Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Selecta Volume E

558 Pages·1997·21.65 MB·English

by  Benoit B. Mandelbrot (auth.)

FRACTALS and
SCALING
in FINANCE
SELECTED WORKS OF BENOIT B. MANDELBROT
REPRINTED, TRANSLATED OR NEW
WITH ANNOTATIONS AND GUEST CONTRIBUTIONS
COMPANION TO THE FRACTAL GEOMETRY OF NATURE
Benoit B. Mandelbrot
FRACTALS and
SCALING
in FINANCE
Discontinuity,
Concentration, Risk
SELECTA VOLUME E
With Foreword by R.E. Gomory
and Contributions by P.H. Cootner, E.F. Fama,
WS. Morris, H.M. Taylor, and others
~ Springer
Benoit B. Mandelbrot
Mathematics Department
Yale University
New Haven, cr 0652~283 USA
http://www·math.yale.edufmandelbrot
IBM T.J. Watson Research Center
Yorktown Heights, NY 10598-0218 USA
Library of Congress Cataloging-in-Publication Data
Mandelbrot, Benoit B.
Fractals and scaling in finance: discontinuity, concentration, risk I Benoit
Mandelbrot.
p. cm.
Includes bibliographical references.
ISBN 978-1-4419-3119-1 ISBN 978-1-4757-2763-0 (eBook)
DOI 10.1007/978-1-4757-2763-0
1. Finance-Statistical methods. 2. Fractals. 3. Scaling (Social sciences)
I. Title.
HG176.5.M36 1997
332'.01'S14742-dc21 97-37930
ISBN 978-1-4419-3119-1 Printed on acid-free paper
C 1997 Springer Science+Business Media New York
Originally published by Benoit B. Mandelbrot in 1997
Softcover reprint of the hardcover I st edition 1997
All rights reserved. This work may not be tmnslated or copied in wbole 01' in part without the written
permission of the publisher, Springer Science+Business Media, LLC,
except for brief excerpts in connection with reviews or scholarly analysis. Use in
connection with any form of information storage and retrieval, electronic adaptation, computer
software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.
The use in this publication of Iladc names, trademarks, service marks and similar terms, even if they
are DOt identified as such, is DOt to be taken as an expression of opinion as to whether or not they are
subject to proprietaIy rights.
987654
springeronlinc.com
To my sons, Laurent and Didier,
I dedicate this intellectual fruit of mine,
their demanding sibling
List of Chapters
In this List of Chapters, the sources given after the titles include (in parentheses)
the letter M followed by the year of publication and by the lower case letter that the
Bibliography uses to distinguish different texts published in the same year. In the
Bibliography, the items reproduced in this book and in Volumes Nand Hare
marked by a star followed by a chapter number, which in some cases is incomplete
or only tentative.
Foreword by Ralph E. Comory ................................................... ix
I. NONMATHEMATICAL PRESENTATIONS
Preface (1996) .................................................................................... 1
E1 Introduction (1996) ......................................................................... 13
E2 Discontinuity and scaling: scope and
likely limitations (1996) ................................................................. 50
E3 New methods in statistical economics (M 1963e) .................... 79
E4 Sources of inspiration and historical background (1996) ...... 105
II. MATHEMATICAL PRESENTATIONS
E5 States of randomness from mild to wild, and
concentration in the short, medium and long run (1996) ..... 117
E6 Self-similarity and panorama of self-affinity (1996) .............. 146
E7 Rank-size plots, Zipf's law, and scaling (1996) ....................... 198
E8 Proportional growth with or without diffusion,
and other explanations of scaling (1996) .
• Appendices (M 19640, M 1974d) ........................................... 219
E9 A case against the lognormal distribution (1996) .................. 252
III • PERSONAL INCOMES AND FIRM SIZES
EI0 L-stable model for the distribution of income (M 1960i) .
• Appendices (M 1963i, M 1963j) ............................................. 271
Ell L-stability and multiplicative variation of income (M 1961e) 307
E12 Scaling distributions and income maximization (M 1962q) .336
E13 Industrial concentration and scaling (1996) ............................ 364
IV. THE M 1963 MODEL OF PRICE VARIATION
E14 The variation of certain speculative prices (M 1963b) .
• Appendices (Fama & Blume 1966, M 1972b, M 1982c) ..... 371
E15 The variation of the price of cotton, wheat, and railroad
stocks, and of some financial rates (M 1967j) ......................... 419
E16 Mandelbrot on price variation (Fama 1963)
(A guest contribution by E. F. Fama) ....................................... 444
E17 Comments by P. H. Cootner, E. Parzen & W. S. Morris
(1960s), and responses (1996) ..................................................... 458
E18 Computation of the L-stable distributions (1996) .................. 466
V. BEYOND THE M 1963 MODEL
E19 Nonlinear forecasts, rational bubbles, and
martingales (M 1966b) ................................................................. 471
E20 Limitations of efficiency and martingales (M 1971e) ............ 492
E21 Self-affine variation in fractal time
(Section 1 is by W. H. Taylor) (M & Taylor 1967, M 1973c) 513
CUMULATIVE BIBLIOGRAPHY .............................................. 526
INDEX ............................................................................................ 542
Foreword
IN
1959-61, while the huge Saarinen-designed research laboratory at
Yorktown Heights was being built, much of IBM's Research was housed
nearby. My group occupied one of the many little houses on the Lamb
Estate complex which had been a sanatorium housing wealthy alcoholics.
The picture below was taken about 1960. It shows from right to left,
T.e. Hu, now at the University of California, Santa Barbara. I am next,
staring at a network I have just written on the blackboard. Then comes
Paul Gilmore, late of the University of British Columbia, then (seated)
Richard Levitan, now retired, and at the left is Benoit Mandelbrot.
x FOREWORD <> <> EF
Even in a Lamb Estate populated exclusively with bright research
oriented people, Benoit always stood out. His thinking was always fresh,
and I enjoyed talking with him about any subject, whether technical, poli
tical, or historical. He introduced me to the idea that distributions having
infinite second moments could be more than a mathematical curiosity and
a source of counter-examples. This was a foretaste of the line of thought
that eventually led to fractals and to the notion that major pieces of the
physical world could be, and in fact could only be, modeled by distrib
utions and sets that had fractional dimensions. Usually these distributions
and sets were known to mathematicians, as they were known to me, as
curiosities and counter-intuitive examples used to show graduate students
the need for rigor in their proofs.
I can remember hearing Benoit assert that day-to-day changes of stock
prices have an infinite second moment. The consequence was that most of
the total price change over a long period was concentrated in a few hectic
days of trading and it was there that fortunes were made and lost. He
also asserted that the historical data on stock prices supported this view,
that as you took longer and longer historical data, the actual second
moments did not converge to any finite number.
His thinking about floods was similar.
Benoit's ideas impressed me enormously, but it was hard to get this
work recognized. Benoit was an outsider to the substantive fields that his
models applied to, for example economics and hydrology, and he received
little support from mathematicians who saw only that he was using
known techniques. Benoit's contribution was to show that these obscure
concepts lie at the roots of a huge range of real world phenomena.
Lack of recognition, however, never daunted Benoit. He stuck to his
ideas and worked steadily to develop them and to broaden their range of
applicability, showing that one phenomenon after another could be
explained by his work. I was very pleased when I was able to get him
named an IBM Fellow, and later was successful in nominating him for the
Barnard Medal. After that the floodgates of recognition started to open
and Benoit today is one of the most visible of scientific figures.
Surely he has earned that visibility, both for his world-changing work,
and for his courage and absolute steadfastness.
Ralph E. Gomory
President, Alfred P. Sloan Foundation