The wealth that can be realized in the various markets (e.g., stock, foreign exchange, commodities) means virtually every aspect of market trading and behavior has been studied (although not necessarily understood). The complexity of markets also means that they are heavily studied by academics. As a result, there is a vast body of mathematical literature on markets, including portfolio theory, option pricing and risk estimation. This mathematics is largely based on Gaussian statistics. Many of the models that have been developed from this mathematical base have proven effective and have earned some practitioners huge fortunes. They have also lost huge fortunes. Market statistics do not follow nice clean Gaussian distributions. Rather, they have "fat tails". This means that the body of the curve is a Gaussian function, but that the tail of the curve follows a power law. In practical terms large losses are more common than the simple Gaussian model suggests.

Power laws (asintotic curves that approach zero) occur in many places, including market statistics. In some cases, the exponent in the power law is related to the fractal dimension. This exponent is referred to as the Hurst exponent, after a hydrologist named Hurst, who discovered it.

The beautiful pictures of fractals, in particular the Mandelbrot fractal, seem to have captured people's imaginations. Chaos and fractal theory have generated a great deal of interest. This interest has been fueled by the intuitive observation that the world around us is filled with fractal shapes. A common example of a fractal shape is a fern frond. Other fractal shapes include the weathered rock of mountains and clouds. Market time series (the series of market prices over time: for example the daily close price of IBM) seem to show fractal structure. A number of people have attempted to use the mathematics of chaotic systems and fractals to analyze market data in an attempt to find structure and predictability. One person who has done some work in this area is Edgar E. Peters, the author of Chaos and Order in the Capital Markets: A New View of Cycles, Prices, and Market Volatility, Second Edition.

There is a vast literature on chaos and fractal theory, from introductory books to advanced graduate level mathematics monographs, not to mention a huge body of research papers. One of the challenges that an author faces when writing about chaos theory applied to financial markets is explaining chaos theory. One book, Footprints of Chaos in the Financial Markets by Richard M. A. Urbach, spends so much space on chaos theory that the actual application of this theory to the market data is largely ignored. Edgar Peters' book Chaos and Order in the Capital Markets does a better job on concentrating on applications to the market and the mathematics requires only a high school math background.

I bought Chaos and Order in the Capital Markets because I was working on software to calculate the Hurst exponent. I am also a computer scientist with a strong interest in finance (I spent two years working at a company that developed financial models). For me, Peters' book was a disappointment.

There are a number of problems with Peters' book. The first edition of the book was writtin in 1990 and the second edition was writtin in 1996. A fairly large body of literature has appeared in the last few years on long memory processes. In fact, Peters sources are dated, even considering the publication date. The fractal techniques that Peters uses to estimate the Hurst exponent and to generate fractional brownian motion (a random walk, biased by the Hurst exponent) are based on an even older 1988 book by Jens Feder, Fractals (Plenum).

Another problem with the book is that Peter's spends the first four chapters arguing against the efficient market hypothesis. Perhaps in 1996 this was a more radical attack that it is today. I suspect that few market practitioners believe in the "strong form" of the efficient market hypothesis. And it is probably safe to say that no short term traders believe in it. As huge databases of market information and high performance computing power have become widely available, statistical evidence against the "strong form" of the efficient market hypothesis has accumulated. The strongest academic attack was launched in A Non-Random Walk Down Wall Street by Andrew W. Lo and A. Craig MacKinlay, Princeton University Press, 1999.

Following the attack on the efficient market hypothesis Peters spends two chapters discussing fractal dimensions, the Hurst exponent and the rescaled range (RS) calculation to estimate the Hurst exponent. If your objective is to get a basic understanding of the Hurst exponent and how it applies to market data, Peters' description is probably sufficient. If you actually want to implement the algorithms, the book falls woefully short. Peters rewrite of Jens Feder's discussion on the Hurst exponent is not nearly as good as the original. Even with Feder's book in hand, I could not understand how to implement the algorithm without reading some of the papers that have been published on the rescaled range technique. For example, one paper that expands on Feder is A Rescaled Range Analysis of Random Events by Fotini Pallikari and Emil Boller, Journal of Scientific Exploration, Vol. 13, No. 1, pp. 25-40, 1999. Despite the application, psychokinesis, I found the description of the algorithm for Hurst exponent calculation clear.

The book comes with a floppy disk which contains the implementation for the algorithms described in the book Microsoft Visual Basic. In a world where C is the lingua franca this is not the best choice. Although I have a relatively recent version of Microsoft Visual C++ (in addition to GNU g++), I don't have Visual Basic, so I can't compile the code. At least in the case of the RS calculation for the Hurst exponent the Visual Basic algorithm does not directly follow the description in the text. This leaves the reader trying to understand the Visual Basic code from the source alone.

Peters applies the Hurst exponent calculation to a few financial time series. He notes that the time series decay toward a random walk and he concludes that this indicates cycles in the time series. As far as I can tell, this conclusion is not supported by Peters' results. Another explaination is that there was a long memory process at work and it has simply decayed.

The chapters that follow the material on the Hurst exponent are no better. Peters explores several other techniques from Chaos theory and applies them out on a few financial time series. There is no indication that these techniques would actually be useful in developing financial models. Peters is an executive at an investment fund, PanAgora Asset Management, and he admits that he would not tells us if he did find an effective technique:

Finally, there are nonlinear elements in our models. True to form, I am not going to go into them in great detail here.

In describing PanAgora, Peters writes

PanAgora manages about $15 billion. We are also true quants, who do not override our models with subjective judgment. It is our conviction that the subjective side of asset management is in formulating the models. Once the models are rigorously tested for robustness, they are strictly followed.

The rigorous testing and gathering of statistics that presumably PanAgora uses in their trading models is lacking in Chaos and Order in the Capital Markets. The chaos and fractal statistics that Peters describes in this book are tried out on a handful of stocks, rather than a broad spectrum, drawn from different market sectors. Peters quotes Feder on accuracty problems with the RS statistics, but he does not provide any real analysis here either. For example, there are no tables showing the error of the RS calculation for various data lengths. Given the dated nature of the book and the lack of any real statistical meat, anyone who who wants more than a general overview will be disappointed.

Afterward

After writing this review I exchanged a few e-mails with the author, Edgar Peters. He points out that Chaos and Order in the Capital Markets is intended to provide an overview of chaos theory applied to the financial markets. His intention was not to provide rigorous algorithmic descriptions and statistical analysis. In fact, in the preface to the first edition, Peters writes:

This book is a conceptual introduction to fractals and chaos theory as applied to investments and, to a lesser degree, economics.

[...]

This book is not meant as a textbook. It is intended to communicate the concepts behind fractals and chaos theory, as they apply to capital markets and economics. I have supplied no proofs of the theorems. those interested in such full mathematical treatments are referred to the bibliography, where an abundance of mathematical and scientific texts and papers is offered.

In his e-mail, Mr. Peters wrote that his book Fractal Market Analysis, 1994, which was published two years after the first edition of this this book, provides more analytical meat. I have not read Fractal Market Analysis so I can comment on this.

The criticism that this review holds the Chaos and Order to a set of standards that it is not intended to address may be true. Peters was probably the first person to write an approachable work on the application of fractals and chaos to financial markets. The book was originally written in 1992, before much of the work on the Hurst exponent and long memory processes, inspired by computer network modeling and analysis appeared.

While this book may have been pioneering in its time, it is still reasonable to ask if this book can be recommended in the year 2003. The fact remains that the book is dated.

I am still left with the question: how do we know what we know? This is a real problem with it comes to the analyzing and modeling markets. There are rarely cases where there are unambiguous answers. Understanding is built from a chain of evidance built from models and statistics. The Hurst exponent and the mathematics of chaos is complex and related to equations in statistics and frequency analysis. In many cases mathematical proofs give us little, because in some cases equations that can be proven correct do not provide good numeric results in practice. For any reader seeking to understand what the equations mean and how they apply, such a broad introduction may not be terribly useful. Without such background information there is also the chance that the author has made errors in analyzing this complex topic.

Ian Kaplan
February 2003
Last updated on: April 2003

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