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

The Kullback-Leibler cluster entropy DC[PHQ] 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 (S&P500, NASDAQ, DJIA, DAX, and FTSEMIB). The Kullback-Leibler functional DC[PHQ] provides complementary perspectives about the stochastic volatility process compared to the Shannon functional S-C[P]. While D-C[PHQ] is maximum at the short timescales, S-C[P] is maximum at the large timescales 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, we build a multiperiod portfolio 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.