Fractals and scaling in finance. Discontinuity, concentration, risk.

*(English)*Zbl 1005.91001
New York, NY: Springer. x, 551 p. (1997).

At statistical analysis of financial time series it is well known that many among them have the properties of statistical self-similarity appeared in the fact that, loosely speaking, the parts are organized the same as the whole. Clearly, that the similar properties require the explanation, and it can be given in the frameworks of general concept of statistical automodelity, carrying into not only such important concepts, as the fractal Brownian motion, the fractal Gaussian noise, but also having rendered the decisive influence on the creation of fractal geometry. Beginning in the 1950s Mandelbrot and others have intensively studied the self-similarity of pathological curves, and they have applied the theory of fractals in modelling natural phenomena. Random fluctuations induce a statistical self-similarity in natural patterns; analysis of these patterns by Mandelbrot’s techniques has been found useful in such diverse fields as fluid mechanics, geomorphology, human physiology, linguistics, and economics. This book incorporates the author’s pioneering contributions to finance, but half is entirely new: some chapters are mathematical; other contain few formulas. It presents and tests three successive rules of variation, tackling fast change and long distribution tails, then long dependence in time, and finally both features simultaneously. Careful modeling is needed because prices do not perform a random walk on the street, to mimic the toss of a coin: they are not tossed around like Brownian motion tosses a small piece of matter. Price change is often subjected to substantial sharp discontinuities, and periods of price activity are far from being uniformly spread over time. The overall theme is that, while prices vary wildly, scaling rules hold ensuring that financial charts are examples of fractal shapes. The recognition of the fractal nature of price variation combines many small mysteries into one very large one. This is the first of a multivolume series of Selecta, and cross-reference are included.

Contents: Foreword by Ralph E. Gomory. Ch.I. Nonmathematical presentations. Preface (1996). Introduction (1996). Discontinuity and scaling: scope and likely limitations (1996). New methods in statistical economics (1963). Sources of inspiration and historical background (1996). Ch.II. Mathematical presentations. States of randomness from mild to wild, and concentration in the short, medium and long run (1996). Self-similarity and panorama of self-affinity (1996). Rank-size plots, Zipf’s law, and scaling (1996). Proportional growth with or without diffusion, and other explanations of scaling (1996). Appendices (1964, 1974). A case against the lognormal distribution (1996). Ch.III. Personal incomes and firm sizes. L-stable model for the distribution of income (1960). Appendices (1963, 1963). L-stability and multiplicative variation of income (1961). Scaling distributions and income maximization (1962). Industrial concentration and scaling (1996). Ch.IV. The M 1963 model of price variation. The variation of certain speculative prices (1963). Appendices (Fama & Blume 1966, 1972, 1982). The variation of the price of cotton, wheat, and railroad stocks, and of some financial rates (1967). Mandelbrot on price variation (Fama 1963). Comments by P. H. Cootner, E. Parzen & W. S. Morris (1960), and responses (1996). Computation of the L-stable distributions (1996). Ch.V. Beyond the M 1963 model. Nonlinear forecasts, rational bubbles, and martingales (1966). Limitations of efficiency and martingales (1971). Self-affine variation in fractal time (Section 1 is by W. H. Taylor) (1967, 1973). Cumulative bibliography. Index. This book is a major contribution to the understanding of haw speculative prices vary in time.

Contents: Foreword by Ralph E. Gomory. Ch.I. Nonmathematical presentations. Preface (1996). Introduction (1996). Discontinuity and scaling: scope and likely limitations (1996). New methods in statistical economics (1963). Sources of inspiration and historical background (1996). Ch.II. Mathematical presentations. States of randomness from mild to wild, and concentration in the short, medium and long run (1996). Self-similarity and panorama of self-affinity (1996). Rank-size plots, Zipf’s law, and scaling (1996). Proportional growth with or without diffusion, and other explanations of scaling (1996). Appendices (1964, 1974). A case against the lognormal distribution (1996). Ch.III. Personal incomes and firm sizes. L-stable model for the distribution of income (1960). Appendices (1963, 1963). L-stability and multiplicative variation of income (1961). Scaling distributions and income maximization (1962). Industrial concentration and scaling (1996). Ch.IV. The M 1963 model of price variation. The variation of certain speculative prices (1963). Appendices (Fama & Blume 1966, 1972, 1982). The variation of the price of cotton, wheat, and railroad stocks, and of some financial rates (1967). Mandelbrot on price variation (Fama 1963). Comments by P. H. Cootner, E. Parzen & W. S. Morris (1960), and responses (1996). Computation of the L-stable distributions (1996). Ch.V. Beyond the M 1963 model. Nonlinear forecasts, rational bubbles, and martingales (1966). Limitations of efficiency and martingales (1971). Self-affine variation in fractal time (Section 1 is by W. H. Taylor) (1967, 1973). Cumulative bibliography. Index. This book is a major contribution to the understanding of haw speculative prices vary in time.

Reviewer: Alexandr Vasil’ev (Odessa)

##### MSC:

91-02 | Research exposition (monographs, survey articles) pertaining to game theory, economics, and finance |

91B84 | Economic time series analysis |

28A80 | Fractals |