Are there Dragon Kings in the Stock Market?

ArXiv ID: 2307.03693 “View on arXiv”

Authors: Unknown

Abstract

We undertake a systematic study of historic market volatility spanning roughly five preceding decades. We focus specifically on the time series of realized volatility (RV) of the S&P500 index and its distribution function. As expected, the largest values of RV coincide with the largest economic upheavals of the period: Savings and Loan Crisis, Tech Bubble, Financial Crisis and Covid Pandemic. We address the question of whether these values belong to one of the three categories: Black Swans (BS), that is they lie on scale-free, power-law tails of the distribution; Dragon Kings (DK), defined as statistically significant upward deviations from BS; or Negative Dragons Kings (nDK), defined as statistically significant downward deviations from BS. In analyzing the tails of the distribution with RV > 40, we observe the appearance of “potential” DK which eventually terminate in an abrupt plunge to nDK. This phenomenon becomes more pronounced with the increase of the number of days over which the average RV is calculated – here from daily, n=1, to “monthly,” n=21. We fit the entire distribution with a modified Generalized Beta (mGB) distribution function, which terminates at a finite value of the variable but exhibits a long power-law stretch prior to that, as well as Generalized Beta Prime (GB2) distribution function, which has a power-law tail. We also fit the tails directly with a straight line on a log-log scale. In order to ascertain BS, DK or nDK behavior, all fits include their confidence intervals and p-values are evaluated for the data points to check if they can come from the respective distributions.

Keywords: Realized Volatility, Power-law Tails, Dragon Kings, Generalized Beta Distribution, S&P 500

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 6.5/10
  • Quadrant: Holy Grail
  • Why: The paper uses advanced statistical and probabilistic models, including modified Generalized Beta distributions and power-law tail fitting, indicating high mathematical complexity. It employs real S&P 500 data over five decades and conducts rigorous statistical tests (p-values, confidence intervals), demonstrating strong empirical rigor.
  flowchart TD
    A["Research Goal:<br>Identify Dragon Kings in S&P500 Volatility"] --> B["Data Source:<br>S&P500 Realized Volatility ~50 Years"]
    B --> C["Methodology:<br>Fit Distribution Tails"]
    C --> D["Fit Models:<br>mGB & GB2 Distributions<br>+ Log-Log Linear Fit"]
    D --> E["Statistical Testing:<br>Confidence Intervals & P-values"]
    E --> F{"Classification"}
    F -->|Power-law fit| G["Black Swans BS"]
    F -->|Statistically significant<br>upward deviation| H["Dragon Kings DK<br>Volatility > 40<br>Abrupt plunge to nDK observed"]
    F -->|Statistically significant<br>downward deviation| I["Negative Dragon Kings nDK"]