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19 pages, 946 KiB

Open AccessArticle

Enhanced Fast Fractional Fourier Transform (FRFT) Scheme Based on Closed Newton-Cotes Rules

by Aubain Nzokem, Daniel Maposa and Anna M. Seimela

Axioms 2025, 14(7), 543; https://doi.org/10.3390/axioms14070543 - 20 Jul 2025

Viewed by 209

The paper presents an enhanced numerical framework for computing the one-dimensional fast Fractional Fourier Transform (FRFT) by integrating closed-form Composite Newton-Cotes quadrature rules. We show that a FRFT of a

Q N

-length weighted sequence can be decomposed analytically into two mathematically [...] Read more.

Q N

-length weighted sequence can be decomposed analytically into two mathematically commutative compositions: one involving the composition of a FRFT of an N-length sequence and a FRFT of a Q-length weighted sequence, and the other in reverse order. The composite FRFT approach is applied to the inversion of Fourier and Laplace transforms, with a focus on estimating probability densities for distributions with complex-valued characteristic functions. Numerical experiments on the Variance-Gamma (VG) and Generalized Tempered Stable (GTS) models show that the proposed scheme significantly improves accuracy over standard (non-weighted) fast FRFT and classical Newton-Cotes quadrature, while preserving computational efficiency. The findings suggest that the composite FRFT framework offers a robust and mathematically sound tool for transform-based numerical approximations, particularly in applications involving oscillatory integrals and complex-valued characteristic functions. Full article

(This article belongs to the Special Issue Numerical Analysis and Applied Mathematics)

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26 pages, 1180 KiB

Open AccessArticle

Stochastic Modeling of Wind Derivatives with Application to the Alberta Energy Market

by Sudeesha Warunasinghe and Anatoliy Swishchuk

Risks 2024, 12(2), 18; https://doi.org/10.3390/risks12020018 - 23 Jan 2024

Cited by 1 | Viewed by 2394

Abstract

Wind-power generators around the world face two risks, one due to changes in wind intensity impacting energy production, and the second due to changes in electricity retail prices. To hedge these risks simultaneously, the quanto option is an ideal financial tool. The natural logarithm of electricity prices of the study will be modeled with a variance gamma (VG) and normal inverse Gaussian (NIG) processes, while wind speed and power series will be modeled with an Ornstein–Uhlenbeck (OU) process. Since the risk from changing wind-power production and spot prices is highly correlated, we must model this correlation as well. This is reproduced by replacing the small jumps of the Lévy process with a Brownian component and correlating it with wind power and speed OU processes. Then, we will study the income of the wind-energy company from a stochastic point of view, and finally, we will price the quanto option of European put style for the wind-energy producer. We will compare quanto option prices obtained from the VG process and NIG process. The novelty brought into this study is the use of a new dataset in a new geographic location and a new Lévy process, VG, apart from NIG. Full article

(This article belongs to the Special Issue Risks: Feature Papers 2023)

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28 pages, 1923 KiB

Open AccessArticle

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

by Aubain Hilaire Nzokem

J. Risk Financial Manag. 2023, 16(1), 55; https://doi.org/10.3390/jrfm16010055 - 16 Jan 2023

Cited by 7 | Viewed by 3895

Abstract

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 [...] Read more.

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

\frac{δ}{σ^{2}} - \sqrt{\frac{δ^{2}}{σ^{4}} + \frac{2}{θ σ^{2}}}

and

\frac{δ}{σ^{2}} + \sqrt{\frac{δ^{2}}{σ^{4}} + \frac{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. Full article

(This article belongs to the Special Issue Mathematical and Empirical Finance)

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