Fractional Calculus and Fractional Processes with Applications to Financial Economics
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Fractional Calculus and Fractional Processes with Applications to Financial Economics

Theory and Application

1st Edition - September 22, 2016

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  • Authors: Hassan Fallahgoul, Sergio Focardi, Frank Fabozzi
  • Hardcover ISBN: 9780128042489
  • eBook ISBN: 9780128042847

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Description

Fractional Calculus and Fractional Processes with Applications to Financial Economics presents the theory and application of fractional calculus and fractional processes to financial data. Fractional calculus dates back to 1695 when Gottfried Wilhelm Leibniz first suggested the possibility of fractional derivatives. Research on fractional calculus started in full earnest in the second half of the twentieth century. The fractional paradigm applies not only to calculus, but also to stochastic processes, used in many applications in financial economics such as modelling volatility, interest rates, and modelling high-frequency data. The key features of fractional processes that make them interesting are long-range memory, path-dependence, non-Markovian properties, self-similarity, fractal paths, and anomalous diffusion behaviour. In this book, the authors discuss how fractional calculus and fractional processes are used in financial modelling and finance economic theory. It provides a practical guide that can be useful for students, researchers, and quantitative asset and risk managers interested in applying fractional calculus and fractional processes to asset pricing, financial time-series analysis, stochastic volatility modelling, and portfolio optimization.

Key Features

  • Provides the necessary background for the book's content as applied to financial economics
  • Analyzes the application of fractional calculus and fractional processes from deterministic and stochastic perspectives

Readership

Graduate students of mathematical finance, statistics and probability, and applied mathematics

Table of Contents

    • Dedication
    • About the Authors
    • Illustrations
    • Part I: Theory
      • 1: Fractional calculus and fractional processes: an overview
        • Abstract
        • 1.1 Fractional calculus
        • 1.2 Fractional processes
      • 2: Fractional Calculus
        • Abstract
        • 2.1 Different definitions for fractional derivatives
        • 2.2 Computation with Matlab
        • Key points of the chapter
      • 3: Fractional Brownian Motion
        • Abstract
        • 3.1 Definition
        • 3.2 Long-Range Dependency
        • 3.3 Self-Similarity
        • 3.4 Existence of Arbitrage
        • Key points of the chapter
      • 4: Fractional Diffusion and Heavy Tail Distributions: Stable Distribution
        • Abstract
        • 4.1 Univariate Stable Distribution
        • 4.2 Multivariate Stable Distribution
        • Key points of the chapter
      • 5: Fractional Diffusion and Heavy Tail Distributions: Geo-Stable Distribution
        • Abstract
        • 5.1 Univariate Geo-stable Distribution
        • 5.2 Multivariate Geo-stable Distribution
        • Key points of the chapter
    • Part II: Applications
      • 6: Fractional Partial Differential Equation and Option Pricing
        • Abstract
        • 6.1 Option Pricing and Brownian Motion
        • 6.2 Option Pricing and the Lévy Process
        • Key points of the chapter
      • 7: Continuous-Time Random Walk and Fractional Calculus
        • Abstract
        • 7.1 Continuous-Time Random Walk
        • 7.2 Fractional Calculus and Probability Density Function
        • 7.3 Applications
        • Key points of the chapter
      • 8: Applications of Fractional Processes
        • Abstract
        • 8.1 Fractionally Integrated Time Series
        • 8.2 Stock-Returns and Volatility Processes
        • 8.3 Interest-Rate Processes
        • 8.4 Order Arrival Processes
        • Key points of the chapter
    • References
    • Index

Product details

  • No. of pages: 118
  • Language: English
  • Copyright: © Academic Press 2016
  • Published: September 22, 2016
  • Imprint: Academic Press
  • Hardcover ISBN: 9780128042489
  • eBook ISBN: 9780128042847

About the Authors

Hassan Fallahgoul

Hasan A. Fallahgoul is currently a postdoctoral researcher at the Swiss Finance Institute @ EPFL, in the team lead by Professor Loriano Mancini. Prior to this

position, he was a postdoctoral researcher at the European Center for Advanced Research in Economics and Statistics (ECARES), Universite Libre de Bruxelles,

Belgium. His research interests are in nancial econometrics, quantitative finance, Levy processes and fractional calculus. Specializing in heavy-tailed distributions

and their applications to finance. Dr. Fallahgoul has published several papers in scientific journals including Quantitative Finance, Applied Mathematics Letters,

and Journal of Statistical Theory and Practice. He holds a PhD and MSc in applied mathematics from the K. N. Toosi University of Technology.

Affiliations and Expertise

Post-Doctoral Researcher, Swiss Finance Institute, École polytechnique fédérale de Lausanne, Switzerland

Sergio Focardi

Sergio M. Focardi is a professor of finance, Director of the Master in Investment Banking and Risk Management and researcher at the Finance Group, ESILV

EMLV of the Pole Universitaire De Vinci, Paris. He is a founding partner of The Intertek Group, Paris. Professor Focardi holds a degree in Electronic Engineering

from the University of Genoa, Italy, and a PhD in Mathematical Finance from the University of Karlsruhe, Germany. A member of the Editorial Advisory Board

of The Journal of Portfolio Management, he has authored numerous articles, monographs, and books.

Affiliations and Expertise

Professor of Finance and Director of the Master in Investment, Banking and Risk Management, Researcher at the Finance Group, ESILV EMLV of the Pole Universitaire De Vinci, Paris, France

Frank Fabozzi

Frank J. Fabozzi is a professor of finance at EDHEC Business School (Nice, France) and a senior scientific adviser at the EDHEC-Risk Institute. He taught

at Yale's School of Management for 17 years and served as a visiting professor at MIT's Sloan School of Management and Princeton University's Department of

Operations Research and Financial Engineering. Professor Fabozzi is the editor of The Journal of Portfolio Management and an associate editor of several journals,

including Quantitative Finance. The author of numerous numerous books and articles on quantitative finance, he holds a doctorate in economics from The

Graduate Center of the City University of New York.

Affiliations and Expertise

Professor of Finance, EDHEC Business School and a member of the EDHEC Risk Institute, USA

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