## Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Selecta Volume EIN 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. |

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### Contents

1 | |

13 | |

scope | 50 |

E3 New methods in statistical economics M 1963e | 79 |

E4 Sources of inspiration and historical background 1996 | 105 |

MATHEMATICAL PRESENTATIONS | 117 |

E6 Selfsimilarity and panorama of selfaffinity 1996 | 146 |

E7 Ranksize plots Zipfs law and scaling 1996 | 198 |

E12 Scaling distributions and income maximization M 1962q | 336 |

E13 Industrial concentration and scaling 1996 | 364 |

e THE M 1963 MODEL OF PRICE VARIATION | 371 |

E15 The variation of the price of cotton wheat and railroad | 419 |

E16 Mandelbrot on price variation Fama 1963 | 444 |

E17 Comments by P H Cootner E Parzen W S Morris | 458 |

E18 Computation of the Lstable distributions 1996 | 466 |

E20 Limitations of efficiency and martingales M 1971e | 492 |

E8 Proportional growth with or without diffusion | 219 |

E9 A case against the lognormal distribution 1996 | 252 |

E10 Lstable model for the distribution of income M 1960i | 270 |

E11 Lstability and multiplicative variation of income M 1961e | 307 |

E21 Selfaffine variation in fractal time | 513 |

526 | |

542 | |

### Other editions - View all

Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Selecta ... Benoit B. Mandelbrot No preview available - 1997 |

Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Selecta ... Benoit B. Mandelbrot No preview available - 2013 |

Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Selecta ... Benoit B. Mandelbrot No preview available - 2010 |

### Common terms and phrases

addends approximation arbitraging argument assumption asymptotically scaling average Bachelier behavior cartoons central limit theorem Chapter E1 concentration concerning Cootner curve decreases defined described distrib economics effect empirical equal example expected exponent exponential fact Figure finance finite firms fractal fractal dimension fractional Brownian motion function Gaussian Gaussian distribution graph hence hypothesis income increases increments independent infinite variance interval invariance L-stable distributions L-stable process L-stable variable large numbers largest Lévy limit theorem linear log p(u log Z(t logarithm lognormal lognormal distribution long-run MANDELBROT martingale mathematical mild moments multifractal nonCaussian observed paper parameters Pareto physics plot Pr{u prediction price changes price variation probability theory problem properties random variables random walk ratio result sample scaling distribution Section self-affine skew spectral speculative statistical statistically independent tail term theory tion trading ution Wiener Brownian motion yields zeta distribution Zipf's law