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.

尽管随机波动率（stochastic volatility）与广义自回归条件异方差（GARCH，generalized autoregressive conditional heteroscedasticity）模型已成功刻画单变量资产收益率的波动动态，但将其扩展至带有动态相关结构的多变量模型时，却因多项核心难题难以推进。其一，若仅使用日度收益率数据，待估参数数量过多，会导致参数估计结果不稳定，解决该问题的思路之一是引入基于日内资产收益率的额外观测数据，例如已实现协方差（realized covariances）。其二，由于多变量资产收益率的交易并非同步，我们只能选取所有资产均有交易的最大时间区间来计算已实现协方差矩阵（realized covariance matrices），但在此场景下，对于交易频率较低的资产，无法充分利用可用的日内信息。其三，难以确保估计所得（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），得到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），充分利用日内信息；（3）确保协方差矩阵始终为正定矩阵；（4）规避了资产收益率排序带来的任意性问题；（5）提出了灵活的相关结构建模框架（例如在必要时可将部分相关系数设为零）；（6）提出了简洁高效的杠杆效应（leverage effect）参数化设定方式。我们将所提模型应用于9只美国股票的日度收益率数据，并结合其已实现波动率（realized volatilities）与两两已实现相关系数开展实证分析，结果表明，在投资组合表现层面，所提模型的性能优于现有同类模型。