Fractional Calculus and Fractional Processes with Applications to Financial Economics: Theory and Application

109 Pages·2017·3.07 MB·English

by  Fabozzi

Fractional Calculus and
Fractional Processes
with Applications to
Financial Economics
Fractional Calculus and
Fractional Processes
with Applications to
Financial Economics
Theory and Application
Hasan A. Fallahgoul
Sergio M. Focardi
Frank J. Fabozzi
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Copyright © 2017 Hasan A. Fallahgoul, Sergio M. Focardi, Frank J. Fabozzi.
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ISBN: 978-0-12-804248-9
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HAF
To my mother Khadijeh and in memory of my father Abobakr Fallahgoul
SMF
To the memory of my parents
FJF
To my family
About the Authors
HasanA.FallahgouliscurrentlyapostdoctoralresearcherattheSwissFinance
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), Universit´e Libre de Bruxelles,
Belgium.Hisresearchinterestsareinfinancialeconometrics,quantitativefinance,
L´evy 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.
Sergio M. Focardi is a professor of finance, director of the Master’s course 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 engi-
neering from the University of Genoa, Italy, and a PhD in mathematical finance
from the University of Karlsruhe, Germany. A member of the Editorial Advi-
sory Board of The Journal of Portfolio Management, he has authored numerous
articles, monographs, and books.
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
OperationsResearchandFinancialEngineering.ProfessorFabozziistheeditorof
The Journal of Portfolio Management andanassociateeditorofseveraljournals,
including Quantitative Finance. The author of numerous books and articles on
quantitativefinance,heholdsadoctorateineconomicsfromTheGraduateCenter
of the City University of New York.
Contents
LisotJ i llustrations xi
PartI Theory
1 Fractional calculuasn d fl'actiopnraolc esse:s an overview 3
1. 1 Fractionalc aluclus 5
1.2 Fractionalp rocesses 7
2 Ft'actionaCla lculus 12
21 Difefrent defitnioins forfr actionald erivatives 12
2.1.1 Riemann-Liouville Fractoina.lD erivtaive 13
2.1.2 Caputo FractoinalD erivaitve 15
2."1.3 Gri.inwald-Letnikov FracitonalD eriavtive 16
2.1.4 Fractional edrivative basde 011 theP ourier trasnform 17
2.2 Computatoni with i\fatlab 18
Keyp oints oft hec haptre 20
3 FractionBarlo wnianM otion 23
3.1 Definition 24
3.2 Long-RangeD ependency 26
3.3 Sel-fSimilarity 28
3.4 Exisentceo fA rbtirage :30
[(eyp oints of the chapter 32
4 FractionaDli ffusion and Heavy TailD istributiso:n Stable
Distribution 33
4.1 Univariate Stabel Distribtuion 33
4.1.1 Homot.opy Perturbaio.nt M.ethod 34
4. .. 21 AdomianD ecomposito.ni Method 35
4.1.3 Varia.tionln Itera.tion IVfethod 35
4.2 I\fultivaria.te Stabel Distribtiuon 36
4.2.1 Homotopy Perturabtion Method 37
4.2.2 AdollliDaellc omposit.olilM ethod 39
4.2.3 Variation",l-I teration Method 39
Keyp oinst oft he hcapter 40
x ConLenLs
5 Fractioln Daiffusiona nd Heavy Tail isDtributiso:n eGo-Stabel
Distribuiton 41
5.1 Univariate Geo-stabel Distribution 41
5.1.1 Homotopy Perturbation Method 43
5.12. AdomianD ecomposiLoillM ethod 47
5.1.:3 Varia.tional i.c.lrat.oin lVlcthod 49
5.2 l\rulitvaiarte Geo-stabel Distribution 52
5.2.1 HomotopyP erturbatoin Method 52
.5.2.2 AdomianD ecomposition Method 53
5.2.3: Varaitio.nal It.erato.ni Method 54
Keyp ointso ft hec hapter 54
PartI I Applications 57
6 Fractionla rPut,ila Differentila Equationa nd OptionP ricing 59
6.1 Option Pricnig andB rownian l\rotion 61
6.1.1 Sotchastic DiffeenrtailE quation 61
6.1.2 PartiDaliff erentialE quatoin 62
6.2 Optio.n Pricnig andt he Levy Process 65
6.2.1 LevyP rocess 66
6.2.2 CGM Y Process 67
6.2.3 Stochastci DifferentailE quation 67
6.2.4 PartiTanlte gro-DifrerentailE quaiton 72
Enropean Option 74
6.2.5 Fracton.ailP aritalD ilTercnt.ialE quati.on 75
Charcateristic Functoin 76
European Option 77
Key poinst oft he hcapter 79
7 Continuosu-Tilne RandOllW1a lk and FractionCaall culsu 81
7.1 Continuous-Time Random \oValk 81
7.2 FractionalC alculus andP robabiliyt DensitFyu nctoin 83
7.2.1 Uncoupled 84
7.2.2C oupled Case 85
7.3 Applica.tions 86
7.3.1 Dynamicso ft he Asset Prices 86
Kcyp oinst oft hc chapter 89
8 Applicatiso onfF ractionParlo cesses 91
8.1 Fractionalyl Integrat.ed Time Series 92
8.2 Stock-Retursn '-\ltVdo lattiyl Pirocesses 93
8.3 lltelrest-RatcP rocesses 95
8.4 OrderA rrivalP rocesses 95
Kcy points oft hcc hapter 96
RefeTences 97
Index 103
Illustrations
2.1 Derivative or integral operator 13
2.2 Ordinary derivative 19
2.3 Fractional derivative 21
2.4 Ordinary derivative and integral 21
2.5 Fractional derivative 22
3.1 Sample Paths for the Fractional Brownian motion 25
3.2 Log-price and log-return for the S&P500 27
3.3 Sample autocorrelation for the: (top) log-return of the S&P500, and
(down) absolute log-return of the S&P500 27
3.4 The connection among LRD, self-similar process, fBm and L´evy process 29
6.1 The probability density function for the stable and Gaussian distribution 60
6.2 The risk-neutral probability density function 65
6.3 Trajectory for the CGMY process 67
6.4 Implied volatility for the CGMY process 71
6.5 Risk-neutral density for the CGMY process 72
7.1 Continous-time random walk 87
1
Fractional calculus and fractional processes:
an overview
In this monograph we discuss how fractional calculus and fractional processes are
used in financial modeling, finance theory, and economics. We begin by giving
an overview of fractional calculus and fractional processes, responding upfront to
two important questions:
1. What is the fractional paradigm for both calculus and stochastic processes?
2. Why is the fractional paradigm important in science in general and in finance
and economics in particular?
Fractional calculus is a generalization of ordinary calculus. Calculus proved
to be a key tool for modern science because it allows the writing of differential
equations that link variables and their rates of change. Differential equations
ushered in modern theoretical quantitative science. A key reason for the success
of differential equations in science is that the description of physical phenomena
simplifies locally. For example, to understand how temperature propagates in a
rod when we apply a source of heat to one of its extremities. While the actual
description of the propagation of temperature can be rather complex in function
of the source of heat, locally, the heat equation is quite easy: at any point on the
rod, the time derivative of the temperature is proportional to the second space
derivative: ∂T = k∂2T. This is the key for the success of differential equations in
∂t ∂x2
science: phenomena simplify locally and one can write a simple, yet extremely
powerful, general law.
However, not all problems are local. Many problems in physics, engineering,
andeconomicsareofaglobalnature.Consider,forexample,variationalproblems,
where the objective is to optimize a functional, that is, to find an optimum in
a space of functions. Though variational problems are not local, it has been a
major mathematical success to translate variational problems into integral or
differential equations. Still the formalism of fractional calculus is a very useful
tool for solving variational problems.
A widely cited example is Abel’s integral equation, which solves the problem
of the tautochrone. A tautochrone is a curve on a vertical plane such that the
time that it takes a point freely sliding on the curve under the sole influence of
gravity to reach the lowest point is a constant independent of its starting point.
Abel’s integral equations can be solved with a fractional derivative.
The problem of the tautochrone is essentially a variational problem. A pro-
found link has been established in several recent papers between the calculus of
http://dx.doi.org/10.1016/B978-0-12-804248-9.50001-2, 3
Copyright © 2017 Hasan A. Fallahgoul, Sergio M. Focardi, Frank J. Fabozzi.
Published by Elsevier Limited. All rights reserved.