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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 ( A V a R α (X) ) on the left side of the distribution is larger than the severity of the profit ( A V a R 1 − 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 ( A V a R (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. [ABSTRACT FROM AUTHOR]

The paper builds a Variance-Gamma (VG) model with five parameters: location (μ), symmetry (δ), volatility (σ), shape (α), and scale (θ); and studies its application to the pricing of European options. The results of our analysis show that the five-parameter VG model is a stochastic volatility model with a Γ (α , θ) Ornstein–Uhlenbeck type process; the associated Lévy density of the VG model is a KoBoL family of order ν = 0 , intensity α , and steepness parameters δ σ 2 − δ 2 σ 4 + 2 θ σ 2 and δ σ 2 + δ 2 σ 4 + 2 θ σ 2 ; and the VG process converges asymptotically in distribution to a Lévy process driven by a normal distribution with mean (μ + α θ δ) and variance α (θ 2 δ 2 + σ 2 θ) . The data used for empirical analysis were obtained by fitting the five-parameter Variance-Gamma (VG) model to the underlying distribution of the daily SPY ETF data. Regarding the application of the five-parameter VG model, the twelve-point rule Composite Newton–Cotes Quadrature and Fractional Fast Fourier (FRFT) algorithms were implemented to compute the European option price. Compared to the Black–Scholes (BS) model, empirical evidence shows that the VG option price is underpriced for out-of-the-money (OTM) options and overpriced for in-the-money (ITM) options. Both models produce almost the same option pricing results for deep out-of-the-money (OTM) and deep-in-the-money (ITM) options. [ABSTRACT FROM AUTHOR]