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 in order 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率动态特征，但将其拓展至带动态相关性的多变量模型时，却因若干核心难题而举步维艰。其一，若仅采用日度收益率数据，待估参数数量过多，易导致估计结果不稳定；对此类问题的一种解决方案是引入基于日内资产收益率的额外观测值，例如已实现协方差（realized covariances）。其二，由于多变量资产收益率并非同步交易，为计算已实现协方差矩阵（realized covariance matrices），需选取最大的时间区间以确保所有资产收益率均被观测到，但该做法在存在低频率交易资产时，无法充分利用可用的日内信息。其三，难以直接保证估计得到（以及已实现的）协方差矩阵为正定矩阵。本文的核心贡献如下：（1）借助已实现测度（realized measures），实现动态相关性模型的稳定参数估计；（2）通过采用成对已实现相关性（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的任意性问题；（5）提出灵活的相关性结构模型（例如，在必要时可将部分相关性设为零）；（6）提出针对杠杆效应的简约设定。本文将所提模型应用于9只美国股票的日度收益率数据，并结合其已实现波动率与成对已实现相关性开展实证分析，结果显示，在投资组合表现方面，本文模型优于现有同类模型。