Last Year in Mandelbrot

On October 19, 1987 the Dow-Jones Industrial Average, the widely-followed proxy for the US stock market, declined by 23%, a move that some observers noted was a 20 standard deviation event. This drop, which was almost twice as large as the famous stock market crash of 1929, is not an isolated incident, but one of a number of large drawdowns and bear markets this century alone, the most recent of which, the October 1997 crash, was only a "modest" 8% drop. Faced with these statistics, most of us would probably be prepared to agree that price changes are not Gaussian or examples of random walk behavior. However, events such as the 1987 crash, World War 1 and the crash of 1929 are extreme and properly regarded as outliers. What about business as usual?

The answer, of course, depends on who you ask. On one hand, the well-regarded semi-popular book on finance, "A Random Walk Down Wall Street" (W.W. Norton, New York, 6^th edition 1996) by Burton Malkiel, takes its title and its theme from the notion that stock price changes follow a Brownian motion. Virtually every textbook on advanced finance takes the Brownian motion description as its starting point, and the celebrated Black-Scholes formula for option prices is based upon this description.

And on the other hand, there is Benoit Mandelbrot. No reader of this journal can be unaware of the enormous impact made by Mandelbrot’s earlier book "The Fractal Geometry of Nature" (Freeman, New York, 1982), which has introduced many to the notions of fractal dimensions, scaling and self-similarity, and which spawned a host of coffee-table imitations. What is perhaps less well-known, however, is that some of Mandelbrot’s earliest forays into fractals involved a detailed analysis of the time series for cotton prices in New York. Mandelbrot’s shocking conclusion, published in 1963, was that the time series was in no way Gaussian: in fact, he argued that the departures from normality could be accounted for by using distribution functions with infinite variance, which are termed L-stable. Mandelbrot examined the convergence in sample number of the variance of the logarithm of the daily price changes and found erratic variation rather than convergence. Subsequently, his student Eugene Fama (who has himself enjoyed a distinguished career in finance) examined the time series for the thirty stocks in the Dow-Jones Industrial Average, finding no exceptions to the long-tailed nature of the distributions observed.

The implications of these and subsequent findings are profound, yet it is fair to say that the work was practically ignored by economists and practitioners of finance. Even today, the problem of "fat tails" is swept under the rug by the vast majority of financial risk managers, even though the phenomenon is sufficiently widespread and well-recognized as to have earned its whimsical name. Mandelbrot’s heirs are primarily physicists entering the field of finance, who recognize the fundamental importance of fat tails and are able to elaborate and extend Mandelbrot’s suggestive results. This is something of an ironic development, as Mandelbrot takes pains to emphasise, and reflects the close intellectual relationship between finance and physics. The discovery of Brownian motion, usually attributed to Einstein’s famous 1905 paper, was in fact anticipated by Louis Bachelier five years earlier in his Ph.D dissertation on finance "Theorie de la speculation", which remained largely ignored by economists until the 1950’s and 60’s. Elements of Mandelbrot’s work in the early 1960’s, which superseded Bachelier’s analysis just as it was becoming widely accepted, arguably anticipate some of the concepts of scaling and renormalization which were a focal point of physics during the 1970’s. The concepts of fractional Brownian motion and multifractals which are still frontier topics of research in physics and academic finance (as practiced by physicists) were introduced by Mandelbrot in the late 1960’s and 1970’s. And most recently, legions of physicists have found gainful employment on Wall Street as "quants", performing intricate calculations of price and risk of derivative securities, using sophisticated detailed models whose underlying premises remain those of Bachelier—Brownian motion (more accurately logBrownian motion).

The present volume, "Fractals and Scaling in Finance", is a characteristically idiosyncratic work. At once a compendium of Mandelbrot’s pioneering work and a sampling of new results, the presentation seems modeled on the brilliant avant-garde movie "Last Year in Marienbad", in which the usual flow of time is suspended, and the plot is gradually revealed by numerous but slightly different repetitions of a few underlying events. As Mandelbrot himself admits in the Preface, the presentation allows the reader unusual freedom of choice in the order in which the book is read. In fact, I enjoyed this work most when I read it in random order, juxtaposing viewpoints and analyses separated in time by 3 decades, and making clear the progression of ideas that Mandelbrot has generated. These include the classification of different forms of randomness, their manifestation in terms of distribution theory, their ability to be represented compactly, the notion of trading time, the importance of discontinuities, the relationship between financial time series and turbulent time series, the pathologies of commonly abused distributions, particularly the log-normal, and a catalogue of the methods used to derive scaling distributions, both honest and fallacious.

Mandelbrot writes with economy and felicitously, and intersperses the more mathematical sections with frank historical anecdotes, such as the events which led up to his work on cotton pricing, and the embarrassment caused by interpreting United States Department of Agriculture data for weekly averages as "Sunday closing prices". There are many fascinating asides on a variety of topics ranging from the importance of computer graphics in science to the distribution of insurance claims resulting from fire damage. In some places, the format of reprinted (but slightly edited) versions of classic papers allows Mandelbrot the surreal luxury of reviewing not only the content, but also the style and presentation of his work. And if this was not enough, there are guest contributions from Eugene Fama, Paul Cootner and others.

This volume is not intended to be a text book of modern finance, and will probably infuriate those seeking a balanced and systematic exposition. Some readers will be irritated by the admitted redundancy of the text and frequent lapses into informality which would not normally be tolerated. My favorite is the caveat on page 232 which reads "Due to time pressure, the algebra in this section has not been checked through and misprints may have evaded attention". Indeed there are many misprints which I noticed in this book, but to criticize the volume on their account would be churlish. The reader who brings to this book a spirit of open-mindedness, and is prepared to indulge one of our more influential and original thinkers, is amply rewarded.

All in all, this is a strange but wonderful book, which will not suit every one’s taste, but will almost surely teach every reader something new. What more can one ask?

Relationship of reviewer to the field

Nigel Goldenfeld is a theoretical physicist and co-founder of NumeriX, a financial software company, specializing in pricing and risk of derivative securities. He is a member of the editorial board of the International Journal of Theoretical and Applied Finance.