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Application of Fractal Processes and Fractional Derivatives in Finance

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A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "General Mathematics, Analysis".

Deadline for manuscript submissions: closed (15 August 2023) | Viewed by 26523

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Dr. Leung Lung Chan

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School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia
Interests: asset pricing models; regime-switching model; volatility derivatives; stochastic volatility models; consumption and investment
Special Issues, Collections and Topics in MDPI journals

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Dear Colleagues,

Over the past four decades, the fractional calculus has represented a rapidly growing research area, both in the theory and applications to practical problems arising in various fields such as econophysics as well as mathematical finance, in which self-similar processes, such as the Brownian motion, the Levy stable process and the fractional Brownian motion, are used. The Brownian motion was firstly introduced and applied in finance by Bachelier (1900). In 1973, the log-price of a stock was modelled as a Brownian motion named the Black–Scholes–Merton model. The Levy stable processes are widely used in financial econometrics to model the dynamics of stock, commodity, currency exchange prices, etc. The fractional Brownian motion was introduced by Kolmogorov in 1940 and later by Mandelbrot in 1965, and was used in hydrology and climatology as well as finance. The dynamics of volatility of asset prices were modelled as a fractional Brownian motion in finance and are called rough volatility models. The applications in finance bring about some new stochastic analysis problems. The fractional diffusion processes are also used to model dynamics of underlying assets. The option price under the fractional diffusion setting induces the fractional partial differential equations involving the fractional derivatives with respect to the time. Some closed-form solutions might be found via transform methods in some cases of applications, and numerical methods to solve fractional partial differential equations are developing.

In this Special Issue, we would like to invite the submission of original research and review articles exploring fractal processes, fractional derivatives and integration and their applications to finance, potential topics including, but not limited to:

The rough volatility model;
Fractal processes applied in finance and other fields;
Fractional differential equations;
Fractional diffusions;
Transform methods applied in fractional differential equations;
Numerical methods for fractional partial differential equations;
Fractional operators.

Dr. Leung Lung Chan
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

rough volatility model
fractal processes applied in finance and other fields
fractional differential equations
fractional diffusions
fractional calculus
transform methods applied in fractional differential equations
numerical methods for fractional partial differential equations
fractional operators

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Application of Fractal Processes and Fractional Derivatives in Finance, 2nd Edition in Fractal and Fractional (2 articles)

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Research

19 pages, 1448 KiB

Open AccessArticle

The Importance of Non-Systemically Important Banks—A Network-Based Analysis for China’s Banking System

by Yong Li

Fractal Fract. 2023, 7(10), 735; https://doi.org/10.3390/fractalfract7100735 - 6 Oct 2023

Cited by 1 | Viewed by 1246

Abstract

There is important theoretical and practical significance to scientifically identifying the systemic importance of banks for effectively preventing and controlling systemic risks in the banking system. Prevalent identification methods are biased because they only pay attention to measuring the systemic risk contribution of individual banks to the whole system in order to determine that bank’s systemic importance. Less attention is paid to the cascade effects of risk spillover among banks. This study proposes a novel method for measuring the cascade effects of risk spillover of banks and their contributions to systemic risks by building up a conditional tail risk network of China’s banking system. Different from previous analyses of systemic risks based on the identification and risk measurement of systemically important banks (SIBs), this paper focuses on analyzing the risk spillover effects of non-SIBs and their contributions to systemic risks by building up a conditional tail risk network of China’s banking system. Our empirical results show that some non-SIBs in China are more vulnerable to the shocks of systemic risk than SIBs, and that they are more likely to act as key intermediaries to transmit risk to SIBs, thereby triggering systemic risk. In view of this, we propose to identify key non-SIBs according to their risk spillover intensity because they are also systemically important. The market regulators not only need to pay attention to SIBs that are too big to fail, but also treat seriously the key intermediaries of “risk spillover too strong to fail” in the network in order to ensure the stability of the banking system. Full article

(This article belongs to the Special Issue Application of Fractal Processes and Fractional Derivatives in Finance)

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31 pages, 18905 KiB

Open AccessArticle

Comparing Market Efficiency in Developed, Emerging, and Frontier Equity Markets: A Multifractal Detrended Fluctuation Analysis

by Min-Jae Lee and Sun-Yong Choi

Fractal Fract. 2023, 7(6), 478; https://doi.org/10.3390/fractalfract7060478 - 15 Jun 2023

Cited by 4 | Viewed by 4122

Abstract

In this article, we investigate the market efficiency of global stock markets using the multifractal detrended fluctuation analysis methodology and analyze the results by dividing them into developed, emerging, and frontier groups. The static analysis results reveal that financially advanced countries, such as Switzerland, the UK, and the US, have more efficient stock markets than other countries. Rolling window analysis shows that global issues dominate the developed country group, while emerging markets are vulnerable to foreign capital movements and political risks. In the frontier group, intensive domestic market issues vary, making it difficult to distinguish similar dynamics. Our findings have important implications for international investors and policymakers. International investors can establish investment strategies based on the degree of market efficiency of individual stock markets. Policymakers in countries with significant fluctuations in market efficiency should consider implementing new regulations to enhance market efficiency. Overall, this study provides valuable insights into the market efficiency of global stock markets and highlights the need for careful consideration by international investors and policymakers. Full article

(This article belongs to the Special Issue Application of Fractal Processes and Fractional Derivatives in Finance)

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15 pages, 501 KiB

Open AccessArticle

An Efficient Numerical Method for Pricing Double-Barrier Options on an Underlying Stock Governed by a Fractal Stochastic Process

by Samuel Megameno Nuugulu, Frednard Gideon and Kailash C. Patidar

Fractal Fract. 2023, 7(5), 389; https://doi.org/10.3390/fractalfract7050389 - 8 May 2023

Cited by 3 | Viewed by 1970

Abstract

After the discovery of the fractal structures of financial markets, enormous effort has been dedicated to finding accurate and stable numerical schemes to solve fractional Black-Scholes partial differential equations. This work, therefore, proposes a numerical scheme for pricing double-barrier options, written on an underlying stock whose dynamics are governed by a non-standard fractal stochastic process. The resultant model is time-fractional and is herein referred to as a time-fractional Black-Scholes model. The presence of the time-fractional derivative helps to capture the time-decaying effects of the underlying stock while capturing the globalized change in underlying prices and barriers. In this paper, we present the construction of the proposed scheme, analyse it in terms of its stability and convergence, and present two numerical examples of pricing double knock-in barrier-option problems. The results suggest that the proposed scheme is unconditionally stable and convergent with order

O (h^{2} + k^{2})

. Full article

(This article belongs to the Special Issue Application of Fractal Processes and Fractional Derivatives in Finance)

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14 pages, 4094 KiB

Open AccessArticle

Stock Index Return Volatility Forecast via Excitatory and Inhibitory Neuronal Synapse Unit with Modified MF-ADCCA

by Luochao Wang and Raymond S. T. Lee

Fractal Fract. 2023, 7(4), 292; https://doi.org/10.3390/fractalfract7040292 - 28 Mar 2023

Cited by 4 | Viewed by 1624

Abstract

Financial prediction persists a strenuous task in Fintech research. This paper introduces a multifractal asymmetric detrended cross-correlation analysis (MF-ADCCA)-based deep learning forecasting model to predict a succeeding day log return via excitatory and inhibitory neuronal synapse unit (EINS) using asymmetric Hurst exponent as input features, with return and volatility increment of Shanghai Stock Exchanges Composite Index (SSECI) from 2014 to 2020 as proxies for analysis. Experimental results revealed that multifractal elements by MF-ADCCA method as input features are applicable to time series forecasting in deep learning than multifractal detrended fluctuation analysis (MF-DFA) method. Further, the proposed biologically inspired EINS model achieved satisfactory performances in effectiveness and reliability in time series prediction compared with prevalent recurrent neural networks (RNNs) such as LSTM and GRU. The contributions of this paper are to (1) introduce a moving-window MF-ADCCA method to obtain asymmetric Hurst exponent sequences used directly as an input feature for deep learning prediction and (2) evaluate performances of various asymmetric multifractal approaches for deep learning time series forecasting. Full article

(This article belongs to the Special Issue Application of Fractal Processes and Fractional Derivatives in Finance)

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19 pages, 2451 KiB

Open AccessArticle

Cross-Correlation Multifractal Analysis of Technological Innovation, Financial Market and Real Economy Indices

by Jinchuan Ke, Yu Duan, Chao Xu and Yue Zhang

Fractal Fract. 2023, 7(3), 267; https://doi.org/10.3390/fractalfract7030267 - 17 Mar 2023

Cited by 3 | Viewed by 1763

Abstract

Technological innovation, the financial market, and the real economy are mutually promoting and restricting. Considering the interference of market-noise information, this paper applies the wavelet-denoising method of the soft- and hard-threshold compromise functions to process the original information so as to eliminate the noise information, and combines multifractal detrended cross-correlation analysis with the sliding-window approach, focusing on the change in the Hurst index and the parameter change in the multifractal spectrum to explore the interaction in between. The research results show that there is a certain cross-correlation among technological-innovation, financial-market, and real-economy indices. Firstly, the cross-correlation among them has significant multifractal characteristics rather than single-fractal characteristics. Secondly, the fractal characteristics reveal the long memory of the interaction among the three indices. Thirdly, there are also obvious differences in the degree of local chaos and volatility of the interaction. Fourthly, the cross-correlation among technological-innovation, financial-market, and real-economy indices has significant multifractal characteristics rather than single-fractal characteristics. In comparison, the cross-correlation multifractal characteristics among technological innovation, the financial market, and the real economy are time-varying, and the cross-correlation multifractal characteristics between the technological-innovation index and the real-economy index are the most obvious. Full article

(This article belongs to the Special Issue Application of Fractal Processes and Fractional Derivatives in Finance)

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23 pages, 12736 KiB

Open AccessArticle

Numerical Investigation of Fractional Step-Down ELS Option

by Xinpei Wu, Shuai Wen, Wei Shao and Jian Wang

Fractal Fract. 2023, 7(2), 126; https://doi.org/10.3390/fractalfract7020126 - 30 Jan 2023

Cited by 1 | Viewed by 1519

Abstract

In this paper, we use the finite difference methods to explore step-down Equity Linked Securities (ELS) options under the fractional Black-Scholes model. We establish Crank-Nicolson scheme under one asset and study the impact of Hurst exponent (H) on return of repayment under fixed stock price. We also explore the impact of stock price on return of repayment under different H. Through numerical experiments, it is found that the return of repayment of options is related to H, and the result of difference scheme will increase with the increase of H. In the case of two assets, we establish implicit scheme, and in the case of three assets, we use operator splitting method (OSM) method to establish semi-implicit scheme. We get the result that the H also influences the return of repayment in two and three assets. We also conduct Greeks analysis. Through Greeks analysis, we find that the long-term correlation of stocks has a huge impact on investment gains or losses. Therefore, we take historical volatility (fractal exponents) into account which can significantly reduce risk and increase revenue for investors. Full article

(This article belongs to the Special Issue Application of Fractal Processes and Fractional Derivatives in Finance)

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18 pages, 902 KiB

Open AccessArticle

An Analysis of the Fractional-Order Option Pricing Problem for Two Assets by the Generalized Laplace Variational Iteration Approach

by Sivaporn Ampun, Panumart Sawangtong and Wannika Sawangtong

Fractal Fract. 2022, 6(11), 667; https://doi.org/10.3390/fractalfract6110667 - 11 Nov 2022

Cited by 1 | Viewed by 1459

Abstract

An option is the right to buy or sell a good at a predetermined price in the future. For customers or financial companies, knowing an option’s pricing is crucial. It is well recognized that the Black–Scholes model is an effective tool for estimating the cost of an option. The Black–Scholes equation has an explicit analytical solution known as the Black–Scholes formula. In some cases, such as the fractional-order Black–Scholes equation, there is no closed form expression for the modified Black–Scholes equation. This article shows how to find the approximate analytic solutions for the two-dimensional fractional-order Black–Scholes equation based on the generalized Riemann–Liouville fractional derivative. The generalized Laplace variational iteration method, which incorporates the generalized Laplace transform with the variational iteration method, is the methodology used to discover the approximate analytic solutions to such an equation. The expression of the two-parameter Mittag–Leffler function represents the problem’s approximate analytical solution. Numerical investigations demonstrate that the proposed scheme is accurate and extremely effective for the two-dimensional fractional-order Black–Scholes Equation in the perspective of the generalized Riemann–Liouville fractional derivative. This guarantees that the generalized Laplace variational iteration method is one of the effective approaches for discovering approximate analytic solutions to fractional-order differential equations. Full article

(This article belongs to the Special Issue Application of Fractal Processes and Fractional Derivatives in Finance)

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20 pages, 3749 KiB

Open AccessArticle

Stock Prediction Model Based on Mixed Fractional Brownian Motion and Improved Fractional-Order Particle Swarm Optimization Algorithm

by Hongwen Hu, Chunna Zhao, Jing Li and Yaqun Huang

Fractal Fract. 2022, 6(10), 560; https://doi.org/10.3390/fractalfract6100560 - 2 Oct 2022

Cited by 8 | Viewed by 2913

Abstract

As one of the main areas of value investing, the stock market attracts the attention of many investors. Among investors, market index movements are a focus of attention. In this paper, combining the efficient market hypothesis and the fractal market hypothesis, a stock prediction model based on mixed fractional Brownian motion (MFBM) and an improved fractional-order particle swarm optimization algorithm is proposed. First, the MFBM model is constructed by adjusting the parameters to mix geometric Brownian motion (GBM) and geometric fractional Brownian motion (GFBM). After that, an improved fractional-order particle swarm optimization algorithm is proposed. The position and velocity formulas of the fractional-order particle swarm optimization algorithm are improved using new fractional-order update formulas. The inertia weight in the update formula is set to be linearly decreasing. The improved fractional-order particle swarm optimization algorithm is used to optimize the coefficients of the MFBM model. Through experiments, the accuracy and validity of the prediction model are proven by combining the error analysis. The model with the improved fractional-order particle swarm optimization algorithm and MFBM is superior to GBM, GFBM, and MFBM models in stock price prediction. Full article

(This article belongs to the Special Issue Application of Fractal Processes and Fractional Derivatives in Finance)

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20 pages, 3529 KiB

Open AccessArticle

Multifractal Characteristics of China’s Stock Market and Slump’s Fractal Prediction

by Yong Li

Fractal Fract. 2022, 6(9), 499; https://doi.org/10.3390/fractalfract6090499 - 5 Sep 2022

Cited by 3 | Viewed by 2324

Abstract

It is necessary to quantitatively describe or illustrate the characteristics of abnormal stock price fluctuations in order to prevent and control financial risks. This paper studies the fractal structure of China’s stock market by calculating the fractal dimension and scaling behavior on the timeline of its eight big slumps, the results show that the slumps have multifractal characteristics, which are correlated with the policy intervention, institutional arrangements, and investors’ rationality. The empirical findings are a perfect match with the anomalous features of the stock prices. The fractal dimensions of the eight stock collapses are between 0.84 and 0.98. The fractal dimension distribution of the slumps is sensitive to market conditions and the active degree of speculative trading. The more mature market conditions and the more risk-averse investors correspond to the higher fractal dimension and the fall which is less deep. Therefore, the fractal characteristics could reflect the evolution characteristics of the stock market and investment philosophy. The parameter set calculated in this paper could be used as an effective tool to foresee the slumps on the horizon. Full article

(This article belongs to the Special Issue Application of Fractal Processes and Fractional Derivatives in Finance)

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9 pages, 299 KiB

Open AccessArticle

Convergence Rate of the High-Order Finite Difference Method for Option Pricing in a Markov Regime-Switching Jump-Diffusion Model

by Jun Liu and Jingzhou Yan

Fractal Fract. 2022, 6(8), 409; https://doi.org/10.3390/fractalfract6080409 - 26 Jul 2022

Cited by 2 | Viewed by 1567

Abstract

The high-order finite difference method for option pricing is one of the most popular numerical algorithms. Therefore, it is of great significance to study its convergence rate. Based on the relationship between this algorithm and the trinomial tree method, as well as the definition of local remainder estimation, a strict mathematical proof is derived for the convergence rate of the high-order finite difference method for option pricing in a Markov regime-switching jump-diffusion model. The theoretical result shows that the convergence rate of this algorithm is

O (Δ τ)

. Moreover, the results also hold in the case of Brownian motion and jump-diffusion models that are specialized forms of the given model. Full article

(This article belongs to the Special Issue Application of Fractal Processes and Fractional Derivatives in Finance)

16 pages, 896 KiB

Open AccessArticle

Uncertain Currency Option Pricing Based on the Fractional Differential Equation in the Caputo Sense

by Qinyu Liu, Ting Jin, Min Zhu, Chenlei Tian, Fuzhen Li and Depeng Jiang

Fractal Fract. 2022, 6(8), 407; https://doi.org/10.3390/fractalfract6080407 - 24 Jul 2022

Cited by 9 | Viewed by 1808

Abstract

The foreign exchange market comprises the largest global volume, so the pricing of foreign exchange options has always been a hot issue in the foreign exchange market. This paper treats the exchange rate as an uncertain process that is described by an uncertain fractional differential equation, and establishes a new uncertain fractional currency model. The uncertain process is driven by Liu process, and, with the application of the Mittag-Leffler function, the solution of the fractional differential equation in a Caputo sense is presented. Then, according to the uncertain fractional currency model, the pricing formulas of European and American currency options are given. Lastly, the two numerical examples of European and American currency options are given; the price of the currency option increased when p changed from 1.0 to 1.1, and prices with different p were all decreasing functions of exercise price K. Full article

(This article belongs to the Special Issue Application of Fractal Processes and Fractional Derivatives in Finance)

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36 pages, 2638 KiB

Open AccessArticle

A New Homotopy Transformation Method for Solving the Fuzzy Fractional Black–Scholes European Option Pricing Equations under the Concept of Granular Differentiability

by Jianke Zhang, Yueyue Wang and Sumei Zhang

Fractal Fract. 2022, 6(6), 286; https://doi.org/10.3390/fractalfract6060286 - 26 May 2022

Cited by 10 | Viewed by 2160

Abstract

The Black–Scholes option pricing model is one of the most significant achievements in modern investment science. However, many factors are constantly fluctuating in the actual financial market option pricing, such as risk-free interest rate, stock price, option underlying price, and security price volatility may be inaccurate in the real world. Therefore, it is of great practical significance to study the fractional fuzzy option pricing model. In this paper, we proposed a reliable approximation method, the Elzaki transform homotopy perturbation method (ETHPM) based on granular differentiability, to solve the fuzzy time-fractional Black–Scholes European option pricing equations. Firstly, the fuzzy function is converted to a real number function based on the horizontal membership function (HMF). Secondly, the specific steps of the ETHPM are given to solve the fuzzy time-fractional Black–Scholes European option pricing equations. Finally, some examples demonstrate that the new approach is simple, efficient, and accurate. In addition, the fuzzy approximation solutions have been visualized at the end of this paper. Full article

(This article belongs to the Special Issue Application of Fractal Processes and Fractional Derivatives in Finance)

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