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23 pages, 867 KiB

Open AccessArticle

Bachelier’s Market Model for ESG Asset Pricing

by Svetlozar Rachev, Nancy Asare Nyarko, Blessing Omotade and Peter Yegon

J. Risk Financial Manag. 2024, 17(12), 553; https://doi.org/10.3390/jrfm17120553 - 10 Dec 2024

Cited by 2 | Viewed by 1583

Environmental, Social, and Governance (ESG) finance is a cornerstone of modern finance and investment, as it changes the classical return-risk view of investment by incorporating an additional dimension to investment performance: the ESG score of the investment. We define the ESG price process [...] Read more.

Environmental, Social, and Governance (ESG) finance is a cornerstone of modern finance and investment, as it changes the classical return-risk view of investment by incorporating an additional dimension to investment performance: the ESG score of the investment. We define the ESG price process and include it in an extension of Bachelier’s market model in both discrete and continuous time, enabling option pricing valuation. Full article

(This article belongs to the Section Economics and Finance)

11 pages, 418 KiB

Open AccessArticle

Fast and Accurate Numerical Integration of the Langevin Equation with Multiplicative Gaussian White Noise

by Mykhaylo Evstigneev and Deniz Kacmazer

Entropy 2024, 26(10), 879; https://doi.org/10.3390/e26100879 - 20 Oct 2024

Cited by 3 | Viewed by 1492

Abstract

A univariate stochastic system driven by multiplicative Gaussian white noise is considered. The standard method for simulating its Langevin equation of motion involves incrementing the system’s state variable by a biased Gaussian random number at each time step. It is shown that the [...] Read more.

A univariate stochastic system driven by multiplicative Gaussian white noise is considered. The standard method for simulating its Langevin equation of motion involves incrementing the system’s state variable by a biased Gaussian random number at each time step. It is shown that the efficiency of such simulations can be significantly enhanced by incorporating the skewness of the distribution of the updated state variable. A new algorithm based on this principle is introduced, and its superior performance is demonstrated using a model of free diffusion of a Brownian particle with a friction coefficient that decreases exponentially with the kinetic energy. The proposed simulation technique proves to be accurate over time steps that are an order of magnitude longer than those required by standard algorithms. The model used to test the new numerical technique is known to exhibit a transition from normal diffusion to superdiffusion as the environmental temperature rises above a certain critical value. A simple empirical formula for the time-dependent diffusion coefficient, which covers both diffusion regimes, is introduced, and its accuracy is confirmed through comparison with the simulation results. Full article

(This article belongs to the Section Statistical Physics)

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29 pages, 3874 KiB

Open AccessArticle

Option Pricing Using a Skew Random Walk Binary Tree

by Yuan Hu, W. Brent Lindquist, Svetlozar T. Rachev and Frank J. Fabozzi

J. Risk Financial Manag. 2024, 17(4), 138; https://doi.org/10.3390/jrfm17040138 - 27 Mar 2024

Cited by 2 | Viewed by 2015

Abstract

We develop a binary tree pricing model with underlying asset price dynamics following Itô–McKean skew Brownian motion. Our work was motivated by the Corns–Satchell, continuous-time, option pricing model. However, the Corns–Satchell market model is incomplete, while our discrete-time market model is defined in [...] Read more.

We develop a binary tree pricing model with underlying asset price dynamics following Itô–McKean skew Brownian motion. Our work was motivated by the Corns–Satchell, continuous-time, option pricing model. However, the Corns–Satchell market model is incomplete, while our discrete-time market model is defined in the natural world, extended to the risk-neutral world under the no-arbitrage condition where derivatives are priced under uniquely determined risk-neutral probabilities, and is complete. The skewness introduced in the natural world is preserved in the risk-neutral world. Furthermore, we show that the model preserves skewness under the continuous-time limit. We provide empirical applications of our model to the valuation of European put and call options on exchange-traded funds tracking the S&P Global 1200 index. Full article

(This article belongs to the Section Economics and Finance)

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39 pages, 1044 KiB

Open AccessEditor’s ChoiceArticle

Option Pricing under a Generalized Black–Scholes Model with Stochastic Interest Rates, Stochastic Strings, and Lévy Jumps

by Alberto Bueno-Guerrero and Steven P. Clark

Mathematics 2024, 12(1), 82; https://doi.org/10.3390/math12010082 - 26 Dec 2023

Viewed by 4574

Abstract

We introduce a novel option pricing model that features stochastic interest rates along with an underlying price process driven by stochastic string shocks combined with pure jump Lévy processes. Substituting the Brownian motion in the Black–Scholes model with a stochastic string leads to [...] Read more.

We introduce a novel option pricing model that features stochastic interest rates along with an underlying price process driven by stochastic string shocks combined with pure jump Lévy processes. Substituting the Brownian motion in the Black–Scholes model with a stochastic string leads to a class of option pricing models with expiration-dependent volatility. Further extending this Generalized Black–Scholes (GBS) model by adding Lévy jumps to the returns generating processes results in a new framework generalizing all exponential Lévy models. We derive four distinct versions of the model, with each case featuring a different jump process: the finite activity lognormal and double–exponential jump diffusions, as well as the infinite activity CGMY process and generalized hyperbolic Lévy motion. In each case, we obtain closed or semi-closed form expressions for European call option prices which generalize the results obtained for the original models. Empirically, we evaluate the performance of our model against the skews of S&P 500 call options, considering three distinct volatility regimes. Our findings indicate that: (a) model performance is enhanced with the inclusion of jumps; (b) the GBS plus jumps model outperform the alternative models with the same jumps; (c) the GBS-CGMY jump model offers the best fit across volatility regimes. Full article

(This article belongs to the Special Issue Financial Mathematics and Applications)

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13 pages, 710 KiB

Open AccessArticle

Derivation of the Fractional Fokker–Planck Equation for Stable Lévy with Financial Applications

by Reem Abdullah Aljethi and Adem Kılıçman

Mathematics 2023, 11(5), 1102; https://doi.org/10.3390/math11051102 - 22 Feb 2023

Cited by 2 | Viewed by 2462

Abstract

This paper aims to propose a generalized fractional Fokker–Planck equation based on a stable Lévy stochastic process. To develop the general fractional equation, we will use the Lévy process rather than the Brownian motion. Due to the Lévy process, this fractional equation can [...] Read more.

This paper aims to propose a generalized fractional Fokker–Planck equation based on a stable Lévy stochastic process. To develop the general fractional equation, we will use the Lévy process rather than the Brownian motion. Due to the Lévy process, this fractional equation can provide a better description of heavy tails and skewness. The analytical solution is chosen to solve the fractional equation and is expressed using the H-function to demonstrate the indicator entropy production rate. We model market data using a stable distribution to demonstrate the relationships between the tails and the new fractional Fokker–Planck model, as well as develop an R code that can be used to draw figures from real data. Full article

(This article belongs to the Special Issue Advanced Research in Mathematical Economics and Financial Modelling)

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15 pages, 5889 KiB

Open AccessArticle

Impact of Geometrical Features on Solute Transport Behavior through Rough-Walled Rock Fractures

by Xihong Chuang, Sanqi Li, Yingtao Hu and Xin Zhou

Water 2023, 15(1), 124; https://doi.org/10.3390/w15010124 - 29 Dec 2022

Viewed by 2059

Abstract

The solute transport in the fractured rock is dominated by a single fracture. The geometric characteristics of single rough-walled fractures considerably influence their solute transport behavior. According to the self-affinity of the rough fractures, the fractal model of single fractures is established based [...] Read more.

The solute transport in the fractured rock is dominated by a single fracture. The geometric characteristics of single rough-walled fractures considerably influence their solute transport behavior. According to the self-affinity of the rough fractures, the fractal model of single fractures is established based on the fractional Brownian motion and the successive random accumulation method. The Navier–Stokes equation and solute transport convective-dispersion equation are employed to analyze the effect of fractal dimension and standard deviation of aperture on the solute transport characteristics. The results show that the concentration front and streamline distribution are inhomogeneous, and the residence time distribution (RTD) curves have obvious tailing. For the larger fractal dimension and the standard deviation of aperture, the fracture surface becomes rougher, aperture distribution becomes more scattered, and the average flow velocity becomes slower. As a result, the average time of solute transport is a power function of the fractal dimension, while the time variance and the time skewness present a negative linear correlation with the fractal dimension. For the standard deviation of aperture, the average time exhibits a linearly decreasing trend, the time variance is increased by a power function, and the skewness is increased logarithmically. Full article

(This article belongs to the Special Issue Hydraulic Engineering and Modelling: Numerical Modelling and Simulation)

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21 pages, 364 KiB

Open AccessFeature PaperArticle

A Measure-on-Graph-Valued Diffusion: A Particle System with Collisions and Its Applications

by Shuhei Mano

Mathematics 2022, 10(21), 4081; https://doi.org/10.3390/math10214081 - 2 Nov 2022

Cited by 1 | Viewed by 2107

Abstract

A diffusion-taking value in probability-measures on a graph with vertex set V,

\sum_{i \in V} x_{i} δ_{i}

is studied. The masses on each vertex satisfy the stochastic differential equation of the form [...] Read more.

A diffusion-taking value in probability-measures on a graph with vertex set V,

\sum_{i \in V} x_{i} δ_{i}

is studied. The masses on each vertex satisfy the stochastic differential equation of the form

d x_{i} = \sum_{j \in N (i)} \sqrt{x_{i} x_{j}} d B_{i j}

on the simplex, where

{B_{i j}}

are independent standard Brownian motions with skew symmetry, and

N (i)

is the neighbour of the vertex i. A dual Markov chain on integer partitions to the Markov semigroup associated with the diffusion is used to show that the support of an extremal stationary state of the adjoint semigroup is an independent set of the graph. We also investigate the diffusion with a linear drift, which gives a killing of the dual Markov chain on a finite integer lattice. The Markov chain is used to study the unique stationary state of the diffusion, which generalizes the Dirichlet distribution. Two applications of the diffusions are discussed: analysis of an algorithm to find an independent set of a graph, and a Bayesian graph selection based on computation of probability of a sample by using coupling from the past. Full article

(This article belongs to the Special Issue Random Combinatorial Structures)

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23 pages, 883 KiB

Open AccessArticle

Asymmetric Lévy Flights Are More Efficient in Random Search

by Amin Padash, Trifce Sandev, Holger Kantz, Ralf Metzler and Aleksei V. Chechkin

Fractal Fract. 2022, 6(5), 260; https://doi.org/10.3390/fractalfract6050260 - 8 May 2022

Cited by 19 | Viewed by 3308

Abstract

We study the first-arrival (first-hitting) dynamics and efficiency of a one-dimensional random search model performing asymmetric Lévy flights by leveraging the Fokker–Planck equation with a

δ

-sink and an asymmetric space-fractional derivative operator with stable index

α

and asymmetry (skewness) parameter

β

. [...] Read more.

We study the first-arrival (first-hitting) dynamics and efficiency of a one-dimensional random search model performing asymmetric Lévy flights by leveraging the Fokker–Planck equation with a

δ

-sink and an asymmetric space-fractional derivative operator with stable index

α

and asymmetry (skewness) parameter

β

. We find exact analytical results for the probability density of first-arrival times and the search efficiency, and we analyse their behaviour within the limits of short and long times. We find that when the starting point of the searcher is to the right of the target, random search by Brownian motion is more efficient than Lévy flights with

β \leq 0

(with a rightward bias) for short initial distances, while for

β > 0

(with a leftward bias) Lévy flights with

α \to 1

are more efficient. When increasing the initial distance of the searcher to the target, Lévy flight search (except for

α = 1

with

β = 0

) is more efficient than the Brownian search. Moreover, the asymmetry in jumps leads to essentially higher efficiency of the Lévy search compared to symmetric Lévy flights at both short and long distances, and the effect is more pronounced for stable indices

α

close to unity. Full article

(This article belongs to the Special Issue Fractional Dynamics: Theory and Applications)

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22 pages, 629 KiB

Open AccessReview

An Intuitive Introduction to Fractional and Rough Volatilities

by Elisa Alòs and Jorge A. León

Mathematics 2021, 9(9), 994; https://doi.org/10.3390/math9090994 - 28 Apr 2021

Cited by 6 | Viewed by 4712

Abstract

Here, we review some results of fractional volatility models, where the volatility is driven by fractional Brownian motion (fBm). In these models, the future average volatility is not a process adapted to the underlying filtration, and fBm is not a semimartingale in general. [...] Read more.

Here, we review some results of fractional volatility models, where the volatility is driven by fractional Brownian motion (fBm). In these models, the future average volatility is not a process adapted to the underlying filtration, and fBm is not a semimartingale in general. So, we cannot use the classical Itô’s calculus to explain how the memory properties of fBm allow us to describe some empirical findings of the implied volatility surface through Hull and White type formulas. Thus, Malliavin calculus provides a natural approach to deal with the implied volatility without assuming any particular structure of the volatility. The aim of this paper is to provides the basic tools of Malliavin calculus for the study of fractional volatility models. That is, we explain how the long and short memory of fBm improves the description of the implied volatility. In particular, we consider in detail a model that combines the long and short memory properties of fBm as an example of the approach introduced in this paper. The theoretical results are tested with numerical experiments. Full article

(This article belongs to the Special Issue Application of Stochastic Analysis in Mathematical Finance)

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17 pages, 591 KiB

Open AccessArticle

On the Cumulants of the First Passage Time of the Inhomogeneous Geometric Brownian Motion

by Elvira Di Nardo and Giuseppe D’Onofrio

Mathematics 2021, 9(9), 956; https://doi.org/10.3390/math9090956 - 25 Apr 2021

Cited by 2 | Viewed by 3038

Abstract

We consider the problem of the first passage time T of an inhomogeneous geometric Brownian motion through a constant threshold, for which only limited results are available in the literature. In the case of a strong positive drift, we get an approximation of [...] Read more.

We consider the problem of the first passage time T of an inhomogeneous geometric Brownian motion through a constant threshold, for which only limited results are available in the literature. In the case of a strong positive drift, we get an approximation of the cumulants of T of any order using the algebra of formal power series applied to an asymptotic expansion of its Laplace transform. The interest in the cumulants is due to their connection with moments and the accounting of some statistical properties of the density of T like skewness and kurtosis. Some case studies coming from neuronal modeling with reversal potential and mean reversion models of financial markets show the goodness of the approximation of the first moment of T. However hints on the evaluation of higher order moments are also given, together with considerations on the numerical performance of the method. Full article

(This article belongs to the Special Issue Stochastic Models with Applications)

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18 pages, 301 KiB

Open AccessArticle

On Weak Limiting Distributions for Random Walks on a Spider

by Youngsoo Seol

Symmetry 2020, 12(12), 2000; https://doi.org/10.3390/sym12122000 - 4 Dec 2020

Cited by 2 | Viewed by 1595

Abstract

In this article, we study random walks on a spider that can be established from the classical case of simple symmetric random walks. The primary purpose of this article is to establish a functional central limit theorem for random walks on a spider [...] Read more.

In this article, we study random walks on a spider that can be established from the classical case of simple symmetric random walks. The primary purpose of this article is to establish a functional central limit theorem for random walks on a spider and to define Brownian spider as the resulting weak limit. In special case, random walks on a spider can be characterized as skew random walks. It is well known for skew Brownian motion as the resulting weak limit of skew random walks. We first will study the tightness and then it will be shown for the convergence of finite dimensional distribution for random walks on a spider. Full article

(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with Their Applications Ⅲ)

17 pages, 927 KiB

Open AccessArticle

An Empirical Investigation to the “Skew” Phenomenon in Stock Index Markets: Evidence from the Nikkei 225 and Others

by Yizhou Bai and Zhiyu Guo

Sustainability 2019, 11(24), 7219; https://doi.org/10.3390/su11247219 - 16 Dec 2019

Cited by 4 | Viewed by 10887

Abstract

The skew processes have recently received much attention, owing to their capacity to describe controlled dynamics. In this paper, we employ the skew geometric Brownian motion (SGBM) to depict nine major stock index markets. The skew process not only shows us where the [...] Read more.

The skew processes have recently received much attention, owing to their capacity to describe controlled dynamics. In this paper, we employ the skew geometric Brownian motion (SGBM) to depict nine major stock index markets. The skew process not only shows us where the “support” and “resistance” levels are, but also how strong the force is. However, the densities of the skew processes make it challenging to estimate the parameters in a convenient manner. For the sake of overcoming this challenge, we adopt a Bayesian approach, which plays an important role in allowing us to estimate the parameters by conditional probability densities without having to evaluate complex integrals. Furthermore, we also propose the likelihood ratio tests and significance tests for the skew probability. In the empirical study, our findings reveal that skew phenomenon exists in the global stock markets and that the SGBM model works better than the traditional GBM model, as well as performing competitively, compared to the GBM-jump model (GBM-J) and Markov regime switching GBM model (GBM-MRS). In addition, we explore the possible reasons behind the skew phenomenon in stock markets, the price clustering phenomenon and herd behaviors can help to explain the skew phenomenon. Full article

(This article belongs to the Special Issue Sustainable Financial Markets)

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17 pages, 679 KiB

Open AccessArticle

A Solution to the Time-Scale Fractional Puzzle in the Implied Volatility

by Hideharu Funahashi and Masaaki Kijima

Fractal Fract. 2017, 1(1), 14; https://doi.org/10.3390/fractalfract1010014 - 25 Nov 2017

Cited by 12 | Viewed by 5602

Abstract

In the option pricing literature, it is well known that (i) the decrease in the smile amplitude is much slower than the standard stochastic volatility models and (ii) the term structure of the at-the-money volatility skew is approximated by a power-law function with [...] Read more.

In the option pricing literature, it is well known that (i) the decrease in the smile amplitude is much slower than the standard stochastic volatility models and (ii) the term structure of the at-the-money volatility skew is approximated by a power-law function with the exponent close to zero. These stylized facts cannot be captured by standard models, and while (i) has been explained by using a fractional volatility model with Hurst index

H > 1 / 2

, (ii) is proven to be satisfied by a rough volatility model with

H < 1 / 2

under a risk-neutral measure. This paper provides a solution to this fractional puzzle in the implied volatility. Namely, we construct a two-factor fractional volatility model and develop an approximation formula for European option prices. It is shown through numerical examples that our model can resolve the fractional puzzle, when the correlations between the underlying asset process and the factors of rough volatility and persistence belong to a certain range. More specifically, depending on the three correlation values, the implied volatility surface is classified into four types: (1) the roughness exists, but the persistence does not; (2) the persistence exists, but the roughness does not; (3) both the roughness and the persistence exist; and (4) neither the roughness nor the persistence exist. Full article

(This article belongs to the Special Issue Fractional Calculus in Economics and Finance)

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14 pages, 677 KiB

Open AccessArticle

A Simulation-Based Study on Bayesian Estimators for the Skew Brownian Motion

by Manuel Barahona, Laura Rifo, Maritza Sepúlveda and Soledad Torres

Entropy 2016, 18(7), 241; https://doi.org/10.3390/e18070241 - 28 Jun 2016

Cited by 4 | Viewed by 5198

Abstract

In analyzing a temporal data set from a continuous variable, diffusion processes can be suitable under certain conditions, depending on the distribution of increments. We are interested in processes where a semi-permeable barrier splits the state space, producing a skewed diffusion that can [...] Read more.

In analyzing a temporal data set from a continuous variable, diffusion processes can be suitable under certain conditions, depending on the distribution of increments. We are interested in processes where a semi-permeable barrier splits the state space, producing a skewed diffusion that can have different rates on each side. In this work, the asymptotic behavior of some Bayesian inferences for this class of processes is discussed and validated through simulations. As an application, we model the location of South American sea lions (Otaria flavescens) on the coast of Calbuco, southern Chile, which can be used to understand how the foraging behavior of apex predators varies temporally and spatially. Full article

(This article belongs to the Special Issue Statistical Significance and the Logic of Hypothesis Testing)

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