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

ArXiv ID: 2409.10543 “View on arXiv”

Authors: Unknown

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.

Keywords: Kullback-Leibler entropy, Realized volatility, Fractional stochastic process, Portfolio optimization, Time-series clustering

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail
• Why: The paper employs advanced mathematical concepts like Kullback-Leibler divergence, coarse-graining, and fractional stochastic processes with Hurst exponents, indicating high math complexity. It also demonstrates strong empirical rigor by applying the methodology to real financial data (five major assets), constructing a portfolio based on the results, and comparing performance against established benchmarks like Markowitz.

  flowchart TD
    A["Research Goal:<br>Quantify Volatility Correlation &<br>Risk Diversity using KL Divergence"] --> B["Data Collection:<br>Realized Volatility of 5 Assets<br>SP500, NASDAQ, DJIA, DAX, FTSEMIB"]
    B --> C["Methodology:<br>Cluster Time-Series &<br>Calculate KL Cluster Entropy<br>vs Shannon Entropy"]
    C --> D["Modeling:<br>Fractional Stochastic Process<br>Power-law distributions H>1/2"]
    C --> E["Portfolio Optimization:<br>Diversity Index derived from<br>KL Entropy Measure"]
    D --> F["Key Findings:<br>1. KL max at short scales<br>2. Shannon max at large scales<br>3. KL-based portfolio<br>outperforms Markowitz"]
    E --> F