Capital Asset Pricing Model and Stochastic Volatility: A Case study of India

Transcription

1 MPRA Munich Personal RePEc Archive Capial Asse Pricing Model and Sochasic Volailiy: A Case sudy of India Ka Wai Terence Fung and Ender Demir and Lu Zhou Beijing Normal Universiy, Hong Kong Bapis Universiy - Unied Inernaional Collge, Isanbul Medeniye Universiy, Ciy College of New York 014 Online a hp://mpra.ub.uni-muenchen.de/56180/ MPRA Paper No , posed 8. May :1 UTC

2 Capial Asse Pricing Model and Sochasic Volailiy: A Case sudy of India Ka Wai Terence Fung Ender Demir & Zhou Lu Absrac Bansal and Yaron (004) demonsrae, by calibraion, ha he Consumpion-Based Capial Asse Pricing Model (CCAPM) can be rescued by assuming ha consumpion growh rae follows a sochasic volailiy model. They show ha he condiional equiy premium is a linear funcion of condiional consumpion and marke reurn volailiies, which can be esimaed handily by various Generalized Auoregressive Condional Heerskedasiciy (GARCH) and Sochasic Volailiy (SV) models. Using he daa from India, we find ha condiional consumpion and marke volailiies are capable of explaining cross-secional reurn differences. Also, he model predicion is consisen wih observed declining equiy premium. JEL Classificaion: E1; G1; G1 Keywords: Financial Economics, Macroeconomics and Moneary Economics, Equiy Premium Puzzle Division of Business and Managemen, Unied Inernaional College- Beijing Normal Universiy. Hong Kong Bapis Universiy, Zhuhai, China econfkw@homail.com Deparmen of Economics, Isanbul Medeniye Universiy, Turkey ender.demir@medeniye.edu.r. Deparmen of Economics and Business, Ciy College of New York, NY luzhou59@gmail.com

3 1. Inroducion The developmen of he famous Consumpion Based Capial Asse Pricing Model (CCAPM) is an early aemp of financial economiss o explore he links beween asse reurns and macroeconomic variables ha capure he sources of sysemaic risk. In a wo period model wih exogenous labor income, he equiy premium is proporional o he aggregae consumpion growh, in which he muliplicaive facor is elasiciy of ineremporal subsiuion of consumpion. Fama and French (1993) advocae a hree facor model - marke reurn, he reurn of small less big socks (SMB), and he reurn on a porfolio of high book-marke value socks less low book-marke value socks (HML). Alhough he Fama and French (1993) model as a resounding success, i is sill no clear how hese facors relae o underlying macroeconomic risk. Acually, he economic inerpreaion of SMB and HML remain as a source of conroversy. Leau and Ludvigson (00) examine CCAPM in a condiional sense. They express he sochasic discoun facor as a condiional or scaled facor model and examine he ime-varing coefficiens by ineracing consumpion growh wih a coinegraing facor - a coinegraing residual beween consumpion, asse (nonhuman) wealh, and labor income (all in log). The parameers in he sochasic discoun facor depend on invesor s expecaions of fuure excess reurn. Leau and Ludvigson (001) demonsrae ha drives ime-variaion in condiional expeced reurn. Using he assumpion ha consumpion growh rae follows a sochasic volailiy model, Bansal and Yaron (004, hereafer referred o as he BY model) show, by calibraion, ha he condiional equiy premium is a linear funcion of condiional consumpion and marke reurn volailiies. Fung, Lau, and Chan (014) proceed o esimae condiional volaiies and hen es he validiy of BY model. Their firs sep is o esimae condiional consumpion and marke volailiies by wo Sochasic Volailiy (SV) models and wo Generalized Auoregressive Condiional Heeroskedasiciy (GARCH) models, namely Exponenial GARCH (EGARCH) and Threshold GARCH (TGARCH). Their second sep is o use he prediced volailiies as facors and apply he Fama-MacBeh approach o es he validiy of he BY model using U.S. 5 Fama-French porfolio reurns sored by size and book-omarke value. Fung, Lau, and Chan find ha he heoreical premium of he BY model ouperforms radiional CAPM ha is based observed marke premium. They can explain 55% variaion of cross-secion reurn difference by using GARCH consumpion and marke volailiies. In his paper, we apply daa from an emrging economy-india-raher han he U.S. daa o esimae condiional volaiies and hen es he validiy of BY model. The firs sep is o esimae condiional consumpion and marke volailiies by wo Sochasic Volailiy (SV) models and by varisou Generalized Auoregressive Condiional Heeroskedasiciy (GARCH) models including Exponenial GARCH (EGARCH), Periodic GARCH (PGARCH), and Threshold GARCH (TGARCH). The second sep is o use he prediced volailiies as facors for a esing of he BY model. This paper will address in he following secions a couple of quesions. Wih he ex-pos marke risk

4 premium being replaced by condiional consumpion and marke reurn volailiies, does i improve he predicive power of Capial-based Asse Pricing Modle (CAPM)? Is his sudy robus o differen sepcificaions of GARCH models? We find ha he Bansal and Yaron heoreical premium significanly ouperforms radiional CAPM using observed marke premium. Using GARCH consumpion and marke volailiy alone can explain mos of variaion of cross-secion reurn difference. I improves he Fama and French model, by replacing he ex-pos marke risk premium wih he Bansal and Yaron (004) premium. This paper is srucured as follows. Secion oulines briefly he derivaion of he he Bansal and Yaron marke premium. Secion 3 models condiional volailiies. Two Sochasic Volailiies and hree ypical GARCH ype volailiies are esimaed: Exponenial GARCH (EGARCH), Periodic GARCH (PGARCH), and Threshold GARCH (TGARCH). Secion 4 delineaes he esimaion equaions. Secion 5 provides he resuls from he India daa. Secion 6 concludes.. Ouline of Bansal and Yaron (004) Model We now consider he Bansal and Yaron (004) model. I shows ha, if consumpion and dividend growh rae conain a small long-run predicable componen, consumpion volailiy is sochasic, and, if he represenaive household has Epsein and Zin preference, he asse and reurn premium will be a linear funcion of condiional consumpion and marke volailiy. The Euler condiion is given by E [ G R R 1 ] = 1 (1 ) 1 a, 1 i, (1) where is he discoun facor, G 1 is gross reurn of consumpion, R a, 1 is he gross reurn on an asse ha delivers aggregae consumpion as is dividends each period, and Ri, 1 is he individual asse reurn. As well-documened in he lieraure, his class of preference disenangles he relaion beween ineremporal elasiciy of subsiuion (IES) 1 and risk aversion. The parameer =, wih 0 as he degree of risk aversion, 1 1 denoes IES. Campbell and Shiller (1988) show ha he log- linearized asse reurn ( r a, 1 ) can be expressed as r z z g () a, 1 = P where 0 and 1 are consans; z = log( ) is he log price-consumpion raio, and g 1 C is he log reurn of consumpion. The log-linearized firs order euler condiion is m 1 = log g1 ( 1) ra, 1 (3) where m 1 is he sochasic discoun facor. When = 1, hen =, and he above equaion is pinned down o he case of Consan Elasiciy of Subsiuion (CES) uiliy 1

5 funcion. Moreover, if = 1 and = 1, we ge he sandard case of log uiliy. In he spiri of neo-classical Real Business Cycle model (RBC), an exogenous i.i.d shock perurbs consumpion and oupu from heir seady pahs. The sysem of shocks is x x e g g = 1 e 1 = 1 x 1 x u d, 1 = d d 1 1 = 1( ) ww 1 e 1, 1, w 1, u 1 ~ N(0,1) This sysem of equaion suggess ha consumpion ( g 1 ) and dividend growh raes( g d, 1 ) are driven by an unobservable process x, and he volailiy of he laer exhibis mean-reversion ( ) bu perurbed by an i.i.d shock ( e 1 ) 1. Bansal and Yaron (004) solve he log price-consumpion raio by mehod of undeermined coefficiens, and find ha z A A x A (4) = [( ) ( A1 1 e ) ] A 1 =, A = 11 (11 1) There are wo noeworhy feaures of his model. Firs, if and are larger han 1, hen is negaive, and a rise in volailiy lowers he price-consumpion raio, since he ineremporal effec dominaes he subsiuion effec. Second, he risk premium is a posiive funcion of he volailiy persisence parameer ; meaning ha he represenaive consumer dislikes a prolonged period of consumpion shocks. Afer some algebra, he marke premium in he presence of ime-varying economic uncerainy akes he form: where and E ( rm, 1 rf, m, e m, e m, w m, w w m, 1 ) = 0.5var ( r ) (5) w are he condiional consumpion and wealh volailiies; is he price of risk, and is he quaniy of risk. The risk premium of any asse, given by CAPM, can be expressed as E ( ri, 1 rf, ) = m, em, e 0.5var ( rm, 1) (6) The BY model calls for esimaion of wo equaions. Equaion (5) saes ha long-run marke risk premium is deermined by condiional consumpion and marke reurn volailiy. In paricular, he coinegraing vecor is ( m, em, e, 0.5). This paper focuses on equaion (6), which explains cross-secional reurn differences by condiional volailiies. The essence of he BY model is ha persisen sochasic volailiy can explain risk premium. Here we provide an empirical es, by regressing cross-secional reurn agains differen varians of condiional sochasic volailiy. Choosing he bes sochasic 1 Wihou w 1, i will become a GARCH model.

6 volailiy model is no he purpose of his paper. Raher, we wan o show ha if equaion (6) can be explained by some common GARCH and SV models, i should provide indirec suppor for he BY model. More imporanly, i provides an alernaive for he Fama-French model. While he independence of Fama-French facors is conroversial, aggregae consumpion and marke reurn volailiies should be uncorreced. Nex secion is devoed o he descripion of various condiional volailiy models. Daa and Mehodology The consumpion daa of India are colleced from he Federal Reserve Bank of S. Louis Economic Research Daabase. We use he quarerly privae aggregae consumpion daa and hen calculae he (log) reurns. The Fama-French facors, marke reurn risk free rae and sored porfolio reurns are available a Agarwalla, Jacob and Varma (013) working paper 3. The sample periold is Q1 of 1993 o Q of 01. Fung, Lau, and Chan (014) used he U.S 5 Fama-French porfolio reurn as he dependen variable. These daa are value-weighed reurns for he inersecion of five size porfolios and five book-o-marke equiy (BE/ME) porfolios on he New York Sock Exchange, he American Sock Exchange, and NASDAQ socks in Compusa. We use daa from Agarwalla, Jacob and Varma (013) who compued he porfolio reurns using Bombay Sock Exchange (BSE) index daa from he CMIE Prowess. This daase is an improvemen over earlier daa. More firms are included; illiquid firms are excluded; he size cu-off poin is redefined; and surivor bias is correed. However, hey only provide six porfolios based on size (measure by marke capializaion, small and big) and value (book/marke raio, growh, neural and value). We conver he original daa from monhly o quarerly series. Due o limied number of cross-secions (in his case, six), we canno adop he Fama- McBeh(1973) procdure. The more appropriae mehod is ime series approach. There is excessive missing observaions of he big-value porfolio. Therefore, i will no be used for esimaion and we have five porfolios. The independen variables are various GARCH models. If he Bansal and Yaron (014) model holds for a less developed counry like India, i should work for differen specificaions of condiional volailiies. The benchmark model is: E ( r r ) e () i, 1 f, = 0 1 where c, 1 and m, 1 are condiional consumpion and marke volailiies respecively. These volailiies will be esimaed by GARCH and he sochasic volailiy models. The lieraure on GARCH ype models are well-documened, ineresed readers can refer o Bollerslev e al. (199), Bollerslev e al. (1994) for a survey. We will also consider he Exponenial GARCH (EGARCH), Threshold GARCH (TGARCH) and Power GARCH (PGARCH) o model asymmery inheren in he series. Sochasic c c, 1 m m, 1 hp://research.slouisfed.org/fred/ 3 hp://

7 Volailiy (SV) models which are reviewed in, for example, Taylor (1999), Ghysels e al. (1996) have been increasingly recognized as a viable alernaive o GARCH models, alhough he laer are sill he sandard in empirical applicaions 4. The sochasic volailiy model considered in his secion follows Harvey e al. (1994) and Mills (1999). = () r = h = ln h () 1 ~ N(0, r is he coninuously compounded reurn of an asse; denoes he volailiy. There is no inercep in he mean equaion. h is always posiive and akes on an AR(1) process. An appropriae mean equaion can be augmened o equaion (). and are assumed o be wo independen errors. This process is nonlinear in naure, which can be ransformed ino a linear funcion by appropriae change of variable. We define y = lnr. I can be shown ha E(ln ) = 1.7 and var(ln ) =. An unobserved componen sae space represenaion for y has he form 1 ~ E[ ] = 0 ) y = 1.7 h, ~ N(0, ) () h = h, N(0, ) () () and are wo independen whie noises. Thes wo equaions will esimaed simulaneously. For a deailed esimaion procedure, see Théore and Racico (010). Resuls The consumpion and BSE marke reurn condiional volailiies are repored in Figure 1-Figure 8. The emporal movemens of he hree GARCH consumpion volailiies are similar, which is consisen wih our regression resuls ha he significance of he consumpion volailiy coefficiens do no depend on model specificaion. One of he possible reason is ha asymmery is no a characerisic of aggregae consumpion series. For GARCH, EGARCH and PGARCH, here are four spikes in volailiy 1997, 001, 003 and 008. However, EGARCH has differen predicion in he las four quarers in he sampel period (Figure.). The sochasive volailiy is repored in Figure 4. Comparing o he GARCH volailiies, here are wo discernible differences. 1. The range of volailiy is smaller only.. The emporal movemen is more smoohed. Tha said, he emporal comovemen is sill similar o hose of GARCH volailiies. 4 For comparison, discussion of meris and deciding rules of hese wo models, see Fleming and Kirby (003), Preminger and Hafner (006), and Heynen and Ka (1994).

8 From Figure 5-Figure 8, i is obvious ha GARCH, EGARCH and sochasic volailiy models are very similar. The range of quarerly volailiy is %. Mos models predic a big drop in he second half of 008. However, he variaion of PGARCH is significanly smaller han oher models. The PGARCH predics no change in some period. The esimaed sochasic volailiy of BSC marke reurn is a modified version of equaion (). We encounered convergence when here is an inercep; herefore, in he final model, he consan is dropped. Tables 1-4 repor he Bansal and Yaron (004) esimaion using various GARCH volailiies. All sandard errors are correced by Newey-Wes (1987). As a usual pracice, we include he inercep erm in all equaions. To conrol for serial correlaion, we add an auo-regression and moverage average erm o he equaion if he coefficien is significan. If he Bansal and Yaron (004) is rue for a developing counry like India, he condiional marke and consumpion volailiy coefficiens should be joinly significan. A he boom of each able, he Chi-square saisic is repored. From Table.1, i is shown ha he Bansal and Yaron (004) model fails miserably when using GARCH volailiies in a ime series seing. For firms wih high marke capializaion, he emporal persisence is capured by he moving average coefficien; for small firms, i is capured by auoregressive coefficien. Neiher he aggregae consumpion growh nor he BSC marke reurn volailiy is significan. They are no joinly significan, eiher. One of he possibiliy is ha marke index reurns are always characerized by asymmery. We proceed o es he Bansal and Yaron (004) wih wo asymmeric volailiies. The performance of Bansal and Yaron (004) model improves significanly when using EGARCH and PGARCH volailiies. As shown in Table., The marke volailiy coefficen is significan in all five porfolios. For insance, a one percen increase in marke volailiy, he reurn of Big-Growh porfolio will increase by.6 poins. The consumpion volailiy coefficiens are significan for wo porfolios -Small-Growh and Small-Value. The null hypohesis of join significance is rejeced in all models. The paern is similar when using PGARCH volailiy (Table. 3). The marke reurn volailiy coefficien is significan in he Big-Neural, Small-Growh and Small-Value porfolios; so is he join hypohesis. However, he consumpion volailiy coefficien is only significan in he Small-Value porfolio. In any case, we show ha once asymmery is accouned for, a linear combinaion of condiional aggregae consumpion and marke volailiy is capable of explaining emporal variaion of size-value sored porfolio reurns. How abou using sochasic volailiy? As shown in Table. 4, he resuls are similar o hose of PGARCH model. There is no significan change. Concluding from hese ables, we found ha he Bansal and Yaron (004) works he bes for small-sized firms. For he case of sochasic volailiy (Table. 4), for insance, a one percen increase in marke volailiy would resul in a 5.5 percen in Small-Growh porfolio. Our resul is consisen wih Fung, Lau, and Chan (014) ha he GARCH ype model, in general, ouperforms he sochasic volaliiy model under he Bansal and Yaron (004) model.

9 References Agarwalla, Sobhesh K., Jacob, J., Varma Jayanh R., 013. Four facor model in Indian Equiies Marke, Working Paper. No , Indian Insiue of Managemen, Ahmedabad. Bansal, R., Yaron, A., 004. Risks for Long-Run: A Poenial Resoluion of Asse Pricing Puzzles. Journal of Finance. 59, Bansal, R., Kiku, D., Yaron, A., 01. An Empirical Evaluaion of he Long-Run risks Model for Asse Prices. Criical Finance Review. 1, Bansal, R., Kiku, D., Shaliasovich, I., Yaron, A, 01. Volailiy, he Macroeconomy and Asse Prices. Naional Bureau of Economic Research Working Paper: Banz, R., The Relaion beween Reurn and Marke Value of Common Socks. Journal of Financial Economics. 9, Black, F., Jensen, M, C., Scholes, M., 197. The capial asse pricing model Some empirical ess, in: Jensen, M. (Eds.), Sudies in he Theory of Capial Markes. Pareger, New York, pp Bollerslev, T., Chou, R., Kroner, K., 199. ARCH Modeling in Finance: A Review of he Theory and Empirical Evidence. Journal of Economerics. 5, Bollerslev, T., Engle, R., Nelson, D., Arch Models, in Engle, R., McFadden, D. (Eds.), Handbook of Economerics, Volume 4. Elsevier Science, Amserdam, pp Campbell, J.Y., A Variance Decomposiion for Sock Reurns. Economic Journal. 101, Campbell, J., Shiller, R., The Dividend-Price Raio and Expecaions of Fuure Dividends and Discoun Facors. Review of Financial Sudies. 1, Chen, N., Roll, R., Ross, S., Economic Forces and he Sock Marke. The Journal of Business. 59, Cochrane, J., A Cross-Secional Tes of Invesmen-Based Asse Pricing Model. Journal of Poliical Economy. 104, Colacio, R., Croce, M., 011. Risks for he Long Run and he Real Exchange Rae. Journal of Poliical Economy. 119, Consaninides, G., Ghosh, A., 011. Asse Pricing Tess wih Long Run Risks in Consumpion Growh. Review of Asse Pricing Sudies. 1,

10 Durbin, J., Koopman, S., Mone Carlo Maximum Likelihood Esimaion of Non- Gaussian Sae Space Model. Biomerika. 84, Fama, E., French K., Common Risk Facors in he Reurns on Socks and Bonds. Journal of Financial Economics. 33, Fama, E., MacBeh, J., Risk, Reurn and Equilibrium: Empirical Tess. Journal of Poliical Economy. 81, Fleming, J., Kirby, C., 003. A Closer look a he Relaion beween GARCH and Sochasic Auoregressive Volailiy. Journal of Financial Economerics. 1, Fung, K.W. T., Lau, C.K.M., Chan, K.H., 014. The Condiional CAPM, Cross- Secion Reurns and Sochasic Volailiy. Economic Modelling. 38, Ghysels, E., Harvey, A., Renaul, E, Sochasic Volailiy, in: Maddala, G., Rao, C. (Eds.), Handbook of Saisics, Volume 14, Saisical Mehods in Finance. Norh-Holland, Amserdam, pp: Hansen, L., Heaon, K., Li, N., 008. Consumpion Srikes Back? Measuring Long Run Risk. Journal of Poliical Economy. 116, Hansen, L., Singleon, K., 198. Generalized Insrumenal Variables Esimaion of Nonlinear Raional Expecaions Models. Economerica. 50, Harvey, A., Ruiz, E., Shephard, N., Mulivariae Sochasic Variance Models. Review of Economic Sudies. 61, Heynen, R., Ka, H., Volailiy Predicion: A Comparison of he Sochasic Volailiy, GARCH (1,1) and EGARCH (1,1) Models. Journal of Derivaive., Jagannahan, R., Wang Z., The Condiional CAPM and he Cross-Secion of Expeced Reurns. The Journal of Finance. 51, Lamon O., Earnings and Expeced Reurns. Journal of Finance. 53, Leau, M. Ludvigson, S., 001. Resurrecing (C)CAPM: A Cross-Secional Tes When Risk Premia are Time-Varying. Journal of Poliical Economy. 109, Marakani, S., 009. A long-run Consumpion Risks: Are They There? Ciy Universiy of Hong Kong Working Paper. Mills, T. C. (1999). The Economeric Modelling of Financial Time Series, nd ediion, Cambridge Universiy Press, Cambridge (UK).

11 Newey, W. K., & Wes, K. D. (1987). Hypohesis esing wih efficien mehod of momens esimaion. Inernaional Economic Review, 8(3), Preminger, A., Hafner, C., 006. Deciding beween GARCH and Socasic Volailiy Via Srong Decision Rules. Monaser Cener of Economic Research, Ben Gurion Universiy of he Negev, working paper no Roll, R., A Criique of he Asse Pricing Theory s Tes: Par 1: On Pas and Poenial Tesabiliy of he Theory. Journal of Financial Economics. 4, Shanken, J., 199. On he Esimaion of Bea-Pricing Models. Review of Financial Sudies. 5, Taylor, S., Modeling Sochasic Volailiy: A Review and Comparaive Sudy. Mahemaical Finance. 4, Théore, R., and Racico, François-Éric., 010. Forecasing sochasic Volailiy using he Kalman filer: an applicaion o Canadian Ineres Raes and Price-Earnings Raio". Published in: Aesimaio. The IEB Inernaional Journal of Finance. 1, 1-0.

12 Appendix Table 1 Bansal-Yaron Esimaion Using GARCH Volailiy Consan (0.595) (0.751) (0.3187) Porfolio Reurns Big-Growh Big-Neural Small- Growh Small- Neural (0.4085) Small- Value (0.4007) Garch Marke Volailiy (0.9473).031 (0.1399) (0.7064) (0.7810) (0.777) Garch Consumpion Volailiy (0.5301) (0.8596) (0.3486) (0.4646) (0.4449) AR(1) (0.014) (0.0374) (0.1567) MA(1) (0.0089) (0.0443) Join Significance (p-value) *indicaes 10% significance **indicaes 5% significance *** indicaes 1% significance Sandard error correced by Newey-Wes

13 Table Bansal-Yaron Esimaion Using EGARCH Volailiy Consan (0.1381) (0.145) (0.6037) Porfolio Reurns Big-Growh Big-Neural Small- Growh Small- Neural (0.6319) Small- Value (0.5679) EGarch Marke Volailiy.617 (0.0554) 3.17 (0.048) 3.69 (0.0151) 3.79 (0.06) 6.57 (0.0691) EGarch Consumpion Volailiy (0.3538) (0.586) ( ) (0.10) -.86 ( ) AR(1) MA(1) (0.058) Join Significance (p-value) *indicaes 10% significance **indicaes 5% significance *** indicaes 1% significance Sandard error correced by Newey-Wes

14 Table 3 Bansal-Yaron Esimaion Using PGARCH Volailiy Consan (0.3473) (0.085) (0.6376) Porfolio Reurns Big-Growh Big-Neural Small- Growh Small- Neural (0.3719) Small- Value (0.490) PGarch Marke Volailiy.18 (0.6397) (0.0043).73 (0.0907) 8.31 (0.1650) (0.086) PGarch Consumpion Volailiy (0.3490) (0.4431) (0.3755) (0.936) ( ) AR(1) MA(1) (0.0175) Join Significance (p-value) *indicaes 10% significance **indicaes 5% significance *** indicaes 1% significance Sandard error correced by Newey-Wes

15 Table 4 Bansal-Yaron Esimaion Using Sochasic Volailiy Consan (0.5676) (0.438) (0.5861) Porfolio Reurns Big-Growh Big-Neural Small- Growh Small- Neural (0.5938) Small- Value (0.566) Sochasic Marke Volailiy (0.541) (0.438) 5.48 (0.0306) -3.7 (0.3096).946 ( ) Sochasic Consumpion Volailiy (0.8333) (0.7064) (0.7908) -.35 (0.664) -6. (0.714) AR(1) (0.0013) (0.01) (0.1157) MA(1) (0.0719) (0.1857) Join Significance (p-value) *indicaes 10% significance **indicaes 5% significance *** indicaes 1% significance Sandard error correced by Newey-Wes

16 GARCH Consumpion Volailiy EGARCH Consumpion Volailiy

17 PGARCH Consumpion Volailiy Sochasic Consumpion Volailiy

18 GARCH Marke Volailiy EGARCH Marke Volailiy

19 PGARCH Marke Volailiy Sochasic Marke Volailiy