双因子非对称已实现SV模型及其实证研究
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摘要
在基本的SV模型中引入包含丰富日内高频信息的已实现测度,同时考虑其偏差修正以及波动率非对称性与长记忆性,构建了双因子非对称已实现SV(2FARSV)模型.进一步基于连续粒子滤波算法,给出了2FARSV模型参数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份指数日内高频数据计算已实现波动率(RV)和已实现极差波动率(RRV),对2FARSV模型进行了实证研究.结果表明:RV和RRV都是真实日度波动率的有偏估计(下偏),但RRV相比RV是更有效的波动率估计量;沪深股市具有强的波动率持续性以及显著的波动率非对称性(杠杆效应与规模效应);2FARSV模型相比其它已实现波动率模型具有更好的数据拟合效果,该模型能够充分地捕获沪深股市波动率的动态特征(时变性、聚集性、非对称性与长记忆性).
Modelling the volatility has attracted a great deal of attention in the literature of financial economics and econometrics.As a measure of risk,volatility modelling is important to researchers,policy makers and regulators as it is closely related to the stability of financial markets,and the SV models are used to model the dynamics of volatility.It has been well-documented in the literature that the volatility exhibits two typical features,namely the asymmetric effects(include leverage and size effects)on volatility caused by previous returns and the long-range dependence in volatility.It is important to accommodate these features in volatility in the SV models.In recent years,high-frequency financial data is available and a number of realized measures of volatility have been introduced,including realized volatility and realized range volatility.These measures are far more informative about the current level of volatility than the daily returns.Standard SV models utilizes daily returns to extract information about the current level of volatility.A shortcoming of conventional SV models is the fact that returns are rather weak signals about the level of volatility.This makes SV models poorly suited for situations where volatility changes rapidly to a new level.Incorporating realized measures into SV models is expected to alleviate this problem.Based on the above analysis,in this paper the two-factor asymmetric realized SV(2 FARSV)model is proposed,which incorporates the basic SV model(which uses daily returns)with asymmetric effects(leverage and size effects),long memory property of the volatility and high-frequency intraday information provided by the realized measure by taking account of the realized measure biases.A maximum likelihood estimation method based on the continuous particle filters is used to estimate the parameters of the2 FARSV model.The monte Carlo simulation study shows that the estimation method performs well.The2 FARSV model is appiled to the realized volatility(RV)and realized range volatility(RRV)computed from the high-frequency intraday data of Shanghai Stock Exchange composite index and Shenzhen Stock Exchange component index.The results show that the RV and RRV are(downward)biased estimators of the true daily volatility.This implies that the effect of non-trading hours is stronger than that of microstructure noise.The RRV is the more effective volatility estimator than the RV,but it leads to more downward bias.In addition,evidence of strong volatility persistence and volatility asymmetry(leverage and size effects)is detected in Shanghai and Shenzhen stock markets.According to the Akaike information criterion(AIC)and Bayesian information criterion(BIC),the 2 FARSV model fits the data better than the one-factor asymmetric realized SV(1 FARSV)model,the two-factor realized SV(2 FRSV)model,the two-factor leverage realized SV(2 FLRSV)model and the realized GARCH model.Model diagnostics suggest that the 2 FARSV model is sufficient to capture the dynamics of the volatility(time-varying volatility,volatility clustering,volatility asymmetry and long memory property of the volatility).This study enriches the empirical research on the volatility modelling based on the high-frequency data.It's also a worthwhile endeavor to further investigate the performance of the 2 FARSV model in asset pricing and value-at-risk calculation.
Modelling the volatility has attracted a great deal of attention in the literature of financial economics and econometrics.As a measure of risk,volatility modelling is important to researchers,policy makers and regulators as it is closely related to the stability of financial markets,and the SV models are used to model the dynamics of volatility.It has been well-documented in the literature that the volatility exhibits two typical features,namely the asymmetric effects(include leverage and size effects)on volatility caused by previous returns and the long-range dependence in volatility.It is important to accommodate these features in volatility in the SV models.In recent years,high-frequency financial data is available and a number of realized measures of volatility have been introduced,including realized volatility and realized range volatility.These measures are far more informative about the current level of volatility than the daily returns.Standard SV models utilizes daily returns to extract information about the current level of volatility.A shortcoming of conventional SV models is the fact that returns are rather weak signals about the level of volatility.This makes SV models poorly suited for situations where volatility changes rapidly to a new level.Incorporating realized measures into SV models is expected to alleviate this problem.Based on the above analysis,in this paper the two-factor asymmetric realized SV(2 FARSV)model is proposed,which incorporates the basic SV model(which uses daily returns)with asymmetric effects(leverage and size effects),long memory property of the volatility and high-frequency intraday information provided by the realized measure by taking account of the realized measure biases.A maximum likelihood estimation method based on the continuous particle filters is used to estimate the parameters of the2 FARSV model.The monte Carlo simulation study shows that the estimation method performs well.The2 FARSV model is appiled to the realized volatility(RV)and realized range volatility(RRV)computed from the high-frequency intraday data of Shanghai Stock Exchange composite index and Shenzhen Stock Exchange component index.The results show that the RV and RRV are(downward)biased estimators of the true daily volatility.This implies that the effect of non-trading hours is stronger than that of microstructure noise.The RRV is the more effective volatility estimator than the RV,but it leads to more downward bias.In addition,evidence of strong volatility persistence and volatility asymmetry(leverage and size effects)is detected in Shanghai and Shenzhen stock markets.According to the Akaike information criterion(AIC)and Bayesian information criterion(BIC),the 2 FARSV model fits the data better than the one-factor asymmetric realized SV(1 FARSV)model,the two-factor realized SV(2 FRSV)model,the two-factor leverage realized SV(2 FLRSV)model and the realized GARCH model.Model diagnostics suggest that the 2 FARSV model is sufficient to capture the dynamics of the volatility(time-varying volatility,volatility clustering,volatility asymmetry and long memory property of the volatility).This study enriches the empirical research on the volatility modelling based on the high-frequency data.It's also a worthwhile endeavor to further investigate the performance of the 2 FARSV model in asset pricing and value-at-risk calculation.
引文
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[2]Yu Jun.Forecasting volatility in the New Zealand stock market[J].Applied Financial Economics,2002,12(3):193-202.
[3]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.
[4]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.
[5]余白敏,吴卫星.基于“已实现”波动率ARFI模型和CAViaR模型的VaR预测比较研究[J].中国管理科学,2015,23(2):50-58.
[6]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.
[7]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.
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[9]Shirota S,Hizu T,Omori Y.Realized stochastic volatility with leverage and long memory[J].Computational Statistics&Data Analysis,2014,76:618-641.
[10]Venter J H,de Jongh P J.Extended stochastic volatility models incorporating realised measures[J].Computational Statistics&Data Analysis,2014,76:687-707.
[11]Zheng Tingguo G,Song Tao.A realized stochastic volatility model with Box-Cox transformation[J].Journal of Business&Economic Statistics,2014,32(4):593-605.
[12]Dobrev D,Szerszen P.The information content of high-frequency data for estimating equity return models and forecasting risk[R].Finance and Economis Discussion Series,et al.Working Paper,2010.
[13]Christoffersen P,Feunou B,Jacobs K,et al.The economic value of realized volatility:Using high-frequency returns for option valuation[J].Journal of Financial and Quantitative Analysis,2014,49(3):663-697.
[14]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.
[15]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[J].中国管理科学,2017,25(3):10-19.
[16]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.
[17]Hansen P R,Huang Zhuo.Exponential GARCH modeling with realized measures of volatility[J].Journal of Business&Economic Statistics,2016,34(2):269-287.
[18]王天一,黄卓.高频数据波动率建模——基于厚尾分布的Realized-GARCH模型[J].数量经济技术经济研究,2012,(5):149-161.
[19]王天一,黄卓.Realized GAS-GARCH及其在VaR预测中的应用[J].管理科学学报,2015,18(5):79-86.
[20]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[J].世界经济文汇,2014,(5):17-30.
[21]唐勇,刘微.加权已实现极差四次幂变差分析及其应用[J].系统工程理论与实践,2013,33(11):2766-2775.
[22]黄友珀,唐振鹏,周熙雯.基于偏t分布realized GARCH模型的尾部风险估计[J].系统工程理论与实践,2015,35(9):2200-2208.
[23]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[J].系统工程学报,2016,31(1):45-54.
[24]Asai M,McAleer M,Medeiros M C.Asymmetry and long memory in volatility modelling[J].Journal of Financial Econometrics,2012,10(3):495-512.
[25]Asai M,McAleer M.Alternative asymmetric stochastic volatility models[J].Econometric Reviews,2011,30(5):548-564.
[26]Gordon N J,Salmond D J,Smith A F M.Novel approach to nonlinear/non-Gaussian Bayesian state estimation[J].IEE Proceedings F-Radar and Signal Processing,1993,140(2):107-113.
[27]Malik S,Pitt M K.Particle filters for continuous likelihood evaluation and maximization[J].Journal of Econometrics,2011,165(2):190-209.
[28]Martens M,van Dijk D.Measuring volatility with the realized range[J].Journal of Econometrics,2007,138(1):181-207.
[29]文凤华,贾俊艳,晁攸丛,等.基于加权已实现极差的中国股市波动特征[J].系统工程,2011,29(9):66-71.
[30]郑挺国,左浩苗.基于极差的区制转移随机波动率模型及其应用[J].管理科学学报,2013,16(9):82-94.
[2]Yu Jun.Forecasting volatility in the New Zealand stock market[J].Applied Financial Economics,2002,12(3):193-202.
[3]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.
[4]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.
[5]余白敏,吴卫星.基于“已实现”波动率ARFI模型和CAViaR模型的VaR预测比较研究[J].中国管理科学,2015,23(2):50-58.
[6]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.
[7]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.
[8]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.
[9]Shirota S,Hizu T,Omori Y.Realized stochastic volatility with leverage and long memory[J].Computational Statistics&Data Analysis,2014,76:618-641.
[10]Venter J H,de Jongh P J.Extended stochastic volatility models incorporating realised measures[J].Computational Statistics&Data Analysis,2014,76:687-707.
[11]Zheng Tingguo G,Song Tao.A realized stochastic volatility model with Box-Cox transformation[J].Journal of Business&Economic Statistics,2014,32(4):593-605.
[12]Dobrev D,Szerszen P.The information content of high-frequency data for estimating equity return models and forecasting risk[R].Finance and Economis Discussion Series,et al.Working Paper,2010.
[13]Christoffersen P,Feunou B,Jacobs K,et al.The economic value of realized volatility:Using high-frequency returns for option valuation[J].Journal of Financial and Quantitative Analysis,2014,49(3):663-697.
[14]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.
[15]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[J].中国管理科学,2017,25(3):10-19.
[16]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.
[17]Hansen P R,Huang Zhuo.Exponential GARCH modeling with realized measures of volatility[J].Journal of Business&Economic Statistics,2016,34(2):269-287.
[18]王天一,黄卓.高频数据波动率建模——基于厚尾分布的Realized-GARCH模型[J].数量经济技术经济研究,2012,(5):149-161.
[19]王天一,黄卓.Realized GAS-GARCH及其在VaR预测中的应用[J].管理科学学报,2015,18(5):79-86.
[20]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[J].世界经济文汇,2014,(5):17-30.
[21]唐勇,刘微.加权已实现极差四次幂变差分析及其应用[J].系统工程理论与实践,2013,33(11):2766-2775.
[22]黄友珀,唐振鹏,周熙雯.基于偏t分布realized GARCH模型的尾部风险估计[J].系统工程理论与实践,2015,35(9):2200-2208.
[23]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[J].系统工程学报,2016,31(1):45-54.
[24]Asai M,McAleer M,Medeiros M C.Asymmetry and long memory in volatility modelling[J].Journal of Financial Econometrics,2012,10(3):495-512.
[25]Asai M,McAleer M.Alternative asymmetric stochastic volatility models[J].Econometric Reviews,2011,30(5):548-564.
[26]Gordon N J,Salmond D J,Smith A F M.Novel approach to nonlinear/non-Gaussian Bayesian state estimation[J].IEE Proceedings F-Radar and Signal Processing,1993,140(2):107-113.
[27]Malik S,Pitt M K.Particle filters for continuous likelihood evaluation and maximization[J].Journal of Econometrics,2011,165(2):190-209.
[28]Martens M,van Dijk D.Measuring volatility with the realized range[J].Journal of Econometrics,2007,138(1):181-207.
[29]文凤华,贾俊艳,晁攸丛,等.基于加权已实现极差的中国股市波动特征[J].系统工程,2011,29(9):66-71.
[30]郑挺国,左浩苗.基于极差的区制转移随机波动率模型及其应用[J].管理科学学报,2013,16(9):82-94.
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