Application of Stochastic Volatility Model to KSE-100

Transcription

1 Pak. j. eng. echnol. sci. Volume 4, No 1, 014, 8-40 ISSN: prin ISSN: online Applicaion of Sochasic Volailiy Model o KSE-100 Syed Monis Jawed* Received on: May0, 013; Acceped on: Aug, 013 ABSTRACT The paper examines he implemenaion of sochasic volailiy (SV) model o he daa of Karachi Sock Exchange 100 index during he period of January 007 o December 011. The Sochasic Volailiy model is compared wih he GARCH (1,1) model for forecasing volailiy. The sochasic volailiy model is basically a parameric approach o observe volailiy ha includes wo noise erms, ends o capure volailiy beer han GARCH (1,1) model. Thus his exercise demonsraes he capabiliy of sochasic volailiy model o forecas volailiy more efficienly for emerging markes such as KSE. 1. INTRODUCTION In finance, we generally encouner rade-off beween reurns and risks. Thus idenifying and gauging risks is an essenial ask for financial decision making. Volailiy is considered a primary ool o figure ou risk. Mahemaically, volailiy is a condiional second momen for a random variable which depends on oher random variable(s). Le Y is a vecor which evolves over ime. I implies ha Y would have observaions [y 1, y,.,,y n ]. Suppose Y is dependen upon some oher vecors X 1,X,.,X m all are indexed wih ime. In his case he condiional disribuion of Y can be depiced by [Y X 1, X,, X m ] and he variance of he above disribuion would be referred o volailiy. Volailiy =Var [Y X 1, X,, X m ] (1) * The auhor is a Co-Operaive Lecurer a Deparmen of Saisics, Universiy of Karachi. He is also pursuing his M. Phil. a Applied Economics Research Cenre, Universiy of Karachi. PJETS Volume 4, No 1, 014 1

2 Wih he developmen of Auoregressive Condiional Heroscedasic Model (ARCH) by Engle (198), he class of condiional heroscedasic models has become insrumenal o observe volailiy. The ARCH model, a he firs lag, ha is ARCH (1) is based upon following equaions y PJETS Volume 4, No 1, 014 y 1 Bollerslev (1986) furher generalized he ARCH model ino Generalized Auoregressive Condiional Heroscedasic(GARCH) model. The GARCH (p,q) model, resembles closely wih he ARMA (p,q) model and i recifies he problem of infinie lag auoregressive model ha can frequenly be observed in ARCH models. The GARCH models for he minimum order, as proposed by Bollerslev (1986), ha is GARCH (1,1), can be presened by he following equaions y y y 1 Since he emergence of GARCH models is varians such as EGARCH, TGARCH, NGARCH and GJR-GARCH have been developed. Bu o dae, he mos frequenly applied model remains GARCH (p,q) model wih p=1 and q=1 (Bollersley (006) in his original work used GARCH (1,1) model). Apar from condiional heroscedasic models, some oher models have also been developed o observe volailiy. Sochasic volailiy models come ino he caegory of such models which are no based on condiional heroscedasiciy in previous ime epochs, raher i akes ino accoun he arrival of informaion in any paricular marke or economy. The marke which is considered in his paper is Karachi Sock Exchange and is all famous KSE-100 has been used as a baromeer of is movemen. I is common among praciioners o calculae he log reurns on he indices of he major equiy marke of any economy as a proxy for he rae of reurns on he economy and o exend he inferences obained from equiy indices analysis o he whole economy. Here we are doing a somewha similar exercise. In he las decade, so many research works have emerged abou he KSE as a resul of sharp growh in is marke capializaion. () (3) (4) (5)

3 Equally imporan ineres has been aken by researchers in he invesigaion of marke crashes ha occurred in his era. The phenomenon of sharp declines has proved o be he foundaion of sudying volailiy a KSE. One recen work abou volailiy of KSE- 100 index has been done by Rafique and Rehman (011)- who have applied GARCH (1,1) model on KSE daa of varying frequencies (daily, monhly and annually) and compared he effec of change in frequency on KSE-100 persisency. The objecive of our sudy is o apply he sochasic volailiy (SV) model on he KSE daa and hen o compare he volailiy modeling of SV model and GARCH (1,1) model and o ascerain which model depics volailiy in a beer way. Several researches including Yu (00) and Krichene (003) have come o he conclusion ha he markes which fall in he caegory of Emerging Markes - he sochasic volailiy models depic heir volailiies in a beer way. The organizaion of his paper is as follows. The nex secion reviews some basics of sochasic volailiy. The hird secion describes he mehodology which we applied here in his paper. The fourh secion shows he resuls of our research. And he las secion is dedicaed o he conclusion of our resuls..the STOCHASTIC VOLATILITY MODEL The idea of Sochasic Volailiy was given by Taylor (1986). The volailiy in GARCH models and in almos all is varians depends upon he volailiy presen in previous ime epochs. Therefore he differen variance series in hose models depend upon he variances of previous ime periods. Sochasic volailiy model gives us he independence from he values of previous ime periods-consequenly he informaion arrival became he only deerminan of volailiy in SV models. SV models are frequenly applied for making projecions in opion pricing bu apar from i, hese models can also be applied o model variaions in he socks. The cornersone for consrucing sochasic volailiy model sems from he Black-Scholes model for esimaing volailiy. Black-Scholes (1973)in heir seminal paper presened heir model for pricing coningen claims. In he Black-Scholes model he reurns on asse are assumed o follow a geomeric Brownian moion.this geomeric Brownian moion ranslaes ino he normaliy of log reurns on he asse. PJETS Volume 4, No 1, 014 3

4 However, various empirical evidences prove conrary o his fac due o higher ails and higher peaks of disribuion of log reurns. This phenomenon indicaes he varying variances of he disribuion of log reurns. The Black-Scholes model can be compued by he following equaion: C S, K, T,, r, S. Nd K Nd (6) 1. Where S is he curren amoun of he underlying asse K is he srike price T is he mauriy ime T is he curren ime R is he risk free rae s is he volailiy of S, and N(.) is he cumulaive normal disribuion Also d 1 log S log K r 0.5 T T (7) d d 1 T (8) The Black-Scholes model which was menioned above does no consider he ime varying volailiy (s). Condiional Heeroscedasic models allow us o capure he volailiy which evolvesover ime. The GARCH (p,q) models can be exended o he sochasic volailiy models as follows y 1 y 1 4 PJETS Volume 4, No 1, 014 (9) 1 (10)

5 Equaion 9 is referred as an observaion equaion of he SV model, whereas equaion 10 is referred as ransiion equaion of he SV model.in equaion 10, all he parameers of sochasic volailiy model are menioned. For compuaional purposes, generally, all he parameers are supposed o belong o vecor q, ha is {w,a,b,s h } Îq. The SV model acually depics he flow of informaion arriving in any dynamic sysem [Anderson (1996)]. One of he unique feaures of SV models is ha i uses wo innovaion erms for error. These erms are e which follows Boh he disurbances (e and h ) may or may no be independen. Preminger and Hafner (006) suggesed ha he inclusion of an addiional error erm, makes SV model more flexible as compared o oher volailiy models. If he disurbances are no independen, or in oher words, correlaed o each oher, hen i implies ha sock price movemens are negaively correlaed wih he volailiy. This phenomenon of correlaion of disurbances is referred by Black(1976) as leverage effec. Thus, in he case of rise in volailiy, he deb o equiy raio will rise for making invesmens and consequenly he invesmens will become more risky. In SV model, h, which is he volailiy of volailiy, used o depic us he leverage effec.alhough SV models provide beer modeling faciliy for reurns, bu esimaion of is parameers is an uphill ask. The complexiy in esimaion of is parameers is amids he inracabiliy of log-likelihood esimaes of SV parameers. The log-likelihood esimaes are difficul o obain, owing o he fac ha SV models are used o compue wo error erms simulaneously i.e.e, he error erm innovaing in he process andh, which is he volailiy of volailiy. Several algorihms such asjacquier, Polson and Rossi (1994), Heson(1993) and Mills (008) have been developed o esimae is parameers. 1. METHODOLOGY As he KSE daa is in he form of ime series, herefore he umos par of hisresearch is o check he saionariy of he ime series. Esablishmen of saionariy ensures ha causal analysis is no going o yield spurious resuls. We have applied he frequenly used Augmened Dickey Fuller (ADF) es for his purpose. The basic parameer of he ADF es isr, whose value, if urns ou o be 1, hen we canno rejec he null hypohesis of non saionariy or we can say,ha he problem of uni roo has emerged in he series. PJETS Volume 4, No 1, 014 5

6 The ADF es can be performed hrough differen forms such as, wih inercep, wihou inercep and wih inclusion of rend facor. However here we are applying he ADF es wih inercep and wihou rend. Following is he equaion of ADF es in his form y y1 1 (11) Afer esing he saionariy, we have firs applied he convenional GARCH (1,1) model on he log reurns of KSE-100. One of he special ineress we have here is o check he persisency of our model. The persisency can be checked in case of GARCH (1,1) model by considering wheher he sum of parameers, a and b is less han or equal o 1 or no. If heir sum exceeds 1, i means ha he model is no persisen and hence no adequae for he paricular daa. Then we urn o he cenral objecive of our research ha is, he applicaion of SV model o he KSE daa. For he purpose of applying SV model, we have followed he algorihm of Mills(008). The echnique of Mills (008) is based on Kalman filering. Barndoff- Nielsen and Shephard (00) described KalmanFiler as eligible for capuring non Gaussian dynamics of volailiy and also provides consisen and asympoically normal se of esimaors. As our model is in a dynamic sysem, he sae in SV model keeps changing. Thus we can model he log reurns of sock indices via he following differenial equaion d LogP d d (1) Where P is he price of sock and indicaes he innovaion of our ime. If we discreize he above equaion hen x can be aken as Dlog(P ), hen he above equaion would become x (13) Tha implies ha V x V (14) 6 PJETS Volume 4, No 1, 014

7 Mills(008) found ha he disribuion of he volailiy of log reurns is lognormally disribued due o he fac ha log(e ) follows a logarhimicc disribuion. The expeced value of log e E(log e ) comes ou o be -1.7 and variance V(log e ) will be equal o This can enable us o esimae he sae equaion. Using Kalman filers we gebes Linear Unbiased Esimaes (BLUE) for volailiy vecor h which depends on he observaion equaion y -1. Afer applying boh GARCH(1,1)model and SV model, menioned in equaions (5) and (10) respecively, we compared boh of hem o ascerain which model is more suiable for modeling KSE reurns. For he purpose of comparison we used wo differen echniques here, so ha we can make our inferences wih more cerainy; firs we compared he roo mean square errors (RMSE) values for boh models and hen compared hese models via Asympoic Relaive Efficiency (ARE) of variance of median over variance of mean. Roo Mean Square Error can simply be obained by aking square roo of Mean Square Error. For he purpose of geing ARE of median over mean, a boosrapping approach is applied on he derived realizaions from GARCH(1,1) and SV models. The larger values of ARE here means ha disribuion is geing heavy ailed and hence giving beer resuls. These comparaive sudies would finally complee our research. RESULTS In his paper we have firs calculaed he descripive saisics for he log reurns on KSE 100 indices as hese descripives are helpful in explaining wha acually is going on in our daa. The following able represens hese descripives PJETS Volume 4, No 1, 014 7

8 Table 5.1 D e s c r i p i v e M e a s u r e s V a l u e s M ea n e M ed i a n S a n d a rd D e v i a i o n S k e w n e s s K u r o s i s J a r q u e B e ra The descripive saisics describe some imporan characerisics of our daa. Firsly, we can see ha he value of he median is much higher as compared o mean which is a depicer of fa ail phenomena in our series. The value of skewness is also slighly deviaing from he sandard normal s skewness. Also he value of kurosis is also higher han ha of sandard normal. This fa ail and higher kurosis is a ypical characerisic of financial ime series. The higher value of Jarque- Bera (which leads o he rejecion of null hypohesis of normaliy) is a naural oucome of hose characerisics which we obained earlier. The figure below shows he plo for he log reurn series. 8 PJETS Volume 4, No 1, 014

9 Now we will consider he saionariy of he log reurns on indices. Though he plo of he log reurns series is depicing saionary paern. However, he convenional ADF es is used for checking he saionary in he series. We have performed his es a level. The following able shows is value. Table 5. -Saisic Prob.* Augmened Dickey-Fuller es saisic I is obvious from Table 5. ha ADF es saisic is highly significan. Therefore we can say ha he given series is saionary.we hen applied he GARCH(1,1) on KSE reurns o model he volailiy from his convenional model. The following able shows he parameer values of GARCH(1,1) PJETS Volume 4, No 1, 014 9

10 Table 5.3 Parameer Value Sig. 8.34e ?? ?? We can see ha he value of a+b<1, i means ha our obained parameers are persisen. These resuls show similar properies as hose repored in he sudy of Rafique and Rehman (011). Our resuls also show verify he volailiy clusering, ha is prevalence of cycles of higher or lower volailiy for longer ime periods, which can be observed via GARCH(1,1) model. Now we come o he applicaion of Sochasic Volailiy model o he reurns of KSE-100 indices. The SV model is implemened here via Kalman filering. The resuls of his model are shown in Table 5.4. Table 5.4 P a r a m e e r V a l u e S i g ?? ?? S V From Table 5.4, i can be observed ha mos of he values of sochasic volailiy parameer vecor qcome ou o be significan. The only insignifican parameer is w. I means ha in he long run variance of he model will be close o 0.The value of a implies long memory of he volailiy process. However he value of, which implies ha he generaed volailiy process is no saionary. Bu negaive sign here informs us abou he presence of asymmeric affec, ha is negaive developmens would have larger impac on KSE-100 index as compared o posiive ones, consequenly he leverage will ge high in he marke.while s h which is he volailiy of he process is also significan which confirms he suiabiliy of SV model for given daa. Furher, he es saisic for ransiion equaion (which is laen) is significan a he highes level. 10 PJETS Volume 4, No 1, 014

11 This shows ha SV model provides good approximaion for he volailiy of an emerging marke, such as KSE. Is value is depicing he magniude of affec ha any shock will yield on KSE-100 index. For he purpose of comparison beween GARCH(1,1) and SV model, we applied wo approaches: he roo mean square error (RMSE) and he asympoic relaive efficiency (ARE) of median over mean. Following able shows is resuls. Table 5.5 RMSE ARE GARCH (1,1) SV As he values of RMSE are very close o zero in boh models, herefore i hins ha boh he models are quie accurae for our daa. Bu as he value of RMSE is slighly small in SV model, i means ha he SV modeling echnique for he KSE100 daa is more appropriae. In ARE (median over mean) however, larger values show beer approximaion of higher kurosis and heavier ails. As he value of ARE is much higher in SV models as compared o GARCH (1,1) model, i means ha according o ARE he SV model is more suiable for our daa. The resuls of our comparison beween GARCH(1,1) and SV model are quie similar o he resuls of Yu (00) and Racico and Theore (010) who found ha SV model provides superior esimaes han GARCH (1,1) model. However hese resuls are conrary o he resuls of Preminger and Hafner (006). The comparison finally complees our analysis. 4. CONCLUSION This research is an aemp o apply SV models on KSE-100 daa. The SV model proved o be beer and more suiable for volailiy exising in he reurns of KSE-100. Thus our research demonsraes he capabiliy of SV model for finding ou he volailiy on reurns of KSE. The research has deep implicaions for invesmen analyss and porfolio managers, who make heir invesmens a Karachi Sock Exchange, as hey can assess he volailiy here via simulaion based SV model apar from ARCH ype models. PJETS Volume 4, No 1,

12 One of he reasons for he success of SV model could be he fac poined ou by Yu(00), which holds mosly due o lack of regularizaion and emerging naure of markes. These effecs in he marke end o follow random movemens, consequenly he simulaion based mehod is beer for modeling he marke movemens. References [1] Anderson, T.G. (1996), Reurn Volailiy and Trading Volume: An Informaion Flow Inerpreaion of Sochasic Volailiy,The Journal of Finance, 51,1, [] Barndorff-Nielsen, O.E. and Shephard, N. (00), Economeric Analysis of realized volailiy and is used in esimaing Sochasic Volailiy Models,Journal of Royal Saisical Sociey, Series B, 64,, [3] Black, F. and Scholes, M. (1973), The Pricing of Opions and Corporae Liabiliies, The Journal of Poliical Economy, 81, 3, [4] Black, F. (1976), The Pricing of Commodiy Conracs, Journal of Financial Economics,3, [5] Bollerslev, T. (1986),Generalized Auoregressive Condiional Heroskedasiciy, Journal of Economerics.31, [6] Engle, R. (198), Auoregressive Condiional Heroscedasiciy wih esimaes of he Variance of Unied Kingdom Inflaion,Economerica.50, [7] Heson, S.L. (1993), A Closed form soluion o Opions wih Sochasic Volailiy wih Applicaions o Bonds and Currency Opions, 6,, [8] Krichene, N.(003), Modeling Sochasic Volailiy wih Applicaions o Sock Reurns, IMF Working Paper, WP/ 03/15. 1 PJETS Volume 4, No 1, 014

13 [9] Jacquier, E., Polson, N.G. and Rossi, P. E. (1994), Bayesian Analysis of Sochasic Volailiy Models, Journal of Business and Economics Saisics, 1, 4, [10] Mills, T. (008), The Economeric Modeling of Financial Time Series, 3 rd Ed., Cambridge Universiy Press, Cambridge (UK). [11] Preminger, A. and Hafner, C.M. (006), Deciding beween GARCH and Sochasic Volailiy Model via Srong Decision Rules, Discussion Paper No, 06-03, Monaser Cener for Economic Research, Ben-Gurion, Universiy of Negev. [1] Racico, F.E. and Theore, R. (010), Forecasing Sochasic Volailiy Using Kalman Filer : An Applicaion o Canadian Ineres Raes and Price - Earnings Raio, The IEB Inernaional Journal of Finance, 1, 1-0 [13] Rafique, A. and Rehman, K. (011), Comparing he persisency of differen frequencies of sock reurns volailiy in an emerging marke: A case sudy of Pakisan, African J ournal of Business Managemen, 5, 1, [14] Taylor, S. J. (1986), Modeling Sochasic Volailiy, A Review and Comparaive Sudy, Mahemaical Finance, 4,, [15] Yu, J. (00), Forecasing Volailiy in he New Zealand Sock Marke, Applied Financial Economics, 1, PJETS Volume 4, No 1,