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Fitting the seven-parameter Generalized Tempered Stable distribution to the financial data

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  • A. H Nzokem

Abstract

The paper proposes and implements a methodology to fit a seven-parameter Generalized Tempered Stable (GTS) distribution to financial data. The nonexistence of the mathematical expression of the GTS probability density function makes the maximum likelihood estimation (MLE) inadequate for providing parameter estimations. Based on the function characteristic and the fractional Fourier transform (FRFT), We provide a comprehensive approach to circumvent the problem and yield a good parameter estimation of the GTS probability. The methodology was applied to fit two heavily tailed data (Bitcoin and Ethereum returns) and two peaked data (S&P 500 and SPY ETF returns). For each index, the estimation results show that the six parameter estimations are statistically significant except for the local parameter ($\mu$). The goodness of fit was assessed through Kolmogorov-Smirnov, Anderson-Darling, and Pearson's chi-squared statistics. While the two-parameter geometric Brownian motion (GBM) hypothesis is always rejected, the Generalized Tempered Sable (GTS) distribution fits significantly with a very high P_value; and outperforms the Kobol, CGMY, and Bilateral Gamma distributions.

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  • A. H Nzokem, 2024. "Fitting the seven-parameter Generalized Tempered Stable distribution to the financial data," Papers 2410.19751, arXiv.org.
    • Handle: RePEc:arx:papers:2410.19751
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    References listed on IDEAS

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    1. A. H. Nzokem, 2022. "Pricing European Options under Stochastic Volatility Models: Case of five-Parameter Variance-Gamma Process," Papers 2201.03378, arXiv.org, revised Jan 2023.
    2. Peter Carr & Hélyette Geman & Dilip B. Madan & Marc Yor, 2003. "Stochastic Volatility for Lévy Processes," Mathematical Finance, Wiley Blackwell, vol. 13(3), pages 345-382, July.
    3. Maashele Kholofelo Metwane & Daniel Maposa, 2023. "Extreme Value Theory Modelling of the Behaviour of Johannesburg Stock Exchange Financial Market Data," IJFS, MDPI, vol. 11(4), pages 1-27, November.
    4. Aubain Hilaire Nzokem, 2023. "Pricing European Options under Stochastic Volatility Models: Case of Five-Parameter Variance-Gamma Process," JRFM, MDPI, vol. 16(1), pages 1-28, January.
    5. A. H. Nzokem & V. T. Montshiwa, 2022. "Fitting Generalized Tempered Stable distribution: Fractional Fourier Transform (FRFT) Approach," Papers 2205.00586, arXiv.org, revised Jun 2022.
    6. A. H. Nzokem, 2023. "European Option Pricing Under Generalized Tempered Stable Process: Empirical Analysis," Papers 2304.06060, arXiv.org, revised Aug 2023.
    7. Marsaglia, George & Tsang, Wai Wan & Wang, Jingbo, 2003. "Evaluating Kolmogorov's Distribution," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 8(i18).
    8. Marsaglia, George & Marsaglia, John, 2004. "Evaluating the Anderson-Darling Distribution," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 9(i02).
    9. Michael Grabchak & Gennady Samorodnitsky, 2010. "Do financial returns have finite or infinite variance? A paradox and an explanation," Quantitative Finance, Taylor & Francis Journals, vol. 10(8), pages 883-893.
    10. repec:dau:papers:123456789/1392 is not listed on IDEAS
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