Fitting the Heston Stochastic Volatility Model to Chinese Stocks

Transcription

1 Inernaional Finance and Banking 1, Vol. 1, No. 1 Fiing he Heson Sochasic Volailiy Model o Chinese Socks Ahme Goncu (Corresponding auhor) Dep. of Mahemaical Sciences, Xi an Jiaoong Liverpool Universiy Renai Road 111, Suzhou, 113, China Tel: Ahme.Goncu@xjlu.edu.cn Hao Yang Dep. of Mahemaical Sciences, Xi an Jiaoong Liverpool Universiy Renai Road 111, Suzhou, 113, China Received: May, 1 Acceped: June 16, 1 Published: June, 1 doi:1.96/ifb.v1i1.71 URL: hp://dx.doi.org/1.96/ifb.v1i1.71 Absrac In his aricle we invesigae he goodness-of-fi of he Heson sochasic volailiy model for he Shanghai composie index and five Chinese socks from differen indusries wih he highes rading volume. We have joinly esimaed he parameers of he Heson sochasic volailiy for he daily, weekly and monhly imescales model by employing a kernel densiy of he empirical reurns o minimize he mean-squared deviaions beween he heoreical and empirical reurn disribuions. We find ha he Heson model is able o characerize he empirical disribuion of Chinese sock reurns a he daily, weekly and monhly imescales. Keywords: Heson sochasic volailiy model, Goodness-of-fi es, Chinese socks, Kernel densiy 7

2 Inernaional Finance and Banking 1, Vol. 1, No Inroducion disribuion of sock reurns are ofen characerized by lepokurosis, which conflics wih he fundamenal assumpion of normaliy of log-reurns in he Black Scholes model. characerisics of sock reurns are ofen beer described by models ha allow for fa ails and high peaks. The use of sochasic processes ha allow for a wider range of shapes for reurn disribuions has been considered in he lieraure, where i is also well documened ha sochasic volailiy models or models based on jump-diffusion and pure jump processes significanly improve he Black-Scholes framework. Among ohers, some examples of hese models are provided by Meron (1976), Madan and Senea (199), Heson (1993), Barndorff-Nielsen (1997, 1998) and Kou (). In his aricle we invesigae he goodness-of-fi of he Heson (1993) model in he Chinese sock marke, which can be characerized by frequen exreme reurns and faer ails compared o developed sock markes. In his respec, he daase we consider also ess he robusness of he Heson model. To he bes of our knowledge here has been no sudy ha invesigaes he goodness-of-fi of he Heson sochasic volailiy model for Chinese socks. Furhermore, we propose he use of a non-parameric kernel densiy in he esimaion of he Heson model, which reduces he esimaion error of model parameers by smoohing he empirical densiy of log-reurns. The Heson model is widely used in modelling sock prices and he pricing of financial derivaives due o hree major advanages: 1) The consan volailiy assumpion of he Black Scholes model is no saisfied, and he implied volailiies of opion prices exhibi volailiy smiles. As he seminal work of Bakshi e al. (1997) has shown, sochasic volailiy should be incorporaed for pricing and inernal consisency, and sochasic volailiy modelling ofen yields he bes performance for hedging; ) The exisence of closed-form opion pricing formulas. In he Heson model, closed-form soluions for vanilla opions are given by he fas Fourier ransform mehod of Carr and Madan (1998), which leads o compuaionally efficien pricing; 3) The probabiliy disribuion for log-reurns under he Heson model is given in closed form by Dragulescu and Yakovenko (), which leads o efficien esimaion of model parameers from hisorical sock reurns. The goodness-of-fi of he Heson model o he hisorical daa of log-reurns has been esed in sudies by Dragulescu and Yakovenko (), Silva and Yakovenko (3) and Daniel e al. (), which ogeher show mixed resuls regarding he performance of he Heson model. Dragulescu and Yakovenko () have derived he closed form of he probabiliy disribuion of log-reurns in he Heson model o show ha he Heson model provides a good fi o he DowJones index reurns a differen ime inervals. However, as Daniel e al. () has poined ou, Dragulescu and Yakovenko () rimmed he daase by removing exreme price movemens. Wihou rimming, Daniel e al. () have shown ha he Heson model does no provide a good fi o he Dow Jones sock marke index. Meanwhile, Silva 7

3 Inernaional Finance and Banking 1, Vol. 1, No. 1 and Yakovenko (3) have verified he goodness-of-fi of he Heson model by explaining he NASDAQ, DowJones and S&P indices and by documening differen resuls for differen daase periods. Our resuls show ha wihou rimming he daase he Heson model fis he empirical disribuion of index and sock reurns in he Chinese sock marke well, especially for he daily log-reurns. Given he daase s frequen exreme evens, someimes he opimizaion rouine used in he parameer esimaion may fail o converge and alernaive iniial values migh be needed. This aricle is organized as follows. In he nex secion we briefly presen he Heson sochasic volailiy model and he probabiliy disribuion of log-reurns. In Secion III we presen he daase, while in Secion IV we discuss he parameer esimaion via disance minimizaion. Secion V presens he goodness-of-fi ess and finally Secion VI offers our conclusions as well as recommendaions for fuure work.. The Heson Model and he Probabiliy Disribuion of Log-Reurns In he Heson (1993) model, sock prices follow he sochasic differenial equaion ds S d S dw where is he drif, is he volailiy and W is a sandard Wiener process. Log-reurns are given by r S log S. Following Dragulescu and Yakovenko (), cenred log-reurns is given by x r while he dynamics of he cenred log-reurns is given by dx d dw. () This equaion denoes he variance by, which obeys he following Ornsein-Uhlenbeck process () d dw d, (3) () where is he long-erm mean of, is he rae of mean reversion, W is a sandard Wiener process and is he variance noise (vol-vol) parameer. In general, he Wiener processes in Equaions 1 and 3 may be correlaed and can be wrien as dw () dw 1 dz, () where Z is a Wiener process independen of W and [1,1 ] is he correlaion coefficien. Dragulescu and Yakovenko () have solved he forward Kolmogorov equaion ha governs he ime evoluion of he join probabiliy P x, ) given he iniial value of he 76 ( i

4 Inernaional Finance and Banking 1, Vol. 1, No. 1 variance ime lag : i o obain he following probabiliy disribuion of cenred log-reurns given a P 1 ip x xf ( px ) ( x) dp xe () wih F ( p x ) ln cosh sinh, (6) where ( px ip x ), ikpx where is he correlaion coefficien beween () wo Wiener processes W and W and,, and are he parameers of he Heson model. 3. Daa To verify he goodness-of-fi of he Heson model, we have uilised daily closing prices of he Shanghai composie index and five socks represening differen indusries wih he highes rading volume. Namely, we consider he sock prices of China Minsheng Banking Corp. (Banking, 616), Sinopec Group (Peroleum and Oil, 68), Jiangsu Sunshine Corp. (Clohing and manufacuring, 6), Shanghai Tongji Science & Technology Indusrial Corp. (Consrucion, 686), Chengdu B-ray Media Corp. (Media, 688). In Table 1 we supply descripive saisics for he daily log-reurns for boh he composie index and five socks wih heir Shanghai Sock Exchange codes. Table 1. Descripive saisics of he daily log-reurns of he Chinese socks and Shanghai composie index Mean SD Skewness Kurosis Min. Max. Shanghai Index Since for each sock he daes of he iniial public offerings differ, we have herefore considered he longes available hisorical daase for each sock. For he Shanghai composie index, we consider he period from 1 January 1998 o 6 May 13. Daily, weekly and monhly log-reurns are given by he following ime series. Daily log-reurns are calculaed as 77

5 Daily Inernaional Finance and Banking 1, Vol. 1, No. 1 Weekly S S for 1,,..., N. Weekly log-reurns we calculae R S / S R / ln 1 ln for 1,6,11,16,..., floor( N / ) and monhly log-reurns are calculaed Monhly as R S / S for 1,3,,..., floor( N / ). No inersecing observaions have ln 1 been used in he above calculaions, and in ranslaion weekly and monhly reurns are reaed as sock price pahs. The advanage of calculaing he reurns as above is discussed in Daniel e al. (). Daniel e al. () has poined ou ha Dragulescu and Yakovenko () rimmed he daase by, for example, discarding observaions ha exceeded % in daily reurns. Since rimming he daase reduces he fairness of he empirical analysis presened in Dragulescu and Yakovenko (), we do no rim he daase by removing exreme values in he daase.. Parameer Esimaion Parameer esimaion can be done uilizing hisorical log-reurns or opions daa or boh. For example in he sudy by Eraker () likelihood based inference is used o esimae sochasic volailiy models joinly using opions and sock price daa. Alhough, opion prices reflec informaion regarding he risk neural densiy, in China here are no raded equiy opions in an organized exchange. Under he Heson model sock reurns a differen imescales are assumed o have he same parameers. Therefore, minimizaion of he disance beween heoreical and empirical disribuions is a suiable approach o fi a single se of parameers for differen imescales joinly. disribuion of log-reurns is ofen calculaed by pariioning he space of log-reurns. As provided by Dragulescu and Yakovenko (), he domain of log-reurns can be pariioned ino equally spaced bins of widh r o allow simple couning of he number of log-reurns belonging o each bin. Relaive frequencies are hen obained by dividing hese frequencies by he oal sample size o yield he empirical disribuion P * ( x ). To obain model parameers, Dragulescu and Yakovenko () minimized he following objecive funcion wih respec o he parameers of he Heson model * Heson ln P ( x) ln P ( x) (7) x, where he sum is aken over all available x and. Since we work wih a smaller daase of log-reurns, we have considered he ime inervals 1,, which correspond o daily, weekly and monhly log-reurns. To improve he convergence of he opimizaion rouine used, we smoohed he empirical disribuion of log-reurns and revised he objecive funcion in Equaion 7 as max,, x, K * Heson P ( x) P ( x) s..,, (8) 78

6 where P * ( x) Inernaional Finance and Banking 1, Vol. 1, No. 1 K is he kernel densiy (Noe 1) calculaed from he empirical disribuion of log-reurns for each ime inerval 1,,. From Equaion 8, a single se of parameers fiing he Heson model o daily, weekly and monhly log-reurns joinly is obained. Tables and 3 supplies he esimaed parameers of he Heson model for he daily and join esimaion a he daily, weekly and monhly inervals. In Figure 1 we plo he fied Heson probabiliy disribuion funcion versus he empirical disribuion for daily, weekly and monhly log-reurns. Figure 1 hus shows ha he Heson model fis he daily log-reurns paricularly well. In Figures, 3, and we plo he fi of he Heson model for he highly raded socks a he Shanghai Sock Exchange. From hese figures i is clear ha he Heson model can characerize he behaviour of empirical log-reurns for a leas he daily ime inerval. Table. Esimaed parameers for he Heson model for daily log-reurns Parameers Shanghai Index Table 3. Esimaed parameers for he Heson model for he join esimaiona daily, weekly and monhly imescales Parameers Shanghai Index If Equaion 8 is maximized for only he daily log-reurns (i.e., = 1), hen all socks considered yield convergence in he opimizaion rouine. Reducing he number of ime inervals in Equaion 8 hus naurally improves he convergence and fi of he Heson model. 79

7 Inernaional Finance and Banking 1, Vol. 1, No. 1 Heson(daily) (daily) Heson Heson(weekly) 3 (weekly) Heson(monhly) 3 3 (monhly) Log-reurn Daily Log-reurn Heson 6 Heson Weekly Log-reurn Monhly Log-reurn Figure 1. Fied probabiliy densiy funcion of he Heson model versus he empirical disribuion of log-reurns for he Shanghai composie index Noe. Firs subplo presens he joinly fied Heson densiy for he daily, weekly, and monhly imescales, whereas oher subplos shows he fi of he Heson model for daily, weekly and monhly reurns separaely. Heson(daily) Heson (daily) Heson(weekly) (weekly) Heson(monhly) 1 1 (monhly) Log-reurn Daily Log-reurn 1. 1 Heson 3. Heson Weekly Log-reurn Monhly Log-reurn Figure. Fied probabiliy densiy funcion of he Heson model versus he empirical disribuion of log-reurns for sock 616 Noe. Firs subplo presens he joinly fied Heson densiy for he daily, weekly, and monhly imescales, whereas oher subplos shows he fi of he Heson model for daily, weekly and monhly reurns separaely. 8

8 Inernaional Finance and Banking 1, Vol. 1, No Heson(daily) Heson (daily) Heson(weekly) (weekly) Heson(monhly) 1 (monhly) Log-reurn Daily Log-reurn 1 1 Heson. Heson Weekly Log-reurn Monhly Log-reurn Figure 3. Fied probabiliy densiy funcion of he Heson model versus he empirical disribuion of log-reurns for sock 68 Noe. Firs subplo presens he joinly fied Heson densiy for he daily, weekly, and monhly imescales, whereas oher subplos shows he fi of he Heson model for daily, weekly and monhly reurns separaely. Heson(daily) Heson (daily) Heson(weekly) (weekly) Heson(monhly) 1 1 (monhly) Log-reurn Daily Log-reurn Heson 3. 3 Heson Weekly Log-reurn Monhly Log-reurn Figure. Fied probabiliy densiy funcion of he Heson model versus he empirical disribuion of log-reurns for sock 688 Noe. Firs subplo presens he joinly fied Heson densiy for he daily, weekly, and monhly imescales, whereas oher subplos shows he fi of he Heson model for daily, weekly and monhly reurns separaely. 81

9 Inernaional Finance and Banking 1, Vol. 1, No. 1. Goodness-of-Fi Tess Once he model parameers were esimaed for he common se of daily, weekly and monhly log-reurns, we esed he goodness-of-fi of he Heson model using via hree saisical ess. In his secion we commen briefly on he implemened ess and presen our resuls. The chi-square goodness-of-fi es is a discree goodness-of-fi es in which he range of observaions is divided ino k bins. For his es we have considered he bins,.,.1,.,.1,.1,.,.1,.,, (,.,.3,.,.17,.1,.,.1,,.1,.,.1,.17,.,.3,., ),.1,.8,.,.,.1,.1,.,.,.8,.1,, for he daily, weekly and monhly ime inervals, respecively. The above inervals were chosen so ha he observed frequencies in each bin would be similar. The degree of freedom of he chi-square es equals dof # bins 1 m, where m is he number of parameers of he model being esed. The corresponding criical values a a 9% confidence level for he chi-square goodness-of-fi es of he Heson model a daily, weekly and monhly ime inervals are given as 11.7, and 1.7, respecively. As he benchmark case for he normal disribuion, he criical values are given as 1.7, 1.3 and 1.1, for daily, weekly and monhly ime inervals, respecively. The Anderson Darling (AD) (19) goodness-of-fi es provides a good measure of disance beween empirical and heoreical densiies. Therefore, a smaller es suggess a beer fi o he daa, whereas he Kolmogorov Smirnov (KS) es measures he maximal discrepancy beween he expeced and observed cumulaive disribuions of log-reurns. To calculae he KS saisic, we used he empirical and heoreical cumulaive disribuion funcions and compued he maximum discrepancy beween hem. Since he esimaors and goodness-of-fi es saisics were calculaed from he same daase, boh AD and KS es saisics are no sufficien o accep he esed model. Chi-square es does no presen his problem, hus criical values are available o es he null hypohesis. In Tables, and 6 we presen he goodness-of-fi es resuls for he daily, weekly and monhly log-reurns, respecively. disribuion, which was used as a benchmark scenario, is rejeced by he chi-square goodness-of-fi es for all socks and he composie index. 8

10 Inernaional Finance and Banking 1, Vol. 1, No. 1 Table. Chi-square, Anderson-Darling and Kolmogorov-Smirnov goodness-of-fi es resuls for daily log-reurns, where criical values a a 9% confidence level for he chi-square goodness-of-fi es of he Heson model and normal disribuion are given as 11.7 and 1.7, respecively Chi-square Anderson Darling Kolmogorov Smirnov Heson Heson Heson Shanghai Index 7.6* * * * * 1.77* * Noe. *Rejeced a he 9% confidence level. Table shows ha he Heson model canno be rejeced a a 9% confidence level for he daily log-reurns, whereas he normal disribuion can clearly be rejeced. AD saisics indicae ha he disance beween he empirical and heoreical disribuions is small. In Table, we presen resuls for he weekly log-reurns, for which he Heson model also provides a good fi excep for he Shanghai composie index. Table 6 also shows ha he Heson model has a good fi for monhly log-reurns. Excep for monhly log-reurns, normal disribuion can be consisenly rejeced for all socks and he composie index. Table. Chi-square, Anderson-Darling and Kolmogorov-Smirnov goodness-of-fi es resuls for weekly log-reurns, where criical values a a 9% confidence level for he chi-square goodness-of-fi es of he Heson model and normal disribuion are given as and 1.3, respecively Chi-square Anderson Darling Kolmogorov Smirnov Heson Heson Heson Shanghai Index 6.61* 31.77* * * * * * Noe. *Rejeced a he 9% confidence level. 83

11 Inernaional Finance and Banking 1, Vol. 1, No. 1 Table 6. Chi-square, Anderson Darling and Kolmogorov Smirnov goodness-of-fi es resuls for monhly log-reurns, where criical values a a 9% confidence level for he chi-square goodness-of-fi es of he Heson model and normal disribuion are given as 1.7 and 1.1, respecively Chi-square Anderson Darling Kolmogorov Smirnov Heson Heson Heson Shanghai Index Noe. *Rejeced a he 9% confidence level. 6. Conclusion and Fuure Work We invesigaed he goodness-of-fi of he Heson sochasic volailiy model o he empirical disribuion of sock and index reurns in he Chinese sock marke. This aricle shows ha he Heson model provides a good fi for Chinese socks, especially for heir daily log-reurns. I should be noed ha we fi a single se of model parameers ha is able o fi o he daily, weekly and monhly log-reurns simulaneously. Overall, he goodness-of-fi es saisics provide evidence ha he Heson model canno be rejeced, especially for daily log-reurns. However, one drawback of using he Heson model is he difficuly in he convergence of parameer esimaion, which may be aribued o frequen exreme movemens in Chinese sock reurns. To improve he parameer esimaion opimizaion, we smoohed he empirical disribuion of log-reurns via a kernel densiy. Fuure work should focus on wha hese resuls imply for risk managemen. Since he Heson model can capure he heavy-ailed empirical disribuions of Chinese socks, i migh also perform well in esimaing quaniles and risk measures, such as he value-a-risk. References Anderson, T. W., & Darling, D. A. (19). Asympoic heory of cerain goodness-of-fi crieria based on sochasic processes. Annals of Mahemaical Saisics, 3, hp://dx.doi.org/1.11/aoms/ Bakshi, G., Cao, C., & Chen, Z. (1997). performance of alernaive opion pricing models. Journal of Finance,, 3 9. hp://dx.doi.org/1.1111/j b79.x Barndorff-Nielsen, O. (1997). inverse Gaussian disribuions and sochasic volailiy modelling. Scandinavian Journal of Saisics,, hp://dx.doi.org/1.1111/ Barndorff-Nielsen, O. (1998). Processes of normal inverse Gaussian ype. Finance and 8

12 Sochasics,, hp://dx.doi.org/1.17/s783 Inernaional Finance and Banking 1, Vol. 1, No. 1 Carr, P., & Madan, D. B. (1998). Opion valuaion using he fas Fourier ransform. Journal of Compuaional Finance,, Daniel, G., Joseph, N. L., & Bree, D. S. (). Sochasic volailiy and he goodness-of-fi of he Heson model. Quaniaive Finance,, hp://dx.doi.org/1.18/ Dragulescu, A. A., & Yakovenko, V. M. (). Probabiliy disribuion of reurns in he Heson model wih sochasic volailiy. Quaniaive Finance,, 3 3. hp://dx.doi.org/1.18/ Eraker, B. (). Do Sock Prices and Volailiy Jump? Reconciling Evidence from Spo and Opion Prices. Journal of Finance, 9, hp://dx.doi.org/1.1111/j x Heson, S. L. (1993). A closed-form soluion for opions wih sochasic volailiy wih applicaions o bond and currency opions. Review of Financial Sudies, 6, hp://dx.doi.org/1.193/rfs/6..37 Kou, S. G. (). A jump-diffusion model for opion pricing. Managemen Science, 8, hp://dx.doi.org/1.187/mnsc Madan, D. B., & Senea, E. (199). On modelling and pricing weaher derivaives. The Journal of Business, 63, 11. hp://dx.doi.org/1.186/9619 Meron, R. (1976). Opion pricing when underlying sock reurns are disconinuous. Journal of Financial Economics, 3, 1 1. hp://dx.doi.org/1.116/3-x(76)9- Silva, A. C., & Yakovenko, V. M. (3). Comparison beween he probabiliy disribuion of reurns in he Heson model and empirical daa for sock indexes. Physica A, 3, hp://dx.doi.org/1.116/s ()193-9 Noe Noe 1. Kernel densiy smoohes he empirical disribuion of log-reurns; he MATLAB funcion ksdensiy(.) is implemened. Once a smoohed empirical probabiliy disribuion is obained, we maximize he objecive funcion in Equaion 8 by employing he consrained maximizaion funcion fmincons(.) in MATLAB. Copyrigh Disclaimer Copyrigh for his aricle is reained by he auhor(s), wih firs publicaion righs graned o he journal. This is an open-access aricle disribued under he erms and condiions of he Creaive Commons Aribuion license (hp://creaivecommons.org/licenses/by/3./). 8