Price movements in financial markets are not random. There are actually clues that allow sophisticated investors to uncover trends and make accurate predictions. The key to discovering this predictability lies in a new set of mathematical techniques --the application of dynamic, non-linear time series. This new science of investment is where chaos theory meets the markets. Richard Urbach offers practical advice and applications on the latest mathematical techniques and examines the opportunities these new techniques can deliver.

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Preface

A dynamical system is a system whose forward evolution is determined by a mathematical function of its present state. Thus, if the system is presently in state s, then its state at some time t in the future is given by fs,t. As far as concepts go, that is pretty simple. However, these systems can exhibit behavior that can be called anything but simple, and paradoxically, even though we can predict in theory exactly where the system will be at any time in the future given its present state, in practice prediction can be very limited.

Classical mathematical analysis of dynamical systems goes back at least to Isaac Newton's development of celestial mechanics and continued into the early 20th century with the work of H. Poincare. But Poincare was perhaps the first to realize the extent and nature of the behavioral complexity possible in these systems. He apparently also despaired at the difficulties in giving a complete analytic description of that complex behavior. The alternative of the numerical approach was also out of reach at the time due to the massive computation resources (man-hours in those days) needed to simulate and analyze the dynamics. Poincare was not mistaken. Analytical results were hard to come by, and progress was slow until the introduction of cheap computing devices circa 1980. From then on research in the subject grew rapidly. The past twenty or so years has brought results of both theoretical and practical importance, especially for systems that exhibit low dimensional chaotic behavior. However, much of what is practical in those results has been slow to emerge in a form that is readily understandable and usable by the general practitioner of time series analysis. The purpose of this book is to make available to the general practitioner the concepts and tools of chaotic dynamical systems analysis developed since the early 1980s. The credit for the work reported here belongs to the many referenced researchers, and of course, the responsibility for any errors in reporting their work rests solely with myself. If I have made a contribution it is in organizing and presenting the material in such a way that it is relatively easy to understand and apply by the non-cognoscenti.

Specifically, this book is about the numerical analysis of chaotic time series originating with nonlinear dynamical systems. The objective of that analysis is to develop a model that explains or predicts the behavior of the 'real' system that generated the time series. A 'real' system is one that has a material presence in the real world. Real systems can hurt you if they run into you, or fall on your toe. But they can also hurt you in less physical ways. For example, a political system can restrict your freedom, and an economic system or market can seriously impact your bank account.

Each chapter of the book treats a different concept in dynamical systems analysis and is organized into three sections. The first section explains the subject with simple ideas and figures that should not be difficult to understand by any quantitatively oriented reader. A reader only wishing to obtain a feeling for the subject could read the first section of each chapter without making contact with the theory. The second section covers enough theoretical material to underpin the intuition of the first section and support the algorithms developed in the third section. With respect to the sections on theory and algorithms, some assumption has to be made concerning the knowledge of the intended audience, and that will be a basic grounding in differential calculus, linear algebra, probability and statistics.

A more detailed description of the topics covered and the way in which they are organized would make little sense here without having first developed at least some concepts that are basic to dynamical systems, but perhaps not familiar to many readers. That will come in the first chapter along with statements of the key operational and technical assumptions that will apply throughout the book. The operational assumptions delimit what we know about the system under study. On the one hand, those assumptions make life difficult in that we will not be in a position to establish with mathematical rigor whether or not many of the theoretical results hold for the systems we are interested in. On the other hand, as the operational assumptions are realistic and the objective is numerical analysis, we have little choice but to just adopt those technical assumptions which are reasonable and lead to tractable computations of quantities that have theoretical meaning. If the quantities so obtained enable us to build a model that explains or predicts the behavior of the system we are studying, then we have arrived at a useful result, and it really does not matter what assumptions we made to get there, if a useful result was the objective. In other words, if a model explains and predicts what we observe in a system, then it fulfills its purpose, regardless of the means used to get it, or whether it duplicates 'nature's' computations.

The examples included in the algorithms section illustrate the effectiveness of each method in meeting the objectives of the algorithm on data generated from chaotic dynamical equations or obtained from measurements on real chaotic systems. Most of those systems have been heavily analyzed and their characteristics are well documented. The reason for including such intensely studied systems as examples is not to prove that they are chaotic, since they are known to be so, but rather to demonstrate the ability of a particular algorithm to reproduce known characteristics. If an algorithm can do that, it is a valuable tool. But, as with any tool used on unfamiliar material, when these methods are applied to less well understood systems, a good deal of the quality of the result will depend on the ingenuity and skill of the user.

There have been a number of papers and books published on the application of dynamical systems analysis to financial time series with the intention of demonstrating the existence or non-existence of determinism in capital markets. As far as I can tell, all the published results are inconclusive or unconvincing. There are a number of reasons why the reader should not be disheartened by reports of failed attempts to discover a deterministic price setting mechanism, in other words, predictability in financial markets. One reason is that research in this area will almost surely suffer from selective publishing. While I am not surprised at seeing inconclusive reports in print, I would be more than mildly surprised to see a report describing a deterministic model of a capital or commodity market. It just beggars the imagination to think that any individual intent on discovering determinism in a market's price setting mechanism would, having found the model, turn around and publish the details. Perhaps I am being too cynical, but I think that if someone has discovered something in this area that has profit potential, then they are quietly exploiting the discovery for their own gain, resisting the temptation for acclaim at least until their memoirs come out.

If the reader does not buy that argument, then there is another reason to remain optimistic. Some problems remain unsolved for long periods of time despite the efforts of the best minds. Then one day someone solves the problem, not necessarily because they had better tools to work with, but often because they saw the problem from a novel perspective or in a different context, or just applied the tools in an unconventional manner.

But if price action is considered to be the product of a random walk only unpredictable news can affect prices, and it is therefore impossible to accurately predict market movement. This line of thought has prompted the growth in passive investing and tracker funds.

Richard Urbach counters this thinking by applying a new set of mathematical techniques to the financial markets, arguing that it is possible to accurately analyze the time series (or 'footprints') of chaotic real systems, map their underlying patterns, and predict future trends.

Many of the techniques of dynamical systems analysis have previously been described only in the dispersed and relatively inaccessible outputs of scientific research. There has been little practical interpretation of non-linear dynamics and its application to financial markets.

Footprints of Chaos is the first book to examine the opportunities presented by this new thinking, and is aimed directly at the investment analysts who can benefit from it.

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**Book Description **Financial Times Prentice Hall, 1999. Condition: New. This item is printed on demand for shipment within 3 working days. Seller Inventory # GM9780273635734