Volatility in financial markets has attracted a great deal of attention from academics, policy makers and practitioners during the past decades, primarily because it plays a crucial role in many financial applications, such as portfolio selection, option pricing and risk management. It has been well-documented in the finance literature that the financial market volatility is not a constant but in fact changes over time and financial time series exhibits volatility clustering. In addition, many empirical researches have indicated that the mean and volatility of asset returns respond asymmetrically to market news. Recently, with the development of information technology, the availability of high frequency data means that it has become possible to measure the latent volatility using the so-called realized volatility (RV) measure. Under some assumptions, the RV is a consistent estimator of the true volatility. In the real market, however, the RV measure computed from high frequency return data suffers from microstructure noise. In this paper, both the threshold effects and realized volatility measure are incorporated into the standard stochastic volatility (SV) model and the threshold realized SV (TRSV) model is proposed to model the volatility of asset returns. The model is able to account for time-varying volatility and volatility clustering and capture simultaneously the mean and volatility asymmetries in asset return data. Also, this model, which uses high-frequency data containing valuable intraday information to extract volatility information, can estimate RV biases and parameters simultaneously. The lack of a closed-form expression of the likelihood function makes the estimation of the SV models being a challenging topic in the literature. In this paper, the efficient importance sampling (EIS) technique is adopted to implement the maximum likelihood (ML) estimation method for our proposed TRSV model. The Monte Carlo simulation study shows that the EIS-ML estimation method can provide appropriate and accurate inference for the parameters of the proposed model. Finally, the TRSV model is applied to the intraday high-frequency data of Shanghai Stock Exchange composite index and Shenzhen Stock Exchange component index of China. Empirical results show that the TRSV model captures the volatility dynamics appropriately and provides better fit to the data compared to the realized SV (RSV) model. Moreover, strong evidence of high persistence of volatility and the mean and volatility asymmetries is detected in Chinese stock markets.
[1] Jacquier E, Polson N G, Rossi P E. Bayesian analysis of stochastic volatility models[J]. Journal of Business & Economic Statistics, 2002,20(1):69-87.
[2] Kim S, Shephard N, Chib S. Stochastic volatility:Likelihood inference and comparison with ARCH models[J]. Review of Economic Studies, 1998, 65(3):361-393.
[3] 江良, 林鸿熙. 随机波动率Hull-White模型参数估计方法[J]. 系统工程学报, 2016, 31(5):633-642.
[4] 周艳丽, 吴洋, 葛翔宇. 一类高新技术企业专利权价值的实物期权评估方法——基于跳扩散过程和随机波动率的美式期权的建模与模拟[J]. 中国管理科学, 2016, 24(6):19-28.
[5] Black F. Studies of stock price volatility changes[C]//Proceedings of the 1976 Meeting of Business and Economic Statistics Section, American Statistical Association, 1976:177-181.
[6] Christie A A. The stochastic behavior of common stock variances[J]. Journal of Financial Economics, 1982, (10):407-432.
[7] Harvey A C, Shephard N. The estimation of an asymmetric stochastic volatility model for asset returns[J]. Journal of Business & Economic Statistics, 1996, 14(4):429-434.
[8] Jacquier E, Polson N G, Rossi P E. Bayesian analysis of stochastic volatility models with fat-tails and correlated errors[J]. Journal of Econometrics, 2004, 122(1):185-212.
[9] Yu Jun. On leverage in a stochastic volatility model[J]. Journal of Econometrics, 2005, 127(2):165-178.
[10] So M K, Li W K, Lam K. A threshold stochastic volatility model[J]. Journal of Forecasting, 2002, 21(7):473-500.
[11] Asai M, McAleer M. Dynamic asymmetric leverage in stochastic volatility models[J]. Econometric Reviews, 2005, 24(3):317-332.
[12] Chen C W S, Liu F C, So M K P. Heavy-tailed distributed threshold stochastic volatility models in financial time series[J]. Australian & New Zealand Journal of Statistics, 2008, 50(1):29-51.
[13] So M K, Choi C Y. A multivariate threshold stochastic volatility model[J]. Mathematics and Computers in Simulation, 2008, 79(3):306-317.
[14] So M K, Choi C Y. A threshold factor multivariate stochastic volatility model[J]. Journal of Forecasting, 2009, 28(8):712-735.
[15] Wu Xinyu, Zhou Hailin. A triple-threshold leverage stochastic volatility model[J]. Studies in Nonlinear Dynamics & Econometrics, 2015, 19(4):483-500.
[16] 吴鑫育, 周海林, 汪寿阳, 等. 双杠杆门限随机波动率模型及其实证研究[J]. 管理科学学报, 2014, 17(7):63-81.
[17] 吴鑫育, 周海林. 中国股票市场的非对称反应[J]. 系统工程, 2015, 33(8):78-83.
[18] 瞿慧, 李洁, 程昕. HAR族模型与GARCH族模型对不同期限波动率的预测精度比较——基于沪深300指数高频价格的实证分析[J]. 系统工程, 2015, 33(3):32-37.
[19] 瞿慧, 程思逸. 考虑成分股联跳与宏观信息发布的沪深300指数已实现波动率模型研究[J]. 中国管理科学, 2016, 24(12):10-19.
[20] Barndorff-Nielsen O E, Shephard N. Non-Gaussian Ornstein-Ulhlenbeck-based models and some of their uses in financial economics[J]. Journal of the Royal Statistical Society, Series B, 2001, 63(2):167-241.
[21] Barndorff-Nielsen O E, Shephard N. Econometric analysis of realized volatility and its use in estimating stochastic volatility models[J]. Journal of the Royal Statistical Society:Series B, 2002, 64(2):253-280.
[22] Takahashi M, Omori Y, Watanabe T. Estimating stochastic volatility models using daily returns and realized volatility simultaneously[J]. Computational Statistics & Data Analysis, 2009, 53(6):2404-2426.
[23] Koopman S J, Scharth M. The analysis of stochastic volatility in the presence of daily realised measures[J]. Journal of Financial Econometrics, 2013, 11(1):76-115.
[24] Shirota S, Hizu T, Omori Y. Realized stochastic volatility with leverage and long memory[J]. Computational Statistics & Data Analysis, 2014, 76:618-641.
[25] Venter J H, de Jongh P J. Extended stochastic volatility models incorporating realised measures[J]. Computational Statistics & Data Analysis, 2014, 76:687-707.
[26] Zheng Tingguo, Song Tao. A realized stochastic volatility model with Box-Cox transformation[J]. Journal of Business and Economic Statistics, 2014, 32(4):593-605.
[27] Takahashi M, Watanabe T, Omori Y. Volatility and quantile forecasts by realized stochastic volatility models with generalized hyperbolic distribution[J]. International Journal of Forecasting, 2016, 32(2):437-457.
[28] Taylor S J. Financial returns modelled by the product of two stochastic processes-a study of daily sugar prices[M]//Anderson O D. Time series analysis:Theory and practice 1. Amsterdam:North Holland, 1982:203-226.
[29] Tong H, Lim K S. Threshold autoregression, limit cycles and cyclical data[J]. Journal of the Royal Statistical Society:Series B, 1980, 42(3):245-292.
[30] 吴鑫育, 马超群, 汪寿阳. 随机波动率模型的参数估计及对中国股市的实证[J]. 系统工程理论与实践, 2014, 34(1):35-44.
[31] Lambert P, Laurent S. Modelling skewness dynamics in series of financial data using skewed location-scale distributions[R]. Discussion Paper, Catholic University of Louvain, 2002.
[32] Broto C, Ruiz E. Estimation methods for stochastic volatility models:A survey[J]. Journal of Economic Surveys, 2004, 18(5):613-649.
[33] Richard J F, Zhang Wei. Efficient high-dimensional importance sampling[J]. Journal of Econometrics, 2007, 141(2):1385-1411.
[34] 赵树然, 姜亚萍, 任培民. 高频波动率矩阵估计的比较分析——基于有噪非同步的金融数据[J]. 中国管理科学, 2015, 23(10):19-29.
[35] 刘志东, 严冠. 基于半鞅过程的中国股市随机波动、跳跃和微观结构噪声统计特征研究[J]. 中国管理科学, 2016, 24(5):18-30.
[36] Hansen P R, Huang Zhuo, 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.
