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Fitting Generalized Tempered Stable Distributions to Financial Returns

Article arXiv papers · Author: Aubain Nzokem et al.
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Summary

This paper presents a method for estimating the seven parameters of a Generalized Tempered Stable distribution when its probability density has no available mathematical expression. That limitation makes direct maximum likelihood estimation unsuitable. Instead, the approach uses the distribution’s characteristic function together with the fractional Fourier transform to obtain parameter estimates.

The method is applied to Bitcoin and Ethereum returns, described as heavily tailed, and to S&P 500 and SPY ETF returns, described as peaked. The fit is assessed with Kolmogorov-Smirnov, Anderson-Darling, and Pearson chi-squared statistics. The reported results favor the GTS model over a two-parameter geometric Brownian motion hypothesis and several alternative distributions. The local parameter estimate is an exception to the reported statistical significance of the other six parameters. These conclusions concern the datasets and comparisons in the study; the description does not provide sample periods, sensitivity checks, or evidence about the distribution’s performance for forecasting or pricing beyond goodness of fit.

Key ideas

  • The GTS density lacks a closed mathematical expression, complicating direct maximum likelihood estimation.
  • The proposed estimation method combines the characteristic function with the fractional Fourier transform.
  • The method is applied to cryptocurrency and equity return data with different distribution shapes.
  • Several goodness-of-fit tests are used to compare the estimated distribution with alternatives.
  • The reported fit is strong in the studied data, though the local parameter is not statistically significant.

Tags

StatisticsCryptoEquitiesDerivatives pricing
Full text
# Fitting the seven-parameter Generalized Tempered Stable distribution to the financial data


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









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, $μ$. 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 GTS distribution fits significantly with a very high p-value; and outperforms the Kobol, Carr-Geman-Madan-Yor, and Bilateral Gamma distributions.

Shown in full with attribution under the source's licence. Licence: abstract CC0

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.