多元GARCH-Ito^模型及其在高维波动率矩阵预测中的应用

doi:10.16381/j.cnki.issn1003-207x.2023.1100

Abstract

Abstract:

A novel multivariate GARCH-It $o^$ model is introduced that integrates the structural framework of multivariate GARCH within a continuous-time diffusion process， providing a unified approach to modeling the dynamic evolution of volatility matrices by jointly utilizing high-frequency and low-frequency data. A quasi-maximum likelihood estimation method for parameter inference is developed and the corresponding asymptotic theory is established. To address the challenges associated with high-dimensional asset spaces， the proposed model is coupled with a factor structure， enabling scalable and efficient prediction of large integrated volatility matrices. Theoretical guarantees for the proposed predictor are provided under high-dimensional settings. Extensive simulation studies are conducted to examine the finite-sample performance of both the estimation and prediction procedures across both low-dimensional and high-dimensional contexts. In an empirical application， 270 constituent stocks of the CSI 300 index are analyzed using minute-level high-frequency data from January 1， 2018， to December 31， 2020. The results demonstrate that the multivariate GARCH-It $o^$ model consistently outperforms several benchmark methods in terms of integrated volatility matrix forecasting and portfolio allocation， offering a flexible and unified framework that incorporates both high- and low-frequency data features for advanced volatility modeling and prediction.

Key words: high-frequency data, dynamic structure, quisi-maximum likelihood method, integrated volatility matrix, factor model

CLC Number:

O212

Xinyu Song,Yuanyuan Deng,Yong Zhou, et al. Multivariate GARCH-It $o^$ Model with Applications in High-dimension Volatility Matrix Prediction[J]. Chinese Journal of Management Science, 2026, 34(7): 33-48.

Figures/Tables 11

$m$	模型参数估计 $θ ̂$ 的平均绝对离差 $× 100$
$m$	$ω 1,1$	$ω 2,1$	$ω 3,1$	$ω 2,2$	$ω 3,2$	$ω 3,3$
240	0.237	0.164	0.132	0.486	0.339	0.803
480	0.185	0.135	0.115	0.353	0.227	0.517
1440	0.127	0.107	0.089	0.253	0.166	0.300
14400	0.052	0.058	0.045	0.139	0.092	0.123
	$γ 1,1$	$γ 2,1$	$γ 3,1$	$γ 2,2$	$γ 3,2$	$γ 3,3$
240	3.942	9.219	8.082	2.198	5.257	12.368
480	3.328	7.392	6.537	1.623	3.493	8.350
1440	2.579	5.472	4.703	0.996	2.089	5.161
14400	1.494	3.164	2.582	0.514	0.893	2.242
	$β 1,1$	$β 2,1$	$β 3,1$	$β 2,2$	$β 3,2$	$β 3,3$
240	2.892	4.948	3.846	1.954	1.849	4.437
480	2.230	4.098	3.259	1.549	1.471	2.945
1440	1.493	3.050	2.287	1.165	1.139	2.106
14400	0.869	1.770	1.352	0.686	0.602	1.001
	$μ 1$	$μ 2$	$μ 3$
240	1.043	2.197	2.577
480	0.823	1.820	2.258
1440	0.626	1.441	1.730
14400	0.332	0.780	0.921

范数	$m$	多元GARCH-It $o ̂$ 模型	PRVM方法	BEKK模型
谱范数 $× 100$	240	2.486	6.365	10.866
	480	1.824	5.432	10.414
	1440	1.078	4.859	9.645
	14400	0.434	3.615	10.047
Frobenius 范数 $× 100$	240	2.766	7.065	11.602
	480	2.037	6.004	11.109
	1440	1.186	5.329	10.355
	14400	0.466	3.931	10.745
最大范数 $× 100$	240	2.240	5.033	8.520
	480	1.651	4.335	7.984
	1440	0.941	3.872	7.627
	14400	0.358	2.838	7.850
相对Frobenius 范数 $× 100$	240	2.955	15.587	47.287
	480	1.500	11.985	48.994
	1440	1.480	8.904	42.427
	14400	0.054	4.826	71.902

$m$	模型参数估计 $θ ̂$ 的平均绝对离差 $× 100$
$m$	$ω 1,1$	$ω 2,1$	$ω 3,1$	$ω 2,2$	$ω 3,2$	$ω 3,3$
240	0.243	0.228	0.222	0.822	0.57	0.629
480	0.215	0.193	0.167	0.664	0.433	0.543
1440	0.203	0.140	0.127	0.494	0.345	0.447
8640	0.198	0.087	0.073	0.293	0.219	0.324
	$γ 1,1$	$γ 2,1$	$γ 3,1$	$γ 2,2$	$γ 3,2$	$γ 3,3$
240	7.157	11.873	12.552	4.339	5.983	16.327
480	5.847	9.094	9.321	2.982	4.941	13.521
1440	4.278	6.803	7.095	2.095	3.675	9.150
8640	2.526	4.174	4.039	1.137	2.204	4.935
	$β 1,1$	$β 2,1$	$β 3,1$	$β 2,2$	$β 3,2$	$β 3,3$
240	3.061	5.537	4.067	2.583	2.046	4.086
480	2.537	4.221	3.331	2.060	1.870	3.507
1440	1.905	3.314	2.757	1.637	1.386	2.338
8640	1.110	2.161	1.710	0.923	0.951	1.415
	$μ 1$	$μ 2$	$μ 3$
240	1.124	2.348	2.948
480	0.973	1.801	2.620
1440	0.714	1.534	1.993
8640	0.450	0.979	1.275

范数	$m$	多元GARCH-It $o ̂$ 模型	因子GARCH-It $o ̂$ 模型	POET方法	PRVM模型	SV-It $o ̂$ 模型
谱范数	240	2.040	8.111	7.300	7.300	3.277
	480	1.628	7.683	6.634	6.634	2.434
	1440	0.991	6.154	5.262	5.258	1.986
	8640	0.600	6.021	4.389	4.381	1.548
Frobenius范数	240	2.333	9.598	8.786	8.937	3.504
	480	1.929	8.850	7.842	7.965	2.874
	1440	1.340	7.063	6.259	6.334	2.478
	8640	1.023	6.604	5.079	5.073	2.172
最大范数	240	0.096	0.366	0.331	0.338	0.148
	480	0.085	0.332	0.292	0.302	0.135
	1440	0.066	0.261	0.232	0.239	0.115
	8640	0.054	0.234	0.182	0.190	0.101
相对Frobenius范数	240	0.534	1.571	1.345	5.297	0.685
	480	0.642	1.213	1.103	3.821	0.756
	1440	0.808	1.048	1.015	2.251	0.811
	8640	0.941	1.014	1.006	0.935	0.968

参数	预测区间	多元GARCH-It $o ̂$					因子 GARCH-It $o ̂$	POET	PRVM	SV-It $o ̂$
参数	预测区间	$r = 1$	$r = 2$	$r = 3$	$r = 4$	$r = 5$	因子 GARCH-It $o ̂$	POET	PRVM	SV-It $o ̂$
$‖ Γ ˜ k - R V k ‖ m a x × 100$	2020年1月	0.317	0.322	0.319	0.317	0.327	0.379	0.372	0.372	0.325
	2020年2月	0.435	0.45	0.451	0.451	0.461	0.399	0.408	0.411	0.431
	2020年3月	0.369	0.404	0.400	0.401	0.409	0.398	0.443	0.443	0.410
	2020年4月	0.242	0.248	0.247	0.246	0.25	0.286	0.277	0.277	0.251
	2020年5月	0.219	0.234	0.220	0.221	0.228	0.295	0.295	0.295	0.234
	2020年6月	0.255	0.253	0.260	0.261	0.265	0.319	0.291	0.291	0.270
	2020年7月	0.544	0.557	0.569	0.569	0.572	0.589	0.591	0.591	0.570
	2020年8月	0.448	0.453	0.459	0.458	0.46	0.374	0.382	0.383	0.465
	2020年9月	0.309	0.313	0.321	0.318	0.322	0.369	0.367	0.366	0.330
	2020年10月	0.295	0.300	0.301	0.301	0.302	0.356	0.353	0.354	0.312
	2020年11月	0.272	0.278	0.278	0.279	0.277	0.268	0.257	0.257	0.270
	2020年12月	0.291	0.295	0.300	0.300	0.296	0.319	0.311	0.312	0.305
$‖ Γ ˜ k - 1 - R V k - 1 ‖$	2020年1月	2683637	2683650	2683670	2683662	2683663	14278552	13865714	4536163	3196557
	2020年2月	1215402	1215431	1215449	1215465	1215465	8825641	7775543	1995877	1723084
	2020年3月	1350027	1350036	1350031	1350036	1350037	6090200	5967906	2067773	1873059
	2020年4月	1531613	1531630	1531631	1531646	1531659	17445132	17749757	2439849	2194860
	2020年5月	1444130	1444160	1444179	1444199	1444193	7510691	7573684	2175766	1906547
	2020年6月	2240633	2240623	2240613	2240609	2240618	10843838	11009881	3713105	2629829
	2020年7月	1784926	1784974	1784984	1785051	1785057	6197296	6182715	3279139	2328066
	2020年8月	2146406	2146426	2146437	2146448	2146456	11184674	11467610	3715848	2695563
	2020年9月	1256194	1256199	1256195	1256211	1256212	30121488	29998299	1988675	1819442
	2020年10月	1728144	1728168	1728177	1728218	1728212	16069270	16447190	2654057	2269224
	2020年11月	2452643	2452663	2452670	2452678	2452692	27518090	128795320	4056123	2904073
	2020年12月	1473267	1473286	1473290	1473300	1473305	15853939	16214797	2187645	1956125

预测误差方差	预测区间	多元GARCH-It $o ̂$					因子GARCH-It $o ̂$	POET	PRVM	SV-It $o ̂$
预测误差方差	预测区间	$r = 1$	$r = 2$	$r = 3$	$r = 4$	$r = 5$	因子GARCH-It $o ̂$	POET	PRVM	SV-It $o ̂$
预测误差方差 $×$ 1000	2020年1月	0.358	0.344	0.318	0.326	0.273	0.344	0.325	0.325	0.323
	2020年2月	0.469	0.351	0.321	0.328	0.278	0.576	0.657	0.657	0.326
	2020年3月	0.544	0.353	0.323	0.323	0.279	0.576	0.68	0.68	0.327
	2020年4月	0.397	0.347	0.321	0.317	0.286	0.367	0.344	0.344	0.325
	2020年5月	0.378	0.347	0.319	0.315	0.29	0.328	0.292	0.292	0.323
	2020年6月	0.363	0.348	0.316	0.313	0.284	0.333	0.285	0.285	0.317
	2020年7月	0.558	0.368	0.317	0.313	0.29	0.738	0.814	0.814	0.319
	2020年8月	0.43	0.356	0.322	0.318	0.296	0.48	0.504	0.504	0.328
	2020年9月	0.405	0.357	0.321	0.317	0.303	0.374	0.364	0.364	0.327
	2020年10月	0.379	0.357	0.32	0.316	0.306	0.356	0.325	0.325	0.327
	2020年11月	0.38	0.356	0.32	0.315	0.31	0.438	0.416	0.416	0.324
	2020年12月	0.379	0.356	0.32	0.314	0.318	0.387	0.377	0.377	0.330

预测区间	多元GARCH-It $o ̂$					因子GARCH- It $o ̂$	POET	PRVM	SV-It $o ̂$
预测区间	$r = 1$	$r = 2$	$r = 3$	$r = 4$	$r = 5$	因子GARCH- It $o ̂$	POET	PRVM	SV-It $o ̂$
2020年1月	0.107	0.093	0.08	0.084	0.088	0.861	0.862	0.61	0.326
2020年2月	0.14	0.145	0.129	0.142	0.156	0.801	0.782	0.944	0.404
2020年3月	0.169	0.177	0.179	0.203	0.221	1.337	1.343	0.245	0.381
2020年4月	0.125	0.109	0.087	0.095	0.096	1.072	1.074	0.259	0.379
2020年5月	0.122	0.114	0.079	0.081	0.085	0.554	0.552	0.156	0.328
2020年6月	0.123	0.101	0.077	0.085	0.081	0.985	0.981	0.23	0.310
2020年7月	0.167	0.19	0.177	0.223	0.207	1.59	1.589	0.393	0.428
2020年8月	0.135	0.151	0.135	0.153	0.148	1.271	1.273	0.337	0.471
2020年9月	0.133	0.117	0.111	0.129	0.118	1.076	1.077	0.284	0.490
2020年10月	0.12	0.096	0.093	0.108	0.095	0.84	0.838	0.68	0.408
2020年11月	0.124	0.109	0.095	0.096	0.102	0.867	0.867	0.192	0.346
2020年12月	0.128	0.093	0.088	0.092	0.093	0.991	0.995	0.316	0.362

References 51

[1]	Engle R F. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation［J］. Econometrica， 1982， 50（4）： 987.
[2]	Bollerslev T. Generalized autoregressive conditional heteroskedasticity［J］. Journal of Econometrics， 1986， 31（3）： 307-327.
[3]	Zhang L， Mykland P A， Aït-Sahalia Y. A tale of two time scales： Determining integrated volatility with noisy high-frequency data［J］. Journal of the American Statistical Association， 2005， 100（472）： 1394-1411.
[4]	Zhang L. Efficient estimation of stochastic volatility using noisy observations： A multi-scale approach［J］. Bernoulli， 2006， 12（6）： 1019-1043.
[5]	Zhang L. Estimating covariation： Epps effect， microstructure noise［J］. Journal of Econometrics， 2011， 160（1）： 33-47.
[6]	Barndorff-Nielsen O E， Hansen P R， Lunde A， et al. Designing realised kernels to measure the ex-post variation of equity prices in the presence of noise［J］. SSRN Electronic Journal， 2008，76： 1481-1536.
[7]	Barndorff-Nielsen O E， Hansen P R， Lunde A， et al. Multivariate realised kernels： Consistent positive semi-definite estimators of the covariation of equity prices with noise and non-synchronous trading［J］. Journal of Econometrics， 2011， 162（2）： 149-169.
[8]	Jacod J， Li Y， Mykland P A， et al. Microstructure noise in the continuous case： The pre-averaging approach［J］. Stochastic Processes and Their Applications， 2009， 119（7）： 2249-2276.
[9]	Christensen K， Kinnebrock S， Podolskij M. Pre-averaging estimators of the ex-post covariance matrix in noisy diffusion models with non-synchronous data［J］. Journal of Econometrics， 2010， 159（1）： 116-133.
[10]	Aït-Sahalia Y， Fan J， Xiu D. High-frequency covariance estimates with noisy and asynchronous financial data［J］. Journal of the American Statistical Association， 2010， 105（492）： 1504-1517.
[11]	Xiu D. Quasi-maximum likelihood estimation of volatility with high frequency data［J］. Journal of Econometrics， 2010， 159（1）： 235-250.
[12]	Ghysels E， Santa-Clara P， Valkanov R. Predicting volatility： Getting the most out of return data sampled at different frequencies［J］. Journal of Econometrics， 2006， 131（1-2）： 59-95.
[13]	Engle R F， Gallo G M. A multiple indicators model for volatility using intra-daily data［J］. Journal of Econometrics， 2006， 131（1-2）： 3-27.
[14]	Corsi F. A simple approximate long-memory model of realized volatility［J］. Journal of Financial Econometrics， 2009， 7（2）： 174-196.
[15]	Shephard N， Sheppard K. Realising the future： Forecasting with high-frequency-based volatility （HEAVY） models［J］. Journal of Applied Econometrics， 2010， 25（2）： 197-231.
[16]	Hansen P R， Huang Z， Shek H H. Realized GARCH： A joint model for returns and realized measures of volatility［J］. Journal of Applied Econometrics， 2012， 27（6）： 877-906.
[17]	Kim D， Wang Y. Unified discrete-time and continuous-time models and statistical inferences for merged low-frequency and high-frequency financial data［J］. Journal of Econometrics， 2016， 194（2）： 220-230.
[18]	Song X Y， Kim D， Yuan H L， et al. Volatility analysis with realized GARCH-Itô models［J］. Journal of Econometrics， 2021， 222（1）： 393-410.
[19]	Yuan H L， Zhou Y， Zhang Z Y， et al. Volatility analysis for the GARCH-Itô model with option data［J］. Canadian Journal of Statistics， 2024， 52（1）： 237-270.
[20]	Yuan H L， Lu K X， Li G D. Volatility analysis with high-frequency and low-frequency historical data， and options-implied information［J］. Statistica Sinica， 2025， 35： 2305-2323.
[21]	Wang Y， Zou J. Vast volatility matrix estimation for high-frequency financial data［J］. The Annals of Statistics， 2010， 38（2）： 943-978.
[22]	Tao M， Wang Y， Chen X. Fast convergence rates in estimating large volatility matrices using high-frequency financial data［J］. Econometric Theory， 2013， 29（4）： 838-856.
[23]	Cai T， Liu W. Adaptive thresholding for sparse covariance matrix estimation［J］. Journal of the American Statistical Association， 2011， 106（494）： 672-684.
[24]	Kim D， Wang Y， Zou J. Asymptotic theory for large volatility matrix estimation based on high-frequency financial data［J］. Stochastic Processes and Their Applications， 2016， 126（11）： 3527-3577.
[25]	穆燕，苑慧玲，周勇. 高频金融数据的高维积分波动率矩阵估计［J］.中国科学：数学，2018，48（2）： 319-344.
	Mu Y， Yuan H L， Zhou Y. High-dimensional integrated volatility matrix estimation for high-frequency financial data［J］. Scientia Sinica （Mathematica）， 2018， 48（2）： 319-344.
[26]	穆燕，周勇. 带跳高频数据下高维积分波动率矩阵估计［J］. 中国科学：数学， 2020， 50（10）： 1455-1486.
	Mu Y， Zhou Y. High-dimensional integrated volatility matrix estimation for high-frequency financial data with jumps［J］. Scientia Sinica （Mathematica）， 2020， 50（10）： 1455-1486.
[27]	Fan J， Fan Y， Lv J. High dimensional covariance matrix estimation using a factor model［J］. Journal of Econometrics， 2008， 147（1）： 186-197.
[28]	Fan J Q， Liao Y， Mincheva M. Large covariance estimation by thresholding principal orthogonal complements［J］. Journal of the Royal Statistical Society Series B： Statistical Methodology， 2013， 75（4）： 603-680.
[29]	Aït-Sahalia Y， Xiu D C. Using principal component analysis to estimate a high dimensional factor model with high-frequency data［J］. Journal of Econometrics， 2017， 201（2）： 384-399.
[30]	Fan J Q， Furger A， Xiu D C. Incorporating global industrial classification standard into portfolio allocation： A simple factor-based large covariance matrix estimator with high-frequency data［J］. Journal of Business & Economic Statistics， 2016， 34（4）： 489-503.
[31]	Dai C， Lu K， Xiu D C. Knowing factors or factor loadings， or neither？ Evaluating estimators of large covariance matrices with noisy and asynchronous data［J］. Journal of Econometrics， 2019， 208（1）： 43-79.
[32]	Kim D， Fan J. Factor GARCH-Itô models for high-frequency data with application to large volatility matrix prediction［J］. Journal of Econometrics， 2019， 208（2）： 395-417.
[33]	Aït-Sahalia Y， Kalnina I， Xiu D C. High-frequency factor models and regressions［J］. Journal of Econometrics， 2020， 216（1）： 86-105.
[34]	Fan J Q， Ke Y， Wang K. Factor-adjusted regularized model selection［J］. Journal of Econometrics， 2020， 216（1）： 71-85.
[35]	Engle R F， Kroner K F. Multivariate simultaneous generalized ARCH［J］. Econometric Theory， 1995， 11（1）： 122-150.
[36]	Bollerslev T， Engle R F， Wooldridge J M. A capital asset pricing model with time-varying covariances［J］. Journal of Political Economy， 1988， 96（1）： 116-131.
[37]	Aït-Sahalia Y， Yu J L. High frequency market microstructure noise estimates and liquidity measures［J］. The Annals of Applied Statistics， 2009， 3（1）： 422-457.
[38]	Aït-Sahalia Y， Xiu D. A Hausman test for the presence of market microstructure noise in high frequency data［J］. Journal of Econometrics， 2019， 211（1）： 176-205.
[39]	Kim D. Statistical inference for unified garch–itô models with high-frequency financial data［J］. Journal of Time Series Analysis， 2016， 37（4）： 513-532.
[40]	Kim D， Liu Y， Wang Y. Large volatility matrix estimation with factor-based diffusion model for high-frequency financial data［J］. Bernoulli， 2018， 24（4B）： 3657-3682.
[41]	Kim D， Song X Y， Wang Y. Unified discrete-time factor stochastic volatility and continuous-time Itô models for combining inference based on low-frequency and high-frequency［J］. Journal of Multivariate Analysis， 2022， 192： 105091.
[42]	Candès E J， Li X， Ma Y W， et al. Robust principal component analysis？［J］. Journal of the ACM， 2011， 58（3）： 1-37.
[43]	Fan J Q， Wang W C， Zhong Y Q. An l ∞ Eigenvector Perturbation Bound and Its Application to Robust Covariance Estimation ［J］. Journal of Machine Learning Resarch， 2018， 18： 1-42.
[44]	Fan J Q， Kim D. Robust high-dimensional volatility matrix estimation for high-frequency factor model［J］. Journal of the American Statistical Association， 2018， 113（523）： 1268-1283.
[45]	Tao M J， Wang Y Z， Zhou H H. Optimal sparse volatility matrix estimation for high-dimensional Itô processes with measurement errors［J］. The Annals of Statistics， 2013， 41（4）： 1816-1864.
[46]	Wang Y Z， Zou J. Volatility analysis in high-frequency financial data［J］. WIREs Computational Statistics， 2014， 6（6）： 393-404.
[47]	Cai T T， Hu J C， Li Y Y， et al. High-dimensional minimum variance portfolio estimation based on high-frequency data［J］. Journal of Econometrics， 2020， 214（2）： 482-494.
[48]	Engle R. Dynamic conditional correlation： A simple class of multivariate generalized autoregressive conditional heteroskedasticity models［J］. Journal of Business & Economic Statistics， 2002， 20（3）： 339-350.
[49]	Dhaene G， Wu J. Incorporating overnight and intraday returns into multivariate GARCH volatility models［J］. Journal of Econometrics， 2020， 217（2）： 471-495.
[50]	Linton O， Wu J. A coupled component DCS-EGARCH model for intraday and overnight volatility［J］. Journal of Econometrics， 2020， 217（1）： 176-201.
[51]	Kim D， Shin M， Wang Y Z. Overnight GARCH-itô volatility models［J］. Journal of Business & Economic Statistics， 2023， 41（4）： 1215-1227.

Multivariate GARCH-It $o^$ Model with Applications in High-dimension Volatility Matrix Prediction

RichHTML

PDF (PC)

Knowledge

Abstract

Cite this article

share this article

Figures/Tables 11

References 51

Related Articles 15

Metrics

Comments

Recommended 0

[1]	Xuanming Ni,Tiantian Zheng,Huimin Zhao,Kangping Wu. Asset Pricing Based on the Optimal Idiosyncratic Return Factor [J]. Chinese Journal of Management Science, 2024, 32(8): 50-60.
[2]	Guang-lin HUANG,Wan-bo LU. High Dimensional Dynamic Higher-order Portfolio Selection Based on the Varying-coefficient Multi-factor Semi-nonparametric Distribution Model [J]. Chinese Journal of Management Science, 2023, 31(12): 272-280.
[3]	LIANG Chao, WEI Yu, MA Feng, LI Xia-fei. Forecasting Volatility of China Gold Futures Price: New Evidence from Model Shrinkage Methods [J]. Chinese Journal of Management Science, 2022, 30(4): 30-41.
[4]	WU Xin-yu, LI Xin-dan, MA Chao-qun. Measuring VaR Based on the Information Content of Option and High-frequency Data [J]. Chinese Journal of Management Science, 2021, 29(8): 13-23.
[5]	QU Hui, ZHANG Yi. The Study of High-dimensional Volatility Estimators and Forecasting Models based on Volatility Timing Performance [J]. Chinese Journal of Management Science, 2020, 28(5): 62-70.
[6]	LIU Zhi-dong, ZHENG Xue-fei. A Study of Stock Price Co-jumps with Hawkes Factor Model [J]. Chinese Journal of Management Science, 2018, 26(7): 18-31.
[7]	ZHAO Ning, YU Fang-kun, YOU Shen, WANG Zhen-shuang. Investment Horizon, System Risk Value and the Sensitive Effect [J]. Chinese Journal of Management Science, 2018, 26(1): 72-80.
[8]	SONG Ya-qiong, WANG Xin-jun. Modeling and Forecasting Volatility of Chinese Stock Market Based on Dynamic Estimation Errors [J]. Chinese Journal of Management Science, 2017, 25(9): 19-27.
[9]	WU Xin-yu, LI Xin-dan, MA Chao-qun. Non-affine Option Pricing in the Presence of Microstructure Noises: An Empirical Study Based on the High-frequency Shanghai 50ETF Options Data [J]. Chinese Journal of Management Science, 2017, 25(12): 99-108.
[10]	LIU Zhi-dong, YAN Guan. Analysis of the Finer Statistical Characteristics of China Stock Market Based on Semimartingales Process [J]. Chinese Journal of Management Science, 2016, 24(5): 18-30.
[11]	ZHAO Jun-li, LIANG Xun. An Improvement on the Estimate of Realized Variance of Stock Yield Based on Transaction Time Sampling [J]. Chinese Journal of Management Science, 2015, 23(7): 26-34.
[12]	GE Jing, TIAN Xin-shi. Models and Empirical Research of Term Structure of Interest Rates in China: Based on No-Arbitrage DNS Model and DNS Model [J]. Chinese Journal of Management Science, 2015, 23(2): 29-38.
[13]	ZHAO Shu-ran, JIANG Ya-ping, REN Pen-min. Comparing Estimators of the High-Frequency Volatility Matrix in the Presence of Non-synchronous Trading and Market Microstructure Noise [J]. Chinese Journal of Management Science, 2015, 23(10): 19-29.
[14]	JIANG Cui-xia, ZHANG Shi-ying. Research on Spillover Effects in Financial Higher Moments Risk [J]. Chinese Journal of Management Science, 2009, 17(1): 17-28.
[15]	HE Yan-lin. An Improvement of Fama French Three-Factor Model Based on State Switch Informations [J]. Chinese Journal of Management Science, 2008, 16(1): 7-15.

Multivariate GARCH-Ito^Model with Applications in High-dimension Volatility Matrix Prediction

RichHTML

PDF (PC)

Knowledge

Abstract

Cite this article

share this article

Figures/Tables 11

References 51

Related Articles 15

Metrics

Comments

Recommended 0

Multivariate GARCH-It $o^$ Model with Applications in High-dimension Volatility Matrix Prediction