavuz Yildirim and Gazanfer Unal Yediepe Universiy 15 November 2010 Online a hps://mpra.ub.uni-muenchen.de/27946/ MPRA Paper No , posed 9 January :10 UTC

2 1 Yavuz Yıldırım Yediepe Universiy Prof. Dr. Gazanfer Ünal Yediepe Universiy From Discree o Coninuous: Modeling Volailiy of he Isanbul Sock Exchange Marke wih GARCH and COGARCH Inroducion The success of he forecas model of Isanbul Sock Exchange (ISE) marke indices has received grea aenion in he pas decade. The reason being ha, any efficien forecasing of he index value would provide he invesors wih profiable reurns. However, he main complicaion in predicion is he volailiy in he ime series. There are several reasons ha one may wan o model and forecas volailiy: for insance o analyze he risk of holding an asse or he value of an opion. Forecas confidence inervals may found o be ime-varying, so ha more accurae inervals can be obained by modeling he variance of he errors. More efficien esimaors can be obained if heeroskedasiciy in he errors is handled properly. I has been raher difficul o decide which model o use in order o make an efficien forecasing. The choice of daa and he seleced period can affec he selecion of an appropriae model. Mos of he models arising from he economeric approach are in discree ime. Paricularly GARCH models and heir exensions have received some aenion as appropriae models o capure cerain empirical facs of he empirical volailiy process [6]. Nelson [10] and Duan [5] aemped o capure he characerisic of financial reurns daa by diffusion approximaions o he discree ime. Klüppelberg [7] adoped he idea of a single noise process and suggesed a new coninuous ime GARCH (COGARCH) model, which capures all he sylised facs as he discree ime GARCH does. As he noise process, any Lévy processes

3 2 are possible, is incremens replacing he innovaions in he discree ime GARCH model. COGARCH based on a single background driving Lévy process, is differen from, hough relaed o, oher coninuous ime sochasic volailiy models ha have been proposed. I generalises he essenial feaures of he discree ime GARCH process in a direc way. Here, we demonsrae he applicabiliy of COGARCH model for modeling he ime-varying volailiy of he ISE100. Maller e. al. [8] have recenly demonsraed how o apply his kind of mehodology o describe he volailiy of he Ausralian sock marke, using i o analyse en years of daily daa, mosly equally spaced in ime for he ASX200 index. Also, Müller e. al. [9] in 2009 analysed he volailiy of sock markes using COGARCH. 1. Mehodology On he discree modeling par, he bes candidae model will be found by considering AIC and BIC values afer saionarising he reurn daa. Then usual ess will be carried ou o check if he model is covariance saionary, wheher i obeys he negaiviy consrains, and wheher he arch effec in he residuals is eliminaed. For coninuous modeling, he parameers from he discree model will be used for coninuous GARCH model (COGARCH). Then simulaions will be carried ou for boh of he models and comparisons will be made. Due o Nelson [10] and ohers, classical diffusion limis have been used in a naural way o sugges coninuous ime limis of discree ime processes, including for he GARCH models. Nelson's model of COGARCH model has wo differen Brownian moions which are independen of each oher., 0 (1), 0 (2) where B (1) and B (2) are independen Brownian moions, and β > 0, η 0, and φ 0 are consans. In Klüppelberg e. al. [7], COGARCH model is a direc analogue of he discree ime GARCH, based on a single background driving Lévy process, and generalises he essenial feaures of he discree ime GARCH process in a naural way.

4 3 The COGARCH process ( G ) 0 is defined in erms of is sochasic differenial dg, such ha where β > 0, η 0, and φ 0 are consans. 0 (3) is he quadraic variaion process of L which is defined as where for 0., > 0 (4) The process G jumps a he same ime as L does, and has jump sizes 0 (6) Klüppelberg [6] shows he ideniy as (5) 0 (7) Deriving a recursive and deerminisic approximaion for he volailiies a he jump imes we ge since σ s is laen and L s is usually no observable, hence using Euler approximaion for he inegral we ge (8) (9) herefore for he volailiy esimaion we end up wih (10) (11)

5 4 The bivariae process ( σ, G ) 0 is Markovian. If ( σ 2 ) 0 is he saionary version 2 2 of he process wihσ 0 = σ, hen ( G ) 0 is a process wih saionary incremens [7, Corrolary 3.1]. 2. Daa We perform he analysis using daily log reurns on ISE100 daily closing index values. We focus on he ime period from 03/01/1994 o 23/06/2010. The daa were obained from ISE. 3. Resuls And Diagnosics The firs sep ino he empirical sudy is o use graphical ools o deec any apparen feaures of he daa. In he case of log reurn of ISE100 daa series; in Error! Reference source no found. i is clear ha he reurn daa is more like a random walk. There is no rend in he log reurn of ISE100 and i is more like a whie noise ype daa series, which suggess ha he ime series is saionary. The saionariy of he daa also suppored by ACF and PACF graphs. This resul will be invesigaed furher by he uni roo ess. 3.1 Resuls Of Uniroo And Saionary Tess According o he p-value of ADF es, 2.713e-41, he null hypohesis ha he daa conains a uni roo can be rejeced. And his is also suppored by KPSS es resul, wih he p-vale , canno rejec he null hypohesis a any significance level ha he daa is saionary around a consan. 3.2 Discree Modeling The bes candidae model is found o be AR(1)~GARCH(1,1) model. The Ljung-Box es wih he p-value of ells us ha here is no auocorrelaion in he model s residuals and he candidae model also removes he ARCH effec in he residuals given he LM Tes s p-value is The model obeys he negaiviy consrain of a GARCH model ha is none of he coefficiens of he parameers are negaive, and i also saisfies he covariance saionariy condiion as he sum of coefficiens is less han 1. All of he coefficiens

6 5 are saisically significan as he -values are greaer han All he resuls show ha he candidae model AR(1)~GARCH(1,1) is a good model for he log reurn of ISE100 ime series. (12) (13) IMKB100 Timeseries Plo Log Reurn of IMKB100 imkb94.10[1:4106, 2] ACF PACF ACF Lag Parial ACF Lag Figure 1: Time series plo of ISE100 index value, Log reurn of ISE100, ACF and PACF of log reurn Source: Own Sudy Table 1 Esimaed Coefficiens of AR(1)~GARCH(1,1) Value Sd.Error value Pr( > ) C e e-008 AR(1) e e-004 A e e-011 ARCH(1) e e+000 GARCH(1) e e+000 Source: Own Sudy

7 6 3.3 Coninuous Modeling Given ha he parameers of COGARCH model is equal o he discree GARCH model's parameers as such β = β, η = lnδ, φ = λ / δ (14) The candidae model's parameers are β = , λ = , δ = (15) The parameers' of COGARCH(1,1) model are η = ln = φ = / = (16) To sar he simulaion we use numerical soluions for dg and 2 dσ in (6) and (11), and we also use a Lévy process driven by compound Poisson process. The compression beween he volailiy of he log reurn daa wih he discree GARCH model and COGARCH's volailiy, Figure 2, shows ha here is a close relaion beween discree and coninuous model and boh models mimic he real daa s volailiy. Volailiy of reurn daa volailiy of COGARCH volailiy of Discree Model Figure 2: Volailiy plos of Log Reurn Daa, coninuous GARCH model, and GARCH model Source: Own Sudy

8 7 Conclusion Log reurn of ISE100 daily closing index value was modeled wih he bes candidae model AR(1)~GARCH(1,1). Then using he parameers from he discree model, coninuous model COGARCH(1,1) was applied o he daa. Volailiy of simulaed daa from discree and coninuous models compared wih he real daa volailiy. We showed ha he simulaed GARCH volailiy and COGARCH volailiy appears o follow he same paern of jumps. Furhermore, boh models imiae he real reurn daa s volailiy. Bibliography 1. Barndorff-Nielsen 0. E., Normal Inverse Gaussian Processes and he Modelling of Sock Reurns, Research Repor 300, Deparmen of Theoreical Saisics, Insiue of Mahemaics, Universiy of Aarhus, Blzsild P., Lecure given a workshop on sochasic processes and financial markes, Personal communicaion, Bollerslev T., Generalised auoregressive condiionally heeroscedasiciy, J. Economerics, 31: , Dickey O.A. and Fuller W.A., Disribuion for he esimaes for auoregressive ime series wih a uni roo, J. Amer. Sais. Assoc., 74: , Duan, J.C., Augmened GARCH(p; q) process and is diffusion limi, J. Economerics,79-97, Engle R.F., Auoregressive condiional heeroscedasiciy wih esimaes of he variance of unied kingdom inflaion, Economerica, 50: , Klüppelberg C., Lindner A., and Maller R., A coninuous ime GARCH process driven by a Levy process: saionariy and second order behavior, J. Appl. Prob.,41(3): , 2004

9 8 8. Maller, R.A., Müller, G. and Szimayer, A., GARCH modelling in coninuous ime for irregularly spaced ime series daa, Bernoulli 14(2) , Müller, G., Durand, R., Maller, R., Klüppelberg, C., Analysis of sock marke volailiy by coninuous-ime GARCH models, [in]: Gregoriou, G.N., Sock Marke Volailiy,Chapman Hall/Taylor and Francis, London, pp , Nelson D. B., ARCH models as diffusion approximaions, J. Economerics, 45:7--38, Taylor S. J., Financial reurns modelled by he produc of wo sochasic processes: a sudy of daily sugar prices In O. D. Anderson, edior, Time Series Analysis:Theory and Pracice, volume 1, pages Norh-Holland, Amserdam, 1982 Summary The objecive of his paper is o model he volailiy of Isanbul Sock Exchange marke, ISE100 Index by ARMA and GARCH models and hen ake a sep furher ino he analysis from discree modeling o coninuous modeling. Through applying uni roo and saionary ess on he log reurn of he index, we found ha log reurn of ISE100 daa is saionary. Bes candidae model chosen was found o be AR(1)~GARCH(1,1) by AIC and BIC crieria. Then using he parameers from he discree model, COGARCH(1,1) was applied as a coninuous model.