Multivariate Stochastic Volatility Model With Realized Volatilities and Pairwise Realized Correlations

Although stochastic volatility and GARCH (generalized autoregressive conditional heteroscedasticity) models have successfully described the volatility dynamics of univariate asset returns, extending them to the multivariate models with dynamic correlations has been difficult due to several major problems. First, there are too many parameters to estimate if available data are only daily returns, which results in unstable estimates. One solution to this problem is to incorporate additional observations based on intraday asset returns, such as realized covariances. Second, since multivariate asset returns are not synchronously traded, we have to use the largest time intervals such that all asset returns are observed to compute the realized covariance matrices. However, in this study, we fail to make full use of the available intraday informations when there are less frequently traded assets. Third, it is not straightforward to guarantee that the estimated (and the realized) covariance matrices are positive definite. Our contributions are the following: (1) we obtain the stable parameter estimates for the dynamic correlation models using the realized measures, (2) we make full use of intraday informations by using pairwise realized correlations, (3) the covariance matrices are guaranteed to be positive definite, (4) we avoid the arbitrariness of the ordering of asset returns, (5) we propose the flexible correlation structure model (e.g., such as setting some correlations to be zero if necessary), and (6) the parsimonious specification for the leverage effect is proposed. Our proposed models are applied to the daily returns of nine U.S. stocks with their realized volatilities and pairwise realized correlations and are shown to outperform the existing models with respect to portfolio performances.随机波动率（stochastic volatility）与广义自回归条件异方差（generalized autoregressive conditional heteroscedasticity, GARCH）模型已成功刻画单变量资产收益的波动动态，但将其扩展至带动态相关结构的多变量模型时，却因若干核心难题难以实现。其一，若仅使用日度收益数据，待估计参数规模过大，会导致估计结果缺乏稳定性。针对该问题的可行方案之一是引入基于日内资产收益的额外观测数据，例如已实现协方差（realized covariances）。其二，由于多变量资产收益并非同步交易，我们仅能选取最大的公共时间区间以确保所有资产收益均可被观测，进而计算已实现协方差矩阵。但本研究中，当存在交易频率较低的资产时，无法充分利用可用的日内信息。其三，难以确保估计得到（以及已实现）的协方差矩阵为正定矩阵。本研究的核心贡献如下：(1) 借助已实现测度（realized measures）获得动态相关模型的稳定参数估计；(2) 通过使用两两已实现相关系数（pairwise realized correlations），充分挖掘日内信息价值；(3) 确保协方差矩阵始终为正定矩阵；(4) 规避了资产收益排序的任意性问题；(5) 提出了灵活的相关结构模型（例如，必要时可将部分相关系数设为零）；(6) 针对杠杆效应提出了简洁的参数设定形式。本研究将所提模型应用于9只美国股票的日度收益数据及其已实现波动率（realized volatilities）与两两已实现相关系数，并通过实证验证，所提模型在投资组合表现上优于现有同类模型。