arXiv subjects

A. H. Nzokem

Publications and source records attributed to A. H. Nzokem.

10 recordsLinked to original sources

arXiv2025

Comparing Bitcoin and Ethereum tail behavior via Q-Q analysis of cryptocurrency returns

The cryptocurrency market presents both significant investment opportunities and higher risks relative to traditional financial assets. This study examines the tail behavior of daily returns for two leading cryptocurrencies, Bitcoin and Ethereum, using seven-parameter estimates from prior research, which applied the Generalized Tempered Stable (GTS) distribution. Quantile-quantile (Q-Q) plots against the Normal distribution reveal that both assets exhibit heavy-tailed return distributions. However, Ethereum consistently shows a greater frequency of extreme values than would be expected under its Bitcoin-modeled counterpart, indicating more pronounced tail risk.

q-fin.ST↗

arXiv2025

Enhanced Fractional Fourier Transform (FRFT) scheme based on closed Newton-Cotes rules

The paper improves the accuracy of the one-dimensional fractional Fourier transform (FRFT) by leveraging closed Newton-Cotes quadrature rules. Using the weights derived from the Composite Newton-Cotes rules of order QN, we demonstrate that the FRFT of a QN-long weighted sequence can be expressed as two composites of FRFTs. The first composite consists of an FRFT of a Q-long weighted sequence and an FRFT of an N-long sequence. Similarly, the second composite comprises an FRFT of an N-long weighted sequence and an FRFT of a Q-long sequence. Empirical results suggest that the composite FRFTs exhibit the commutative property and maintain consistency both algebraically and numerically. The proposed composite FRFT approach is applied to the inversion of Fourier and Laplace transforms, where it outperforms both the standard non-weighted FRFT and the Newton-Cotes integration method, though the improvement over the latter is less pronounced.

math.NA↗

arXiv2025

European Option Pricing Under Generalized Tempered Stable Process: Empirical Analysis

The paper investigates the performance of the European option price when the log asset price follows a rich class of Generalized Tempered Stable (GTS) distribution. The GTS distribution is an alternative to Normal distribution and $α$-stable distribution for modeling asset return and many physical and economic systems. The data used in the option pricing computation comes from fitting the GTS distribution to the underlying S\&P 500 Index return distribution. The Esscher transform method shows that the GTS distribution preserves its structure. The extended Black-Scholes formula and the Generalized Black-Scholes Formula are applied in the study. The 12-point rule Composite Newton-Cotes Quadrature and the Fractional Fast Fourier (FRFT) algorithms were implemented, and they yield the same European option price at two decimal places. Compared to the option price under the GTS distribution, the Black-Scholes (BS) model is underpriced for the Near-The-Money (NTM) and the in-the-money (ITM) options. However, the BS model and GTS European options yield the same option price for the deep out-of-the-money (OTM) and the deep-in-the-money (ITM) options.

q-fin.PR↗

arXiv2024

Self-Decomposable Laws Associated with General Tempered Stable (GTS) Distribution and their Simulation Applications

The paper describes the self-decomposable distribution and the background driving Lévy process (BDLP) associated with the Generalized Tempered Stable (GTS) distribution. Two distributions are provided: the background driving Lévy process (BDLP) of the GTS distribution and the self-decomposable distribution generated by the GTS distribution as BDLP. The derived self-decomposable distribution and the GTS distribution are used as stationary distribution in the Ornstein-Uhlenbeck type process. A simulation method, based on sampling the random integral representation, is applied to mimic S&P 500 Index and Bitcoin daily cumulative return process.

math.PR↗

arXiv2023

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

The S&P 500 index is considered the most popular trading instrument in financial markets. With the rise of cryptocurrencies over the past years, Bitcoin has also grown in popularity and adoption. The paper aims to analyze the daily return distribution of the Bitcoin and S&P 500 index and assess their tail probabilities through 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 (FRFT) algorithm and the 12-point rule Composite Newton-Cotes Quadrature. The findings show that peakedness is the main characteristic of the S&P 500 return distribution, whereas heavy-tailedness is the main characteristic of the 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 ($AVaR_α(X)$) on the left side of the distribution is larger than the severity of the profit ($AVaR_{1-α}(X)$) 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 a risk level ($α$) the average value-at-risk ($AVaR(X)$) of the bitcoin returns at one significant figure is four times larger than that of the S&P 500 index returns at the same risk.

q-fin.ST↗

arXiv2023

Pricing European Options under Stochastic Volatility Models: Case of five-Parameter Variance-Gamma Process

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 $\fracδ{σ^2} - \sqrt{\frac{δ^2}{σ^4}+\frac{2}{θσ^2}}$ and $\fracδ{σ^2}+ \sqrt{\frac{δ^2}{σ^4}+\frac{2}{θσ^2}}$; and the VG process converges asymptotically in distribution to a Lévy process driven by a normal distribution with mean $(μ+ αθδ)$ and variance $α(θ^2δ^2 + σ^2θ)$. 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

q-fin.PR↗

arXiv2022

Fitting Generalized Tempered Stable distribution: Fractional Fourier Transform (FRFT) Approach

The paper investigates the rich class of Generalized Tempered Stable distribution, an alternative to Normal distribution and the $α$-Stable distribution for modelling asset return and many physical and economic systems. Firstly, we explore some important properties of the Generalized Tempered Stable (GTS) distribution. The theoretical tools developed are used to perform empirical analysis. The GTS distribution is fitted using S&P 500, SPY ETF and Bitcoin BTC. The Fractional Fourier Transform (FRFT) technique evaluates the probability density function and its derivatives in the maximum likelihood procedure. Based on the results from the statistical inference and the Kolmogorov-Smirnov (K-S) goodness-of-fit, the GTS distribution fits the underlying distribution of the SPY ETF return. The right side of the Bitcoin BTC return, and the left side of the S&P 500 return underlying distributions fit the Tempered Stable distribution; while the left side of the Bitcoin BTC return and the right side of the S&P 500 return underlying distributions are modelled by the compound Poisson process

q-fin.ST↗

arXiv2022

Variance-Gamma (VG) model: Fractional Fourier Transform (FRFT)

The paper examines the Fractional Fourier Transform (FRFT) based technique as a tool for obtaining the probability density function and its derivatives, and mainly for fitting stochastic model with the fundamental probabilistic relationships of infinite divisibility. The probability density functions are computed, and the distributional proprieties are reviewed for Variance-Gamma (VG) model. The VG model has been increasingly used as an alternative to the Classical Lognormal Model (CLM) in modelling asset prices. The VG model was estimated by the FRFT. The data comes from the SPY ETF historical data. The Kolmogorov-Smirnov (KS) goodness-of-fit shows that the VG model fits the cumulative distribution of the sample data better than the CLM. The best VG model comes from the FRFT estimation.

stat.ME↗

arXiv2021

Fitting Infinitely divisible distribution: Case of Gamma-Variance Model

The paper examines the Fractional Fourier Transform (FRFT) based technique as a tool for obtaining probability density function and its derivatives, and mainly for fitting stochastic model with the fundamental probabilistic relationships of infinite divisibility. The probability density functions are computed and the distributional proprieties such as leptokurtosis, peakedness, and asymmetry are reviewed for Variance-Gamma (VG) model and Compound Poisson with Normal Compounding model. The first and second derivatives of probability density function of the VG model are also computed in order to build the Fisher information matrix for the Maximum likelihood method. The VG model has been increasingly used as an alternative to the Classical Lognormal Model (CLM) in modelling asset price. The VG model with fives parameters was estimated by the FRFT. The data comes from the daily SPY ETF price data. The Kolmogorov-Smirnov (KS) goodness-of-fit shows that the VG model fits better the empirical cumulative distribution than the CLM. The best VG model comes from the FRFT estimation.

stat.ME↗

arXiv2021

SIS Epidemic Model: Birth-and-Death Markov Chain Approach

We are interested in describing the infected size of the SIS Epidemic model using Birth-Death Markov process. The Susceptible-Infected-Susceptible (SIS) model is defined within a population of constant size $M$; the size is kept constant by replacing each death with a newborn healthy individual. The life span of each individual in the population is modelled by an exponential distribution with parameter $α$; and the disease spreads within the population is modelled by a Poisson process with a rate $λ_{I}$. $λ_{I}=βI(1-\frac{I}{M}) $ is similar to the instantaneous rate in the logistic population growth model. The analysis is focused on the disease outbreak, where the reproduction number $R=\fracβα $ is greater than one. As methodology, we use both numerical and analytical approaches. The analysis relies on the stationary distribution for Birth and Death Markov process. The numerical approach creates sample path simulations into order show the infected size dynamics, and the relationship between infected size and $R$. As $M$ becomes large, some stable statistical characteristics of the infected size distribution can be deduced. And the infected size is shown analytically to follow a normal distribution with mean $(1-\frac{1}{R}) M$ and Variance $\frac{M}{R} $.

q-bio.PE↗