Kullback-Leibler cluster entropy to quantify volatility correlation and risk diversity

L. Ponta, A. Carbone
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Abstract

The Kullback-Leibler cluster entropy $\mathcal{D_{C}}[P \| Q] $ is evaluated for the empirical and model probability distributions $P$ and $Q$ of the clusters formed in the realized volatility time series of five assets (SP\&500, NASDAQ, DJIA, DAX, FTSEMIB). The Kullback-Leibler functional $\mathcal{D_{C}}[P \| Q] $ provides complementary perspectives about the stochastic volatility process compared to the Shannon functional $\mathcal{S_{C}}[P]$. While $\mathcal{D_{C}}[P \| Q] $ is maximum at the short time scales, $\mathcal{S_{C}}[P]$ is maximum at the large time scales leading to complementary optimization criteria tracing back respectively to the maximum and minimum relative entropy evolution principles. The realized volatility is modelled as a time-dependent fractional stochastic process characterized by power-law decaying distributions with positive correlation ($H>1/2$). As a case study, a multiperiod portfolio built on diversity indexes derived from the Kullback-Leibler entropy measure of the realized volatility. The portfolio is robust and exhibits better performances over the horizon periods. A comparison with the portfolio built either according to the uniform distribution or in the framework of the Markowitz theory is also reported.
库尔巴克-莱伯勒聚类熵量化波动相关性和风险多样性
针对五种资产(SP/&500、NASDAQ、DJIA、DAX、FTSEMIB)的已实现波动率时间序列中形成的簇的经验和模型概率分布 $P$ 和 $Q$,评估了库尔巴克-莱布勒簇熵 $\mathcal{D_{C}}[P \| Q] $。与香农函数 $\mathcal{S_{C}}[P]$ 相比,Kullback-Leibler 函数 $\mathcal{D_{C}}[P\| Q] $ 为随机波动过程提供了互补的视角。虽然 $\mathcal{D_{C}}[P \| Q] $ 在短时间尺度上是最大的,$\mathcal{S_{C}}[P]$ 在大时间尺度上是最大的,从而导致互补的优化标准分别追溯到最大和最小相对熵演化原理。已实现的波动率被模拟为一个随时间变化的分数随机过程,其特征是具有正相关性的幂律衰减分布($H>1/2$)。作为案例研究,一个多周期投资组合建立在从已实现波动率的库尔巴克-莱伯勒熵度量得出的多样性指数上。该投资组合是稳健的,并且在各期限内表现较好。报告还比较了根据均匀分布或马科维茨理论框架建立的投资组合。
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