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 Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1800, San Diego, CA 92101-4495, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom Copyright © 2017 Hasan A. Fallahgoul, Sergio M. Focardi, Frank J. Fabozzi. Published by Elsevier Limited. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. 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Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-804248-9 For information on all Academic Press publications visit our website at https://www.elsevier.com/ Publisher: Nikki Levy Acquisition Editor: Glyn Jones Editorial Project Manager: Anna Valutkevich Production Project Manager: Poulouse Joseph Cover Designer: Mark Rogers Typeset by SPi Global, India 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.