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15 pages, 276 KB

Open AccessFeature PaperArticle

by Michael Grabchak

Mathematics 2025, 13(6), 907; https://doi.org/10.3390/math13060907 - 8 Mar 2025

Viewed by 1013

A common approach to simulating a Lévy process is to truncate its shot-noise representation. We focus on subordinators and introduce the remainder process, which represents the jumps that are removed by the truncation. We characterize when these processes are self-similar and show that, [...] Read more.

A common approach to simulating a Lévy process is to truncate its shot-noise representation. We focus on subordinators and introduce the remainder process, which represents the jumps that are removed by the truncation. We characterize when these processes are self-similar and show that, in the self-similar case, they can be indexed by a parameter

α \in (- \infty, 1)

. When

α \in (0, 1)

, they correspond to

α

-stable distributions, and when

α = 0

, they correspond to certain generalizations of the Dickman distribution. Thus, the Dickman distribution plays the role of a 0-stable distribution in this context. Full article

(This article belongs to the Section D1: Probability and Statistics)

18 pages, 1269 KB

Open AccessArticle

Beyond the Traditional VIX: A Novel Approach to Identifying Uncertainty Shocks in Financial Markets

by Ayush Jha, Abootaleb Shirvani, Svetlozar T. Rachev and Frank J. Fabozzi

J. Risk Financial Manag. 2025, 18(1), 11; https://doi.org/10.3390/jrfm18010011 - 29 Dec 2024

Viewed by 5142

Abstract

We introduce a new identification strategy for uncertainty shocks to explain macroeconomic volatility in financial markets. The Chicago Board Options Exchange Volatility Index (VIX) measures the market expectations of future volatility, but traditional methods based on second-moment shocks and the time-varying volatility of [...] Read more.

We introduce a new identification strategy for uncertainty shocks to explain macroeconomic volatility in financial markets. The Chicago Board Options Exchange Volatility Index (VIX) measures the market expectations of future volatility, but traditional methods based on second-moment shocks and the time-varying volatility of the VIX often do not effectively to capture the non-Gaussian, heavy-tailed nature of asset returns. To address this, we constructed a revised VIX by fitting a double-subordinated Normal Inverse Gaussian Lévy process to S&P 500 log returns, to provide a more comprehensive measure of volatility that captures the extreme movements and heavy tails observed in financial data. Using an axiomatic framework, we developed a family of risk–reward ratios that, when computed with our revised VIX and fitted to a long-memory time series model, provide a more precise identification of uncertainty shocks in financial markets. Full article

(This article belongs to the Special Issue Financial Markets, Financial Volatility and Beyond, 3rd Edition)

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19 pages, 2444 KB

Open AccessEditor’s ChoiceArticle

Fractional Telegrapher’s Equation under Resetting: Non-Equilibrium Stationary States and First-Passage Times

by Katarzyna Górska, Francisco J. Sevilla, Guillermo Chacón-Acosta and Trifce Sandev

Entropy 2024, 26(8), 665; https://doi.org/10.3390/e26080665 - 5 Aug 2024

Cited by 8 | Viewed by 1862

Abstract

We consider two different time fractional telegrapher’s equations under stochastic resetting. Using the integral decomposition method, we found the probability density functions and the mean squared displacements. In the long-time limit, the system approaches non-equilibrium stationary states, while the mean squared displacement saturates [...] Read more.

We consider two different time fractional telegrapher’s equations under stochastic resetting. Using the integral decomposition method, we found the probability density functions and the mean squared displacements. In the long-time limit, the system approaches non-equilibrium stationary states, while the mean squared displacement saturates due to the resetting mechanism. We also obtain the fractional telegraph process as a subordinated telegraph process by introducing operational time such that the physical time is considered as a Lévy stable process whose characteristic function is the Lévy stable distribution. We also analyzed the survival probability for the first-passage time problem and found the optimal resetting rate for which the corresponding mean first-passage time is minimal. Full article

(This article belongs to the Special Issue Theory and Applications of Hyperbolic Diffusion and Shannon Entropy)

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21 pages, 5564 KB

Open AccessArticle

Bitcoin Volatility and Intrinsic Time Using Double-Subordinated Lévy Processes

by Abootaleb Shirvani, Stefan Mittnik, William Brent Lindquist and Svetlozar Rachev

Risks 2024, 12(5), 82; https://doi.org/10.3390/risks12050082 - 20 May 2024

Cited by 4 | Viewed by 3500

Abstract

We propose a doubly subordinated Lévy process, the normal double inverse Gaussian (NDIG), to model the time series properties of the cryptocurrency bitcoin. By using two subordinated processes, NDIG captures both the skew and fat-tailed properties of, as well as the intrinsic time [...] Read more.

We propose a doubly subordinated Lévy process, the normal double inverse Gaussian (NDIG), to model the time series properties of the cryptocurrency bitcoin. By using two subordinated processes, NDIG captures both the skew and fat-tailed properties of, as well as the intrinsic time driving, bitcoin returns and gives rise to an arbitrage-free option pricing model. In this framework, we derive two bitcoin volatility measures. The first combines NDIG option pricing with the Chicago Board Options Exchange VIX model to compute an implied volatility; the second uses the volatility of the unit time increment of the NDIG model. Both volatility measures are compared to the volatility based on the historical standard deviation. With appropriate linear scaling, the NDIG process perfectly captures the observed in-sample volatility. Full article

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19 pages, 339 KB

Open AccessArticle

Fractional Equations for the Scaling Limits of Lévy Walks with Position-Dependent Jump Distributions

by Vassili N. Kolokoltsov

Mathematics 2023, 11(11), 2566; https://doi.org/10.3390/math11112566 - 3 Jun 2023

Cited by 1 | Viewed by 1790

Abstract

Lévy walks represent important modeling tools for a variety of real-life processes. Their natural scaling limits are known to be described by the so-called material fractional derivatives. So far, these scaling limits have been derived for spatially homogeneous walks, where Fourier and Laplace [...] Read more.

Lévy walks represent important modeling tools for a variety of real-life processes. Their natural scaling limits are known to be described by the so-called material fractional derivatives. So far, these scaling limits have been derived for spatially homogeneous walks, where Fourier and Laplace transforms represent natural tools of analysis. Here, we derive the corresponding limiting equations in the case of position-depending times and velocities of walks, where Fourier transforms cannot be effectively applied. In fact, we derive three different limits (specified by the way the process is stopped at an attempt to cross the boundary), leading to three different multi-dimensional versions of Caputo–Dzherbashian derivatives, which correspond to different boundary conditions for the generators of the related Feller semigroups and processes. Some other extensions and generalizations are analyzed. Full article

(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications)

24 pages, 781 KB

Open AccessArticle

Valuing Exchange Options under an Ornstein-Uhlenbeck Covariance Model

by Enrique Villamor and Pablo Olivares

Int. J. Financial Stud. 2023, 11(2), 55; https://doi.org/10.3390/ijfs11020055 - 27 Mar 2023

Cited by 2 | Viewed by 2427

Abstract

In this paper we study the pricing of exchange options between two underlying assets whose dynamic show a stochastic correlation with random jumps. In particular, we consider a Ornstein-Uhlenbeck covariance model, with Levy Background Noise Processes driven by Inverse Gaussian subordinators. We use [...] Read more.

In this paper we study the pricing of exchange options between two underlying assets whose dynamic show a stochastic correlation with random jumps. In particular, we consider a Ornstein-Uhlenbeck covariance model, with Levy Background Noise Processes driven by Inverse Gaussian subordinators. We use expansions in terms of Taylor polynomials and cubic splines to approximately compute the price of the derivative contract. Our findings show that the later approach provides an efficient way to compute the price when compared with a Monte Carlo method, while maintaining an equivalent degree of accuracy. Full article

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9 pages, 266 KB

Open AccessArticle

The Speed of Convergence of the Threshold Estimator of Ruin Probability under the Tempered α-Stable Lévy Subordinator

by Yuan Gao and Honglong You

Mathematics 2021, 9(21), 2654; https://doi.org/10.3390/math9212654 - 20 Oct 2021

Cited by 2 | Viewed by 1616

Abstract

In this paper, a nonparametric estimator of ruin probability is introduced in a spectrally negative Lévy process where the jump component is a tempered

α

-stable subordinator. Given a discrete record of high-frequency data, a threshold technique is proposed to estimate the mean [...] Read more.

In this paper, a nonparametric estimator of ruin probability is introduced in a spectrally negative Lévy process where the jump component is a tempered

α

-stable subordinator. Given a discrete record of high-frequency data, a threshold technique is proposed to estimate the mean of the jump size and use the Fourier transform and the Pollaczek–Khinchin formula to construct the estimator of ruin probability. The convergence rate of the integrated squared error for the estimator is studied. Full article

17 pages, 340 KB

Open AccessArticle

Lévy Processes Linked to the Lower-Incomplete Gamma Function

by Luisa Beghin and Costantino Ricciuti

Fractal Fract. 2021, 5(3), 72; https://doi.org/10.3390/fractalfract5030072 - 17 Jul 2021

Cited by 4 | Viewed by 2438

Abstract

We start by defining a subordinator by means of the lower-incomplete gamma function. This can be considered as an approximation of the stable subordinator, easier to be handled in view of its finite activity. A tempered version is also considered in order to [...] Read more.

We start by defining a subordinator by means of the lower-incomplete gamma function. This can be considered as an approximation of the stable subordinator, easier to be handled in view of its finite activity. A tempered version is also considered in order to overcome the drawback of infinite moments. Then, we study Lévy processes that are time-changed by these subordinators with particular attention to the Brownian case. An approximation of the fractional derivative (as well as of the fractional power of operators) arises from the analysis of governing equations. Finally, we show that time-changing the fractional Brownian motion produces a model of anomalous diffusion, which exhibits a sub-diffusive behavior. Full article

(This article belongs to the Special Issue Fractional and Anomalous Diffusions on Regular and Irregular Domains)

29 pages, 2358 KB

Open AccessArticle

Slices of the Anomalous Phase Cube Depict Regions of Sub- and Super-Diffusion in the Fractional Diffusion Equation

by Richard L. Magin and Ervin K. Lenzi

Mathematics 2021, 9(13), 1481; https://doi.org/10.3390/math9131481 - 24 Jun 2021

Cited by 7 | Viewed by 2724

Abstract

Fractional-order time and space derivatives are one way to augment the classical diffusion equation so that it accounts for the non-Gaussian processes often observed in heterogeneous materials. Two-dimensional phase diagrams—plots whose axes represent the fractional derivative order—typically display: (i) points corresponding to distinct [...] Read more.

Fractional-order time and space derivatives are one way to augment the classical diffusion equation so that it accounts for the non-Gaussian processes often observed in heterogeneous materials. Two-dimensional phase diagrams—plots whose axes represent the fractional derivative order—typically display: (i) points corresponding to distinct diffusion propagators (Gaussian, Cauchy), (ii) lines along which specific stochastic models apply (Lévy process, subordinated Brownian motion), and (iii) regions of super- and sub-diffusion where the mean squared displacement grows faster or slower than a linear function of diffusion time (i.e., anomalous diffusion). Three-dimensional phase cubes are a convenient way to classify models of anomalous diffusion (continuous time random walk, fractional motion, fractal derivative). Specifically, each type of fractional derivative when combined with an assumed power law behavior in the diffusion coefficient renders a characteristic picture of the underlying particle motion. The corresponding phase diagrams, like pages in a sketch book, provide a portfolio of representations of anomalous diffusion. The anomalous diffusion phase cube employs lines of super-diffusion (Lévy process), sub-diffusion (subordinated Brownian motion), and quasi-Gaussian behavior to stitch together equivalent regions. Full article

(This article belongs to the Special Issue Fractional Calculus in Magnetic Resonance)

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12 pages, 273 KB

Open AccessArticle

Hadamard-Type Fractional Heat Equations and Ultra-Slow Diffusions

by Alessandro De Gregorio and Roberto Garra

Fractal Fract. 2021, 5(2), 48; https://doi.org/10.3390/fractalfract5020048 - 23 May 2021

Cited by 8 | Viewed by 2857

Abstract

In this paper, we study diffusion equations involving Hadamard-type time-fractional derivatives related to ultra-slow random models. We start our analysis using the abstract fractional Cauchy problem, replacing the classical time derivative with the Hadamard operator. The stochastic meaning of the introduced abstract differential [...] Read more.

In this paper, we study diffusion equations involving Hadamard-type time-fractional derivatives related to ultra-slow random models. We start our analysis using the abstract fractional Cauchy problem, replacing the classical time derivative with the Hadamard operator. The stochastic meaning of the introduced abstract differential equation is provided, and the application to the particular case of the fractional heat equation is then discussed in detail. The ultra-slow behaviour emerges from the explicit form of the variance of the random process arising from our analysis. Finally, we obtain a particular solution for the nonlinear Hadamard-diffusive equation. Full article

(This article belongs to the Special Issue Fractional and Anomalous Diffusions on Regular and Irregular Domains)

27 pages, 854 KB

Open AccessArticle

Pricing, Risk and Volatility in Subordinated Market Models

by Jean-Philippe Aguilar, Justin Lars Kirkby and Jan Korbel

Risks 2020, 8(4), 124; https://doi.org/10.3390/risks8040124 - 17 Nov 2020

Cited by 13 | Viewed by 4528

Abstract

We consider several market models, where time is subordinated to a stochastic process. These models are based on various time changes in the Lévy processes driving asset returns, or on fractional extensions of the diffusion equation; they were introduced to capture complex phenomena [...] Read more.

We consider several market models, where time is subordinated to a stochastic process. These models are based on various time changes in the Lévy processes driving asset returns, or on fractional extensions of the diffusion equation; they were introduced to capture complex phenomena such as volatility clustering or long memory. After recalling recent results on option pricing in subordinated market models, we establish several analytical formulas for market sensitivities and portfolio performance in this class of models, and discuss some useful approximations when options are not far from the money. We also provide some tools for volatility modelling and delta hedging, as well as comparisons with numerical Fourier techniques. Full article

(This article belongs to the Special Issue Risks: Feature Papers 2020)

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21 pages, 368 KB

Open AccessArticle

Skellam Type Processes of Order k and Beyond

by Neha Gupta, Arun Kumar and Nikolai Leonenko

Entropy 2020, 22(11), 1193; https://doi.org/10.3390/e22111193 - 22 Oct 2020

Cited by 14 | Viewed by 3933

Abstract

In this article, we introduce the Skellam process of order k and its running average. We also discuss the time-changed Skellam process of order k. In particular, we discuss the space-fractional Skellam process and tempered space-fractional Skellam process via time changes in [...] Read more.

In this article, we introduce the Skellam process of order k and its running average. We also discuss the time-changed Skellam process of order k. In particular, we discuss the space-fractional Skellam process and tempered space-fractional Skellam process via time changes in Skellam process by independent stable subordinator and tempered stable subordinator, respectively. We derive the marginal probabilities, Lévy measures, governing difference-differential equations of the introduced processes. Our results generalize the Skellam process and running average of Poisson process in several directions. Full article

(This article belongs to the Special Issue Fractional Calculus and the Future of Science)

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21 pages, 340 KB

Open AccessArticle

Ruin Time and Severity for a Lévy Subordinator Claim Process: A Simple Approach

by Claude Lefèvre and Philippe Picard

Risks 2013, 1(3), 192-212; https://doi.org/10.3390/risks1030192 - 13 Dec 2013

Cited by 1 | Viewed by 5378

Abstract

This paper is concerned with an insurance risk model whose claim process is described by a Lévy subordinator process. Lévy-type risk models have been the object of much research in recent years. Our purpose is to present, in the case of a subordinator, [...] Read more.

This paper is concerned with an insurance risk model whose claim process is described by a Lévy subordinator process. Lévy-type risk models have been the object of much research in recent years. Our purpose is to present, in the case of a subordinator, a simple and direct method for determining the finite time (and ultimate) ruin probabilities, the distribution of the ruin severity, the reserves prior to ruin, and the Laplace transform of the ruin time. Interestingly, the usual net profit condition will be essentially relaxed. Most results generalize those known for the compound Poisson claim process. Full article

(This article belongs to the Special Issue Application of Stochastic Processes in Insurance)

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