Search for Articles:
Title / Keyword
Author / Affiliation / Email
Journal
Article Type
 
 
Advanced Search
 
Section
Special Issue
Volume
Issue
Number
Page
 
4.6
3.6
Journals
Fractal Fract
Special Issues
Application of Fractal Processes and Fractional Derivatives in Finance, 2nd...
share announcement

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

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 (28 February 2025) | Viewed by 7234

Special Issue Editor

Dr. Leung Lung Chan

E-Mail Website
Guest Editor
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

Special Issue Information

Dear Colleagues,

Over the past four decades, fractional calculus has represented a rapidly advancing research area, both in its theory and application in practical problems arising in various fields, such as econophysics and mathematical finance, in which self-similar processes, such as the Brownian motion, the Levy stable process and the fractional Brownian motion, are employed. 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 employed 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 has been applied in hydrology and climatology as well as finance. The dynamics of the volatility of asset prices have been modelled as a fractional Brownian motion in finance and are called rough volatility models. Its applications in finance engender several novel stochastic analysis problems. Fractional diffusion processes are also applied to model the 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 applications, and numerical methods to solve fractional partial differential equations are developing.

In this Special Issue, we welcome the submission of original research and review articles exploring fractal processes, fractional derivatives and integration, and their applications in finance. The scope of this Special Issue includes, but is not limited to, the following topics:

  • 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

Benefits of Publishing in a Special Issue

  • Ease of navigation: Grouping papers by topic helps scholars navigate broad scope journals more efficiently.
  • Greater discoverability: Special Issues support the reach and impact of scientific research. Articles in Special Issues are more discoverable and cited more frequently.
  • Expansion of research network: Special Issues facilitate connections among authors, fostering scientific collaborations.
  • External promotion: Articles in Special Issues are often promoted through the journal's social media, increasing their visibility.
  • e-Book format: Special Issues with more than 10 articles can be published as dedicated e-books, ensuring wide and rapid dissemination.

Further information on MDPI's Special Issue policies can be found here.

Related Special Issue

Published Papers (6 papers)

Download All Papers
Order results
Result details
Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:

Research

25 pages, 11115 KiB  
Article
Enhancing Banking Transaction Security with Fractal-Based Image Steganography Using Fibonacci Sequences and Discrete Wavelet Transform
by Alina Iuliana Tabirca, Catalin Dumitrescu and Valentin Radu
Fractal Fract. 2025, 9(2), 95; https://doi.org/10.3390/fractalfract9020095 - 2 Feb 2025
Viewed by 1045
Abstract
The growing reliance on digital banking and financial transactions has brought significant security challenges, including data breaches and unauthorized access. This paper proposes a robust method for enhancing the security of banking and financial transactions. In this context, steganography—hiding information within digital media—is [...] Read more.
The growing reliance on digital banking and financial transactions has brought significant security challenges, including data breaches and unauthorized access. This paper proposes a robust method for enhancing the security of banking and financial transactions. In this context, steganography—hiding information within digital media—is valuable for improving data protection. This approach combines biometric authentication, using face and voice recognition, with image steganography to secure communication channels. A novel application of Fibonacci sequences is introduced within a direct-sequence spread-spectrum (DSSS) system for encryption, along with a discrete wavelet transform (DWT) for embedding data. The secret message, encrypted through Fibonacci sequences, is concealed within an image and tested for effectiveness using the Mean Square Error (MSE) and Peak Signal-to-Noise Ratio (PSNR). The experimental results demonstrate that the proposed method achieves a high PSNR, particularly for grayscale images, enhancing the robustness of security measures in mobile and online banking environments. Full article
(This article belongs to the Special Issue Application of Fractal Processes and Fractional Derivatives in Finance, 2nd Edition)
►▼ Show Figures

Figure 1

19 pages, 2883 KiB  
Article
Nonlinear Analysis of the U.S. Stock Market: From the Perspective of Multifractal Properties and Cross-Correlations with Comparisons
by Chenyu Han and Yingying Xu
Fractal Fract. 2025, 9(2), 73; https://doi.org/10.3390/fractalfract9020073 - 24 Jan 2025
Viewed by 689
Abstract
This study investigates the multifractal properties of daily returns of the Standard and Poor’s 500 Index (SPX), the Dow Jones Industrial Average (DJI), and the Nasdaq Composite Index (IXIC), the three main indices representing the U.S. stock market, from 1 January 2005 to [...] Read more.
This study investigates the multifractal properties of daily returns of the Standard and Poor’s 500 Index (SPX), the Dow Jones Industrial Average (DJI), and the Nasdaq Composite Index (IXIC), the three main indices representing the U.S. stock market, from 1 January 2005 to 1 November 2024. The multifractal detrended fluctuation analysis (MF-DFA) method is applied in this study. The origins of the multifractal properties of these returns are both long-range correlation and fat-tail distribution properties. Our findings show that the SPX exhibits the highest multifractal degree, and the DJI exhibits the lowest for the whole sample. This study also examines the multifractal behaviors of cross-correlations among the three major indices through the multifractal detrended cross-correlation analysis (MF-DCCA) method. It is concluded that the indices are cross-correlated and the cross-correlations also exhibit multifractal properties. Meanwhile, these returns exhibit different multifractal properties in different stages of the market, which shows some asymmetrical dynamics of the multifractal properties. These empirical results may have some important managerial and academic implications for investors, policy makers, and other market participants. Full article
(This article belongs to the Special Issue Application of Fractal Processes and Fractional Derivatives in Finance, 2nd Edition)
►▼ Show Figures

Figure 1

24 pages, 4168 KiB  
Article
Multifractal Characteristics and Information Flow Analysis of Stock Markets Based on Multifractal Detrended Cross-Correlation Analysis and Transfer Entropy
by Wenjuan Zhou, Jingjing Huang and Maofa Wang
Fractal Fract. 2025, 9(1), 14; https://doi.org/10.3390/fractalfract9010014 - 30 Dec 2024
Cited by 1 | Viewed by 840
Abstract
Understanding cross-correlation and information flow between stocks is crucial for stock market analysis. However, traditional methods often struggle to capture financial markets’ complex and multifaceted dynamics. This paper presents a robust combination of techniques, integrating three advanced methods: Multifractal Detrended Cross-Correlation Analysis (MFDCCA), [...] Read more.
Understanding cross-correlation and information flow between stocks is crucial for stock market analysis. However, traditional methods often struggle to capture financial markets’ complex and multifaceted dynamics. This paper presents a robust combination of techniques, integrating three advanced methods: Multifractal Detrended Cross-Correlation Analysis (MFDCCA), transfer entropy (TE), and complex networks. To address inherent non-stationarity and noise in financial data, we employ Ensemble Empirical Mode Decomposition (EEMD) for preprocessing, which helps reduce noise and handle non-stationary effects. The application and effectiveness of this combination of techniques are demonstrated through examples, uncovering significant multifractal properties and long-range cross correlations among the stocks studied. This combination of techniques also captures the magnitude and direction of information flow between stocks. This holistic analysis provides valuable insights for investors and policymakers, enhancing their understanding of stock market behavior and supporting better-informed portfolio decisions and risk management strategies. Full article
(This article belongs to the Special Issue Application of Fractal Processes and Fractional Derivatives in Finance, 2nd Edition)
►▼ Show Figures

Figure 1

27 pages, 4051 KiB  
Article
Fractal-Based Robotic Trading Strategies Using Detrended Fluctuation Analysis and Fractional Derivatives: A Case Study in the Energy Market
by Ekaterina Popovska and Galya Georgieva-Tsaneva
Fractal Fract. 2025, 9(1), 5; https://doi.org/10.3390/fractalfract9010005 - 26 Dec 2024
Viewed by 1104
Abstract
This paper presents an integrated robotic trading strategy developed for the day-ahead energy market that includes different methods for time series analysis and forecasting, such as Detrended Fluctuation Analysis (DFA), Rescaled Range Analysis (R/S analysis), fractional derivatives, Long Short-Term Memory (LSTM) Networks, and [...] Read more.
This paper presents an integrated robotic trading strategy developed for the day-ahead energy market that includes different methods for time series analysis and forecasting, such as Detrended Fluctuation Analysis (DFA), Rescaled Range Analysis (R/S analysis), fractional derivatives, Long Short-Term Memory (LSTM) Networks, and Seasonal Autoregressive Integrated Moving Average (SARIMA) models. DFA and R/S analysis may capture the long-range dependencies and fractal features inherited by the nature of the electricity price time series and give information about persistence and variability in their behavior. Given this, fractional derivatives can be used to analyze price movements concerning the minor changes in price and time acceleration for that change, which makes the proposed framework more flexible for quickly changing market conditions. LSTM, from their perspective, may capture complex and non-linear dependencies, while SARIMA models may help handle seasonal trends. This integrated approach improves market signal interpretation and optimizes the market risk through adjustable stop-loss and take-profit levels which could lead to better portfolio performance. The proposed integrated strategy is based on actual data from the Bulgarian electricity market for the years 2017–2024. Findings from this research show how the combination of fractals with statistical and machine learning models can improve complex trading strategies implementation for the energy markets. Full article
(This article belongs to the Special Issue Application of Fractal Processes and Fractional Derivatives in Finance, 2nd Edition)
►▼ Show Figures

Figure 1

10 pages, 1899 KiB  
Article
Application of the Fractal Brownian Motion to the Athens Stock Exchange
by John Leventides, Evangelos Melas, Costas Poulios, Maria Livada, Nick C. Poulios and Paraskevi Boufounou
Fractal Fract. 2024, 8(8), 454; https://doi.org/10.3390/fractalfract8080454 - 31 Jul 2024
Cited by 1 | Viewed by 1329
Abstract
The Athens Stock Exchange (ASE) is a dynamic financial market with complex interactions and inherent volatility. Traditional models often fall short in capturing the intricate dependencies and long memory effects observed in real-world financial data. In this study, we explore the application of [...] Read more.
The Athens Stock Exchange (ASE) is a dynamic financial market with complex interactions and inherent volatility. Traditional models often fall short in capturing the intricate dependencies and long memory effects observed in real-world financial data. In this study, we explore the application of fractional Brownian motion (fBm) to model stock price dynamics within the ASE, specifically utilizing the Athens General Composite (ATG) index. The ATG is considered a key barometer of the overall health of the Greek stock market. Investors and analysts monitor the index to gauge investor sentiment, economic trends, and potential investment opportunities in Greek companies. We find that the Hurst exponent falls outside the range typically associated with fractal Brownian motion. This, combined with the established non-normality of increments, disfavors both geometric Brownian motion and fractal Brownian motion models for the ATG index. Full article
(This article belongs to the Special Issue Application of Fractal Processes and Fractional Derivatives in Finance, 2nd Edition)
►▼ Show Figures

Figure 1

13 pages, 2004 KiB  
Article
Forward Starting Option Pricing under Double Fractional Stochastic Volatilities and Jumps
by Sumei Zhang, Haiyang Xiao and Hongquan Yong
Fractal Fract. 2024, 8(5), 283; https://doi.org/10.3390/fractalfract8050283 - 8 May 2024
Viewed by 1179
Abstract
This paper aims to provide an effective method for pricing forward starting options under the double fractional stochastic volatilities mixed-exponential jump-diffusion model. The value of a forward starting option is expressed in terms of the expectation of the forward characteristic function of log [...] Read more.
This paper aims to provide an effective method for pricing forward starting options under the double fractional stochastic volatilities mixed-exponential jump-diffusion model. The value of a forward starting option is expressed in terms of the expectation of the forward characteristic function of log return. To obtain the forward characteristic function, we approximate the pricing model with a semimartingale by introducing two small perturbed parameters. Then, we rewrite the forward characteristic function as a conditional expectation of the proportion characteristic function which is expressed in terms of the solution to a classic PDE. With the affine structure of the approximate model, we obtain the solution to the PDE. Based on the derived forward characteristic function and the Fourier transform technique, we develop a pricing algorithm for forward starting options. For comparison, we also develop a simulation scheme for evaluating forward starting options. The numerical results demonstrate that the proposed pricing algorithm is effective. Exhaustive comparative experiments on eight models show that the effects of fractional Brownian motion, mixed-exponential jump, and the second volatility component on forward starting option prices are significant, and especially, the second fractional volatility is necessary to price accurately forward starting options under the framework of fractional Brownian motion. Full article
(This article belongs to the Special Issue Application of Fractal Processes and Fractional Derivatives in Finance, 2nd Edition)
►▼ Show Figures

Figure 1

Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:
 
Displaying articles 1-6
Back to TopTop