混合正态双因子已实现SV模型及其实证研究
|
摘要
传统的波动率模型(如GARCH模型和SV模型)采用日度收益率数据对波动率建模,忽略了日内高频数据包含的丰富信息。同时,资产收益率的波动率往往展现长记忆性,资产收益率的分布展现非正态性(尖峰、厚尾),传统的波动率模型不能很好地刻画现实资产收益率的这些特征。将包含丰富日内高频信息的已实现波动率引入传统低频SV模型中,同时考虑已实现波动率偏差修正、波动率长记忆性和资产收益率的非正态性(尖峰、厚尾),构建混合正态双因子已实现SV(2FRSV-MN)模型。为了估计2FRSV-MN模型的参数,给出灵活且易于实现的基于连续粒子滤波的极大似然估计方法。蒙特卡罗模拟实验表明,给出的估计方法是有效的。采用上证综合指数和深证成分指数5分钟高频交易数据,对提出的2FRSV-MN模型进行实证检验。研究结果表明,已实现波动率是真实日度波动率的有偏估计(下偏),沪深股市非交易时间效应强于微观结构噪声效应;沪深股市具有强的波动率持续性和显著的杠杆效应,且杠杆效应只存在于短记忆波动率因子过程中,长记忆波动率因子过程中存在反向杠杆效应;根据赤池信息准则,2FRSV-MN模型比其他模型具有更好的数据拟合效果。对风险值的估计结果表明,2FRSV-MN模型能较好地测量金融市场风险,但并非一定是具有最好的风险测量效果的模型,这取决于选取的概率和数据。研究结果不仅为投资者和监管机构提供了信息和决策参考,也丰富了基于高频数据的波动率建模和市场风险测量的研究。
The traditional volatility models,such as the GARCH model and SV model,use only daily returns data to model volatility. However,they do not take advantage of additional information provided by high-frequency intra-day data. In addition,it has been well documented that the volatility of asset returns exhibits the property of long-range dependence and the empirical distribution of asset returns exhibits non-normality( leptokurtosis and heavy-tails). The traditional volatility models fail to account for these empirical stylized facts of realistic asset returns.This paper incorporates the low-frequency SV model with high-frequency intra-day information provided by the realized volatility( RV) and proposes the two-factor realized SV model with mixture of normals( 2 FRSV-MN model). The model takes the RV biases,long memory property of the volatility and non-normality of the asset returns into consideration. To estimate the parameters of the model,the maximum likelihood estimation method based on the continuous particle filters is developed. Monte Carlo simulation study shows that the estimation method performs well. We apply the 2 FRSV-MN model to the 5-min high-frequency intra-day data of Shanghai Stock Exchange composite( SSE) index and Shenzhen Stock Exchange component( SZSE) index.The results show that the RV is a( downward) biased estimator of the true daily volatility. This implies that the effect of non-trading hours is stronger than that of microstructure noise. Evidence of strong volatility persistence and significant leverage effect is detected in Shanghai and Shenzhen stock markets. Moreover,we do find the strong evidence of leverage effect in the short-memory volatility process,while reverse leverage effect is found in the long-memory volatility process. According to the Akaike information criterion( AIC),the 2 FRSV-MN model fits the data better than the others. The empirical findings,based on the Value at Risk( VaR) estimates,indicate that the 2 FRSV-MN model performs well. However,the 2 FRSV-MN model is not necessary to be the best model,and the risk measurement performance of the model is sensitive to the choice of the probability and data.This paper not only provides some information and decision-making reference for the investors and regulators,but also enriches the empirical research on the volatility modelling and market risk measurement based on the high-frequency data.
The traditional volatility models,such as the GARCH model and SV model,use only daily returns data to model volatility. However,they do not take advantage of additional information provided by high-frequency intra-day data. In addition,it has been well documented that the volatility of asset returns exhibits the property of long-range dependence and the empirical distribution of asset returns exhibits non-normality( leptokurtosis and heavy-tails). The traditional volatility models fail to account for these empirical stylized facts of realistic asset returns.This paper incorporates the low-frequency SV model with high-frequency intra-day information provided by the realized volatility( RV) and proposes the two-factor realized SV model with mixture of normals( 2 FRSV-MN model). The model takes the RV biases,long memory property of the volatility and non-normality of the asset returns into consideration. To estimate the parameters of the model,the maximum likelihood estimation method based on the continuous particle filters is developed. Monte Carlo simulation study shows that the estimation method performs well. We apply the 2 FRSV-MN model to the 5-min high-frequency intra-day data of Shanghai Stock Exchange composite( SSE) index and Shenzhen Stock Exchange component( SZSE) index.The results show that the RV is a( downward) biased estimator of the true daily volatility. This implies that the effect of non-trading hours is stronger than that of microstructure noise. Evidence of strong volatility persistence and significant leverage effect is detected in Shanghai and Shenzhen stock markets. Moreover,we do find the strong evidence of leverage effect in the short-memory volatility process,while reverse leverage effect is found in the long-memory volatility process. According to the Akaike information criterion( AIC),the 2 FRSV-MN model fits the data better than the others. The empirical findings,based on the Value at Risk( VaR) estimates,indicate that the 2 FRSV-MN model performs well. However,the 2 FRSV-MN model is not necessary to be the best model,and the risk measurement performance of the model is sensitive to the choice of the probability and data.This paper not only provides some information and decision-making reference for the investors and regulators,but also enriches the empirical research on the volatility modelling and market risk measurement based on the high-frequency data.
引文
[1]ANDERSEN T G,BOLLERSLEV T,DIEBOLD F X,et al.The distribution of realized stock return volatility.Journal of Financial Economics,2001,61(1):43-76.
[2]BARNDORFF-NIELSEN O E,SHEPHARD N.Econometric analysis of realized volatility and its use in estimating stochastic volatility models.Journal of the Royal Statistical Society:Series B,2002,64(2):253-280.
[3]TAKAHASHI M,OMORI Y,WATANABE T.Estimating stochastic volatility models using daily returns and realized volatility simultaneously.Computational Statistics&Data Analysis,2009,53(6):2404-2426.
[4]KOOPMAN S J,SCHARTH M.The analysis of stochastic volatility in the presence of daily realised measures.Journal of Financial Econometrics,2013,11(1):76-115.
[5]SHIROTA S,HIZU T,OMORI Y.Realized stochastic volatility with leverage and long memory.Computational Statistics&Data Analysis,2014,76:618-641.
[6]VENTER J H,DE JONGH P J.Extended stochastic volatility models incorporating realised measures.Computational Statistics&Data Analysis,2014,76:687-707.
[7]ZHENG T,SONG T.A realized stochastic volatility model with Box-Cox transformation.Journal of Business&Economic Statistics,2014,32(4):593-605.
[8]CHRISTOFFERSEN P,FEUNOU B,JACOBS K,et al.The economic value of realized volatility:using high-frequency returns for option valuation.Journal of Financial and Quantitative Analysis,2014,49(3):663-697.
[9]TAKAHASHI M,WATANABE T,OMORI Y.Volatility and quantile forecasts by realized stochastic volatility models with generalized hyperbolic distribution.International Journal of Forecasting,2016,32(2):437-457.
[10]吴鑫育,李心丹,马超群.门限已实现随机波动率模型及其实证研究.中国管理科学,2017,25(3):10-19.WU Xinyu,LI Xindan,MA Chaoqun.Threshold realized stochastic volatility model and its empirical test.Chinese Journal of Management Science,2017,25(3):10-19.(in Chinese)
[11]HANSEN P R,HUANG Z,SHEK H H.Realized GARCH:a joint model for returns and realized measures of volatility.Journal of Applied Econometrics,2012,27(6):877-906.
[12]HANSEN P R,HUANG Z.Exponential GARCH modeling with realized measures of volatility.Journal of Business&Economic Statistics,2016,34(2):269-287.
[13]王天一,赵晓军,黄卓.利用高频数据预测沪深300指数波动率:基于Realized GARCH模型的实证研究.世界经济文汇,2014(5):17-30.WANG Tianyi,ZHAO Xiaojun,HUANG Zhuo.Forecasting volatility of CSI300 index with high frequency data:empirical evidence from Realized GARCH model.World Economic Papers,2014(5):17-30.(in Chinese)
[14]黄友珀,唐振鹏,周熙雯.基于偏分布realized GARCH模型的尾部风险估计.系统工程理论与实践,2015,35(9):2200-2208.HUANG Youpo,TANG Zhenpeng,ZHOU Xiwen.Estimation of tail risk based on realized GARCH model with skew-t distribution.Systems Engineering-Theory&Practice,2015,35(9):2200-2208.(in Chinese)
[15]黄友珀,唐振鹏,唐勇.基于藤copula-已实现GARCH的组合收益分位数预测.系统工程学报,2016,31(1):45-54.HUANG Youpo,TANG Zhenpeng,TANG Yong.Portfolio quantile forecasts based on vine copula and realized GARCH.Journal of Systems Engineering,2016,31(1):45-54.(in Chinese)
[16]HUANG Z,WANG T,HANSEN P R.Option pricing with the realized GARCH model:an analytical approximation approach.The Journal of Futures Markets,2017,37(4):328-358.
[17]王鹏,吕永健.基于不同记忆性异方差模型的中国股票市场波动率预测.中国管理科学,2013,21(S1):270-275.WANG Peng,LYU Yongjian.Comparison of forecasting ability of heteroskedastic volatility models with different memory.Chinese Journal of Management Science,2013,21(S1):270-275.(in Chinese)
[18]郑艺,梁循,孙晓蕾.基于RV的LMSV模型在中国股市中的实证研究.中国管理科学,2014,22(S1):212-221.ZHENG Yi,LIANG Xun,SUN Xiaolei.An empirical study of LMSV model in China stock market based on realized volatility.Chinese Journal of Management Science,2014,22(S1):212-221.(in Chinese)
[19]刘志东,刘雯宇.Lévy过程驱动的非高斯OU随机波动模型及其贝叶斯参数统计推断方法研究.中国管理科学,2015,23(8):1-9.LIU Zhidong,LIU Wenyu.The non Ornstein-Uhlenbeck models driven by the general Lévy process and its Bayesian inference.Chinese Journal of Management Science,2015,23(8):1-9.(in Chinese)
[20]ASAI M,MCALEER M J,MEDEIROS M C.Asymmetry and long memory in volatility modeling.Journal of Financial E-conometrics,2012,10(3):495-512.
[21]ASAI M,CHANG C,MCALEER M.Realized stochastic volatility with general asymmetry and long memory.Journal of Econometrics,2017,199(2):202-212.
[22]ASAI M,MCALEER M,PEIRIS S.Realized stochastic volatility models with generalized Gegenbauer long memory.Netherlands:Tinbergen Institute,2017.
[23]孔继红.基于非对称扩散跳跃过程的利率模型研究.数量经济技术经济研究,2014,31(11):103-117,145.KONG Jihong.A research on interest rate model based on asymmetric diffusion jump process.The Journal of Quantitative&Technical Economics,2014,31(11):103-117,145.(in Chinese)
[24]DURHAM G B.SV mixture models with application to S&P500 index returns.Journal of Financial Economics,2007,85(3):822-856.
[25]苏理云,彭相武,王杰,等.基于状态空间SV-T-MN模型的股指波动率预测.数理统计与管理,2016,35(5):929-942.SU Liyun,PENG Xiangwu,WANG Jie,et al.Forecasting volatility in stock market using SV-T-MN based on state space models.Journal of Applied Statistics and Management,2016,35(5):929-942.(in Chinese)
[26]王鹏,杨兴林.基于时变波动率与混合对数正态分布的50ETF期权定价.管理科学,2016,29(4):149-160.WANG Peng,YANG Xinglin.Option pricing of mixture of lognormal distributions with time-varying volatility in 50ETFoption.Journal of Management Science,2016,29(4):149-160.(in Chinese)
[27]BEKIERMAN J,GRIBISCH B.Estimating stochastic volatility models using realized measures.Studies in Nonlinear Dynamics&Econometrics,2016,20(3):279-300.
[28]CORSI F,RENR.Discrete-time volatility forecasting with persistent leverage effect and the link with continuous-time volatility modeling.Journal of Business&Economic Statistics,2012,30(3):368-380.
[29]苏云鹏,杨宝臣.随机波动HJM框架下可违约债券市场波动结构的实证研究.管理科学,2015,28(1):122-132.SU Yunpeng,YANG Baochen.Empirical research on volatility structure of defaultable bond market under HJM framework with stochastic volatility.Journal of Management Science,2015,28(1):122-132.(in Chinese)
[30]瞿慧,刘烨.沪深300指数收益率及已实现波动联合建模研究.管理科学,2012,25(6):101-110.QU Hui,LIU Ye.A joint model for CSI300 index return and realized volatility.Journal of Management Science,2012,25(6):101-110.(in Chinese)
[31]吕永健,王鹏.基于时变高阶矩模型的贵金属市场风险测度研究.管理科学,2015,28(1):133-143.LYU Yongjian,WANG Peng.Risk measurements on precious metal market given the context of time-varying high order moments.Journal of Management Science,2015,28(1):133-143.(in Chinese)
[32]GORDON N J,SALMOND D J,SMITH A F M.Novel approach to nonlinear/non-Gaussian Bayesian state estimation.IEE Proceedings F-Radar and Signal Processing,1993,140(2):107-113.
[33]CALVET L E,FEARNLEY M,FISHER A J,et al.What is beneath the surface?Option pricing with multifrequency latent states.Journal of Econometrics,2015,187(2):498-511.
[34]HURN A S,LINDSAY K A,MCCLELLAND A J.Estimating the parameters of stochastic volatility models using option price data.Journal of Business&Economic Statistics,2015,33(4):579-594.
[35]PITT M K,MALIK S,DOUCET A.Simulated likelihood inference for stochastic volatility models using continuous particle filtering.Annals of the Institute of Statistical Mathematics,2014,66(3):527-552.
[36]MALIK S,PITT M K.Particle filters for continuous likelihood evaluation and maximisation.Journal of Econometrics,2011,165(2):190-209.
[37]YU J.A semiparametric stochastic volatility model.Journal of Econometrics,2012,167(2):473-482.
[38]WU X Y,ZHOU H L.A triple-threshold leverage stochastic volatility model.Studies in Nonlinear Dynamics&Econometrics,2015,19(4):483-500.
[39]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.系统工程理论与实践,2014,34(1):35-44.WU Xinyu,MA Chaoqun,WANG Shouyang.Estimation of stochastic volatility models:an empirical study of China's stock market.Systems Engineering-Theory&Practice,2014,34(1):35-44.(in Chinese)
[40]LIU Y,LI J,NG A.Option pricing under GARCH models with Hansen's skewed-t distributed innovations.North American Journal of Economics and Finance,2015,31:108-125.
[2]BARNDORFF-NIELSEN O E,SHEPHARD N.Econometric analysis of realized volatility and its use in estimating stochastic volatility models.Journal of the Royal Statistical Society:Series B,2002,64(2):253-280.
[3]TAKAHASHI M,OMORI Y,WATANABE T.Estimating stochastic volatility models using daily returns and realized volatility simultaneously.Computational Statistics&Data Analysis,2009,53(6):2404-2426.
[4]KOOPMAN S J,SCHARTH M.The analysis of stochastic volatility in the presence of daily realised measures.Journal of Financial Econometrics,2013,11(1):76-115.
[5]SHIROTA S,HIZU T,OMORI Y.Realized stochastic volatility with leverage and long memory.Computational Statistics&Data Analysis,2014,76:618-641.
[6]VENTER J H,DE JONGH P J.Extended stochastic volatility models incorporating realised measures.Computational Statistics&Data Analysis,2014,76:687-707.
[7]ZHENG T,SONG T.A realized stochastic volatility model with Box-Cox transformation.Journal of Business&Economic Statistics,2014,32(4):593-605.
[8]CHRISTOFFERSEN P,FEUNOU B,JACOBS K,et al.The economic value of realized volatility:using high-frequency returns for option valuation.Journal of Financial and Quantitative Analysis,2014,49(3):663-697.
[9]TAKAHASHI M,WATANABE T,OMORI Y.Volatility and quantile forecasts by realized stochastic volatility models with generalized hyperbolic distribution.International Journal of Forecasting,2016,32(2):437-457.
[10]吴鑫育,李心丹,马超群.门限已实现随机波动率模型及其实证研究.中国管理科学,2017,25(3):10-19.WU Xinyu,LI Xindan,MA Chaoqun.Threshold realized stochastic volatility model and its empirical test.Chinese Journal of Management Science,2017,25(3):10-19.(in Chinese)
[11]HANSEN P R,HUANG Z,SHEK H H.Realized GARCH:a joint model for returns and realized measures of volatility.Journal of Applied Econometrics,2012,27(6):877-906.
[12]HANSEN P R,HUANG Z.Exponential GARCH modeling with realized measures of volatility.Journal of Business&Economic Statistics,2016,34(2):269-287.
[13]王天一,赵晓军,黄卓.利用高频数据预测沪深300指数波动率:基于Realized GARCH模型的实证研究.世界经济文汇,2014(5):17-30.WANG Tianyi,ZHAO Xiaojun,HUANG Zhuo.Forecasting volatility of CSI300 index with high frequency data:empirical evidence from Realized GARCH model.World Economic Papers,2014(5):17-30.(in Chinese)
[14]黄友珀,唐振鹏,周熙雯.基于偏分布realized GARCH模型的尾部风险估计.系统工程理论与实践,2015,35(9):2200-2208.HUANG Youpo,TANG Zhenpeng,ZHOU Xiwen.Estimation of tail risk based on realized GARCH model with skew-t distribution.Systems Engineering-Theory&Practice,2015,35(9):2200-2208.(in Chinese)
[15]黄友珀,唐振鹏,唐勇.基于藤copula-已实现GARCH的组合收益分位数预测.系统工程学报,2016,31(1):45-54.HUANG Youpo,TANG Zhenpeng,TANG Yong.Portfolio quantile forecasts based on vine copula and realized GARCH.Journal of Systems Engineering,2016,31(1):45-54.(in Chinese)
[16]HUANG Z,WANG T,HANSEN P R.Option pricing with the realized GARCH model:an analytical approximation approach.The Journal of Futures Markets,2017,37(4):328-358.
[17]王鹏,吕永健.基于不同记忆性异方差模型的中国股票市场波动率预测.中国管理科学,2013,21(S1):270-275.WANG Peng,LYU Yongjian.Comparison of forecasting ability of heteroskedastic volatility models with different memory.Chinese Journal of Management Science,2013,21(S1):270-275.(in Chinese)
[18]郑艺,梁循,孙晓蕾.基于RV的LMSV模型在中国股市中的实证研究.中国管理科学,2014,22(S1):212-221.ZHENG Yi,LIANG Xun,SUN Xiaolei.An empirical study of LMSV model in China stock market based on realized volatility.Chinese Journal of Management Science,2014,22(S1):212-221.(in Chinese)
[19]刘志东,刘雯宇.Lévy过程驱动的非高斯OU随机波动模型及其贝叶斯参数统计推断方法研究.中国管理科学,2015,23(8):1-9.LIU Zhidong,LIU Wenyu.The non Ornstein-Uhlenbeck models driven by the general Lévy process and its Bayesian inference.Chinese Journal of Management Science,2015,23(8):1-9.(in Chinese)
[20]ASAI M,MCALEER M J,MEDEIROS M C.Asymmetry and long memory in volatility modeling.Journal of Financial E-conometrics,2012,10(3):495-512.
[21]ASAI M,CHANG C,MCALEER M.Realized stochastic volatility with general asymmetry and long memory.Journal of Econometrics,2017,199(2):202-212.
[22]ASAI M,MCALEER M,PEIRIS S.Realized stochastic volatility models with generalized Gegenbauer long memory.Netherlands:Tinbergen Institute,2017.
[23]孔继红.基于非对称扩散跳跃过程的利率模型研究.数量经济技术经济研究,2014,31(11):103-117,145.KONG Jihong.A research on interest rate model based on asymmetric diffusion jump process.The Journal of Quantitative&Technical Economics,2014,31(11):103-117,145.(in Chinese)
[24]DURHAM G B.SV mixture models with application to S&P500 index returns.Journal of Financial Economics,2007,85(3):822-856.
[25]苏理云,彭相武,王杰,等.基于状态空间SV-T-MN模型的股指波动率预测.数理统计与管理,2016,35(5):929-942.SU Liyun,PENG Xiangwu,WANG Jie,et al.Forecasting volatility in stock market using SV-T-MN based on state space models.Journal of Applied Statistics and Management,2016,35(5):929-942.(in Chinese)
[26]王鹏,杨兴林.基于时变波动率与混合对数正态分布的50ETF期权定价.管理科学,2016,29(4):149-160.WANG Peng,YANG Xinglin.Option pricing of mixture of lognormal distributions with time-varying volatility in 50ETFoption.Journal of Management Science,2016,29(4):149-160.(in Chinese)
[27]BEKIERMAN J,GRIBISCH B.Estimating stochastic volatility models using realized measures.Studies in Nonlinear Dynamics&Econometrics,2016,20(3):279-300.
[28]CORSI F,RENR.Discrete-time volatility forecasting with persistent leverage effect and the link with continuous-time volatility modeling.Journal of Business&Economic Statistics,2012,30(3):368-380.
[29]苏云鹏,杨宝臣.随机波动HJM框架下可违约债券市场波动结构的实证研究.管理科学,2015,28(1):122-132.SU Yunpeng,YANG Baochen.Empirical research on volatility structure of defaultable bond market under HJM framework with stochastic volatility.Journal of Management Science,2015,28(1):122-132.(in Chinese)
[30]瞿慧,刘烨.沪深300指数收益率及已实现波动联合建模研究.管理科学,2012,25(6):101-110.QU Hui,LIU Ye.A joint model for CSI300 index return and realized volatility.Journal of Management Science,2012,25(6):101-110.(in Chinese)
[31]吕永健,王鹏.基于时变高阶矩模型的贵金属市场风险测度研究.管理科学,2015,28(1):133-143.LYU Yongjian,WANG Peng.Risk measurements on precious metal market given the context of time-varying high order moments.Journal of Management Science,2015,28(1):133-143.(in Chinese)
[32]GORDON N J,SALMOND D J,SMITH A F M.Novel approach to nonlinear/non-Gaussian Bayesian state estimation.IEE Proceedings F-Radar and Signal Processing,1993,140(2):107-113.
[33]CALVET L E,FEARNLEY M,FISHER A J,et al.What is beneath the surface?Option pricing with multifrequency latent states.Journal of Econometrics,2015,187(2):498-511.
[34]HURN A S,LINDSAY K A,MCCLELLAND A J.Estimating the parameters of stochastic volatility models using option price data.Journal of Business&Economic Statistics,2015,33(4):579-594.
[35]PITT M K,MALIK S,DOUCET A.Simulated likelihood inference for stochastic volatility models using continuous particle filtering.Annals of the Institute of Statistical Mathematics,2014,66(3):527-552.
[36]MALIK S,PITT M K.Particle filters for continuous likelihood evaluation and maximisation.Journal of Econometrics,2011,165(2):190-209.
[37]YU J.A semiparametric stochastic volatility model.Journal of Econometrics,2012,167(2):473-482.
[38]WU X Y,ZHOU H L.A triple-threshold leverage stochastic volatility model.Studies in Nonlinear Dynamics&Econometrics,2015,19(4):483-500.
[39]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.系统工程理论与实践,2014,34(1):35-44.WU Xinyu,MA Chaoqun,WANG Shouyang.Estimation of stochastic volatility models:an empirical study of China's stock market.Systems Engineering-Theory&Practice,2014,34(1):35-44.(in Chinese)
[40]LIU Y,LI J,NG A.Option pricing under GARCH models with Hansen's skewed-t distributed innovations.North American Journal of Economics and Finance,2015,31:108-125.