# Are There Dragon Kings in the Stock Market?

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## Abstract

**:**

## 1. Introduction

## 2. Time Series of Realized Volatility

## 3. Generalized Beta Distribution Function

## 4. Fitting Distribution of Realized Volatility

#### 4.1. Methodology

#### 4.2. Results

- Full data CDF fit with mGB and GB2 and LF of the tails;
- Same as above shown for $RV>40$;
- p-values of all three fits for $RV>40$, with $p<0.05$ indicating DK and $p>0.95$ nDK;
- LF with its CI;
- GB2 fit with its CI;
- mGB fit with its CI.

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Clauset, A.; Shalizi, C.R.; Newman, M.E. Power-Law Distributions in Empirical Data. SIAM Rev.
**2009**, 51, 661–703. [Google Scholar] [CrossRef] - Gabaix, X. Power laws in economics and finance. Annu. Rev. Econ.
**2009**, 1, 255–294. [Google Scholar] [CrossRef] - Saichev, A.; Malevergne, Y.; Sornette, D. Theory of Zipf’s Law and Beyond; Lecture Notes in Economics and Mathematical Systems; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar]
- Janczura, J.; Weron, R. Black swans or dragon-kings? A simple test for deviations from the power law. Eur. Phys. J. Spec. Top.
**2012**, 205, 79–93. [Google Scholar] [CrossRef] - Pisarenko, V.F.; Sornette, D. Robust statistical tests of Dragon-Kings beyond power law distribution. Eur. Phys. J. Spec. Top.
**2012**, 205, 95–115. [Google Scholar] [CrossRef] - Wheatley, S.; Sornette, D. Multiple Outlier Detection in Samples with Exponential Pareto Tails: Redeeming the Inward Approach Detecting Dragon Kings. arXiv
**2015**, arXiv:1507.08689. [Google Scholar] [CrossRef] - Sornette, D. Dragon-Kings, Black Swans and the Prediction of Crises. Int. J. Terraspace Sci. Eng.
**2009**, 2, 1–18. [Google Scholar] [CrossRef] - Sornette, D.; Ouillon, G. Dragon-kings: Mechanisms, statistical methods and empirical evidence. Eur. Phys. J. Spec. Top.
**2012**, 205, 1–26. [Google Scholar] [CrossRef] - Golosovsky, M.; Solomon, S. Runaway events dominate the heavy tail of citation distributions. Eur. Phys. J. Spec. Top.
**2012**, 205, 303–311. [Google Scholar] [CrossRef] - Medina, J.A. Extreme reaction times determine fluctuation scaling in human color vision. Phys. A
**2016**, 461, 125–132. [Google Scholar] [CrossRef] - Johansen, A.; Sornette, D. Stock market crashes are outliers. Eur. Phys. J. B
**1998**, 1, 141–143. [Google Scholar] [CrossRef] - Johansen, A.; Sornette, D. Large Stock Market Price Drawdowns Are Outliers. J. Risk
**2001**, 4, 69–110. [Google Scholar] [CrossRef] - Sornette, D.; Johansen, A. Significance of log-periodic precursors to financial crashes. Quant. Financ.
**2001**, 1, 452–471. [Google Scholar] [CrossRef] - Filimonov, V.; Sornette, D. Power law scaling and “Dragon-Kings” in distributions of intraday financial drawdowns. Chaos Solitons Fractals
**2015**, 74, 27–45. [Google Scholar] [CrossRef] - CBOE VIX Index. Available online: https://www.cboe.com/tradable_products/vix/ (accessed on 1 January 2024).
- VIX Options and Futures Historical Data. Available online: http://www.cboe.com/products/vix-index-volatility/vix-options-and-futures/vix-index/vix-historical-data (accessed on 1 January 2024).
- Demeterfi, K.; Derman, E.; Kamal, M.; Zou, J. A guide to volatility and variance swaps. J. Deriv.
**1999**, 6, 9–32. [Google Scholar] [CrossRef] - The CBOE Volatility Index—VIX. Previous Location of VIX White Paper, Appears to Have Been Since Removed. 2003. Available online: https://www.cboe.com/micro/vix/vixwhite.pdf (accessed on 1 January 2024).
- Chrstensen, B.J.; Prabhala, N.R. The Relation Between Implied and Realized Volaility. J. Financ. Econ.
**1998**, 50, 125–150. [Google Scholar] [CrossRef] - Vodenska, I.; Chambers, W.J. Understanding the Relationship between VIX and the S&P 500 Index Volatility. In Proceedings of the 26th Australasian Finance and Banking Conference, Sydney, Australia, 17–19 December 2013. [Google Scholar]
- Kownatzki, C. How good is the VIX as a predictor of market risk? J. Account. Financ.
**2016**, 16, 39–60. [Google Scholar] - Russon, M.D.; Vakil, A.F. On the non-linear relationship between VIX and realized SP500 volatility. Invest. Manag. Financ. Innov.
**2017**, 14, 200–206. [Google Scholar] [CrossRef] - Dashti Moghaddam, M.; Liu, Z.; Serota, R.A. Distributions of Historic Market Data—Implied and Realized Volatility. Appl. Econ. Financ.
**2019**, 6, 104–130. [Google Scholar] [CrossRef] - Dashti Moghaddam, M.; Liu, J.; Serota, R.A. Implied and Realized Volatility: A Study of Distributions and Distribution of Difference. Int. J. Financ. Econ.
**2021**, 26, 2581–2594. [Google Scholar] [CrossRef] - Liu, J.; Serota, R.A. Rethinking Generalized Beta family of distributions. Eur. Phys. J. B
**2023**, 96, 24. [Google Scholar] [CrossRef] - McDonald, J.B.; Xu, Y.J. A generlazition of the beta distribution with applications. J. Econom.
**1996**, 66, 133–152. [Google Scholar] [CrossRef] - Risken, H. The Fokker-Planck Equation; Springer: Berlin/Heidelberg, Germany, 1996. [Google Scholar]
- Hertzler, G. “Classical” Probability Distributions for Stochastic Dynamic Models. In Proceedings of the 47th Annual Conference of the Australian Agricultural and Resource Economics Society, Fremantle, Australia, 12–14 February 2003. [Google Scholar]
- Jacobs, K. Stochastic Processes for Physicists; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Dashti Moghaddam, M.; Serota, R.A. Combined Mutiplicative-Heston Model for Stochastic Volatility. Phys. A Stat. Mech. Its Appl.
**2021**, 561, 125263. [Google Scholar] [CrossRef] - NIST Digital Library of Mathematical Functions. Available online: https://dlmf.nist.gov (accessed on 6 February 2024).
- Massey, F.J. The Kolmogorov-Smirnov Test for Goodness of Fit. J. Am. Stat. Assoc.
**1985**, 80, 954–958. [Google Scholar] [CrossRef] - Liu, Z.; Dashti Moghaddam, M.; Serota, R.A. Distributions of Historic Market Data—Stock Returns. Eur. Phys. J. B
**2019**, 92, 60. [Google Scholar] [CrossRef] - Dashti Moghaddam, M.; Liu, Z.; Serota, R.A. Distributions of historic market data: Relaxation and correlations. Eur. Phys. J. B
**2021**, 94, 83. [Google Scholar] [CrossRef] - Liu, J.; Farahani, H.; Serota, R.A. Exploring distributions of housing prices and housing prices index. arXiv
**2023**, arXiv:2312.14325. [Google Scholar]

**Figure 1.**Time series of RV for S&P500, with $RV>17$ shown. From top to bottom $n=1,7,21$. Black dots indicate $RV>{10}^{1.75}\approx 56$.

**Figure 2.**Snapshots of $n=1$ for S&P500 RV time series in Figure 1 around Black Monday, Tech Bubble, Financial Crisis and COVID-19 pandemic.

**Figure 4.**Linear, GB2 and mGB fits of S&P500 RV (

**top**), tail $RV>40$ (

**middle**), and p-values (

**bottom**) for $n=1$.

**Figure 5.**Same as the middle Figure 4 (tail parts of fits of S&P500 RV for $n=1$) with the respective CI, “potential” DK (up triangles) and nDK (down triangles).

**Figure 6.**Linear, GB2 and mGB fits of S&P500 RV (

**top**), tail $RV>40$ (

**middle**), and p-values (

**bottom**) for $n=5$.

**Figure 7.**Same as the middle Figure 6 (tail parts of fits of S&P500 RV for $n=5$) with the respective CI, “potential” DK (up triangles) and nDK (down triangles).

**Figure 8.**Linear, GB2 and mGB fits of S&P500 RV (

**top**), tail $RV>40$ (

**middle**), and p-values (

**bottom**) for $n=7$.

**Figure 9.**Same as the middle Figure 8 (tail parts of fits of S&P500 RV for $n=7$) with the respective CI, “potential” DK (up triangles) and nDK (down triangles).

**Figure 10.**Linear, GB2 and mGB fits of S&P500 RV (

**top**), tail $RV>40$ (

**middle**), and p-values (

**bottom**) for $n=17$.

**Figure 11.**Same as the middle Figure 10 (tail parts of fits of S&P500 RV for $n=17$) with the respective CI, “potential” DK (up triangles) and nDK (down triangles).

**Figure 12.**Linear, GB2 and mGB fits of S&P500 RV (

**top**), tail $RV>40$ (

**middle**), and p-values (

**bottom**) for $n=21$.

**Figure 13.**Same as the middle Figure 12 (tail parts of fits of S&P500 RV for $n=21$) with the respective CI, “potential” DK (up triangles) and nDK (down triangles).

**Figure 14.**From top to bottom, LF of the tail of S&P500 RV, $RV>40$, for $n=7,17,21$ with end values $>0.9max\left\{RV\right\}$ excluded from LF.

**Figure 15.**As a function of n: linear slope values of linear (stars) and GB2 (squares) fits (also listed in Table 3)—(

**left**); KS statistic values indicating relative goodness of fit of mGB (stars) and GB2 (squares)—(

**right**).

n | Parameters of $\mathbf{mGB}(\mathit{\alpha},{\mathit{\beta}}_{1},{\mathit{\beta}}_{2},\mathit{p},\mathit{q})$ | Parameters of $\mathbf{GB}2(\mathit{\alpha},\mathit{\beta},\mathit{p},\mathit{q})$ |
---|---|---|

1 | $(1.5500,399.9009,27.4233,0.6519,1.7828)$ | $(1.5289,27.4156,0.6545,2.7780)$ |

5 | $(2.3708,200.5519,10.7210,1.7255,0.4456)$ | $(2.3749,10.7480,1.7205,1.4409)$ |

7 | $(2.4744,180.8711,9.7136,2.1430,0.3848)$ | $(2.4681,9.7169,2.1371,1.3796)$ |

17 | $(2.3026,120.110,6.3561,6.1121,0.5403)$ | $(2.2797,6.3471,6.0946,1.5334)$ |

21 | $(2.4016,104.9925,6.3853,6.3429,0.5043)$ | $(2.3850,6.2193,6.5117,1.4457)$ |

n | SE of Parameters of $\mathbf{mGB}\phantom{\rule{3.33333pt}{0ex}}(\mathit{\alpha},{\mathit{\beta}}_{1},{\mathit{\beta}}_{2},\mathit{p},\mathit{q})$ | SE of Parameters of $\mathbf{GB}2\phantom{\rule{3.33333pt}{0ex}}(\mathit{\alpha},\mathit{\beta},\mathit{p},\mathit{q})$ |
---|---|---|

1 | $(0.2049,5.6537,0.3884,0.1196,0.3171)$ | $(0.0079,0.0018,0.0018,0.0026)$ |

5 | $(0.0690,0.0127,0.0152,0.0175,0.0922)$ | $(0.0018,0.0004,0.0004,0.0006)$ |

7 | $(0.1193,0.0246,0.0348,0.0246,0.0281)$ | $(0.0021,0.0005,0.0005,0.0007)$ |

17 | $(0.0614,0.0220,0.0150,0.0288,0.0325)$ | $(0.0055,0.0013,0.0013,0.0018)$ |

21 | $(0.6300,0.2807,0.0114,0.3419,0.2451)$ | $(0.0085,0.0020,0.0020,0.0028)$ |

n | $\mathbf{GB}2$ | LF |
---|---|---|

1 | −4.25 | −3.01 |

5 | −3.42 | −2.95 |

7 | −3.41 | −2.84 |

17 | −3.50 | −2.34 |

21 | −3.45 | −2.28 |

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**MDPI and ACS Style**

Liu, J.; Dashti Moghaddam, M.; Serota, R.A.
Are There Dragon Kings in the Stock Market? *Foundations* **2024**, *4*, 91-113.
https://doi.org/10.3390/foundations4010008

**AMA Style**

Liu J, Dashti Moghaddam M, Serota RA.
Are There Dragon Kings in the Stock Market? *Foundations*. 2024; 4(1):91-113.
https://doi.org/10.3390/foundations4010008

**Chicago/Turabian Style**

Liu, Jiong, Mohammadamin Dashti Moghaddam, and Rostislav A. Serota.
2024. "Are There Dragon Kings in the Stock Market?" *Foundations* 4, no. 1: 91-113.
https://doi.org/10.3390/foundations4010008