Fractional Calculus and Fractional Processes with Applications to Financial Economics

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.

• 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

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

• 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