arXiv 5 Sep 2022 · Finance — Statistical Finance · publishedThe European Physical Journal B (2023) · 7 citations (OpenAlex)
arXiv:2209.05225 · PDF · DOI · OpenAlex · Extracted main text
We approach the Generalized Beta (GB) family of distributions using a mean-reverting stochastic differential equation (SDE) for a power of the variable, whose steady-state (stationary) probability density function (PDF) is a modified GB (mGB) distribution. The SDE approach allows for a lucid explanation of Generalized Beta Prime (GB2) and Generalized Beta (GB1) limits of GB distribution and, further down, of Generalized Inverse Gamma (GIGa) and Generalized Gamma (GGa) limits, as well as describe the transition between the latter two. We provide an alternative form to the "traditional" GB PDF to underscore that a great deal of usefulness of GB distribution lies in its allowing a long-range power-law behavior to be ultimately terminated at a finite value. We derive the cumulative distribution function (CDF) of the "traditional" GB, which belongs to the family generated by the regularized beta function and is crucial for analysis of the tails of the distribution. We analyze fifty years of historical data on realized market volatility, specifically for S&P500, as a case study of the use of GB/mGB distributions and show that its behavior is consistent with that of negative Dragon Kings.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | J. B. McDonald, Y. J. Xu, A generlazition of the beta distributionwi… (1996) 133–152 | 1.000 | 6 | 3 | 100% |
| 2 | G. Hertzler, "classical" probability distributions for stochastic dy… (2003) | 0.928 | 4 | 3 | 100% |
| 3 | J. Liu, R. A. Serota, Are there dragon kings in the stoock market?,… (2022) | 0.928 | 4 | 3 | 100% |
| 4 | V. F. Pisarenko, D. Sornette, Robust statistical tests of dragon-kin… (2012) 95–115 | 0.928 | 4 | 3 | 100% |
| 5 | M. Dashti Moghaddam, R. Serota, Combined mutiplicative-heston model… (2021) 125263 | 0.874 | 7 | 2 | 100% |
| 6 | D. Chotikapanjch (Ed.), Modeling Income Distributions and Lorenz Cur… (2008) | 0.843 | 3 | 3 | 100% |
| 7 | J. B. McDonald, Some generalized functions for the size distribution… (1984) 647–665 | 0.843 | 3 | 3 | 100% |
| 8 | N. Eugene, C. Lee, F. Famoye, Beta-normal distribution and its appli… (2002) 497–512 | 0.737 | 3 | 2 | 100% |
| 9 | T. Ma, R. Serota, A model for stock returns and volatility, Physica… (2014) 89–115 | 0.737 | 3 | 2 | 100% |
| 10 | C. Alexander, G. M. Cordeiro, O. E. M. M, J. M. Sarabia, Generalized… (2012) 1880–1897 | 0.644 | 2 | 2 | 100% |
Showing the top 10 of 42 scored citations.