Stable and Tempered Stable Distributions

Rachev, Svetlozar T.; Kim, Young S.; Bianchi, Michele Leonardo; Fabozzi, Frank J.

doi:10.1002/9781118182635.efm0101

Cited by 2 publications

(1 citation statement)

References 13 publications

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“…Additionally, λ + and λ − are also skewness parameters. If λ + > λ − (λ + < λ − ), then the distribution is skewed to the left (right), and if λ + = λ − , then it is symmetric [27,28]. α and λ are related to the degree of peakedness and thickness of the distribution.…”

Section: Generalized Tempered Stable (Gts) Process: Overviewmentioning

confidence: 99%

Bitcoin versus S&P 500 Index: Return and Risk Analysis

Nzokem,

Maposa

2024

MCA

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The S&P 500 Index is considered the most popular trading instrument in financial markets. With the rise of cryptocurrencies over the past few years, Bitcoin has grown in popularity and adoption. This study analyzes the daily return distribution of Bitcoin and the S&P 500 Index and assesses their tail probabilities using two financial risk measures. As a methodology, we use Bitcoin and S&P 500 Index daily return data to fit the seven-parameter General Tempered Stable (GTS) distribution using the advanced fast fractional Fourier transform (FRFT) scheme developed by combining the fast fractional Fourier transform algorithm and the 12-point composite Newton–Cotes rule. The findings show that peakedness is the main characteristic of the S&P 500 Index return distribution, whereas heavy-tailedness is the main characteristic of Bitcoin return distribution. The GTS distribution shows that 80.05% of S&P 500 returns are within −1.06% and 1.23% against only 40.32% of Bitcoin returns. At a risk level (α), the severity of the loss (AVaRα(X)) on the left side of the distribution is larger than the severity of the profit (AVaR1−α(X)) on the right side of the distribution. Compared to the S&P 500 Index, Bitcoin has 39.73% more prevalence to produce high daily returns (more than 1.23% or less than −1.06%). The severity analysis shows that, at α risk level, the average value-at-risk (AVaR(X)) of Bitcoin returns at one significant figure is four times larger than that of the S&P 500 Index returns at the same risk.

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Section: Generalized Tempered Stable (Gts) Process: Overviewmentioning

confidence: 99%

Bitcoin versus S&P 500 Index: Return and Risk Analysis

Nzokem,

Maposa

2024

MCA

View full text Add to dashboard Cite

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Fitting the Seven-Parameter Generalized Tempered Stable Distribution to Financial Data

Nzokem,

Maposa

2024

JRFM

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This 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 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 heavy-tailed data (Bitcoin and Ethereum returns) and two peaked data (S&P 500 and SPY ETF returns). For each historical data, the estimation results show that 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.

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Stable and Tempered Stable Distributions

Cited by 2 publications

References 13 publications

Bitcoin versus S&P 500 Index: Return and Risk Analysis

Bitcoin versus S&P 500 Index: Return and Risk Analysis

Fitting the Seven-Parameter Generalized Tempered Stable Distribution to Financial Data

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