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

Abstract

The Kullback-Leibler cluster entropy ${\mathcal{D}}_{\mathcal{C}}[P\Vert Q]$ 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 ${\mathcal{D}}_{\mathcal{C}}[P\Vert Q]$ provides complementary perspectives about the stochastic volatility process compared to the Shannon functional ${\mathcal{S}}_{\mathcal{C}}[P]$. While ${\mathcal{D}}_{\mathcal{C}}[P\Vert Q]$ is maximum at the short time scales, ${\mathcal{S}}_{\mathcal{C}}[P]$ 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.

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