2002

Table of Contents Introduction………………………………………………………….. 1 1. What’s markets? A Brief History of Trade………………………….. 2 2. Philosophy of Chaos………………………………………………… 5 3. Chaos Theory………………………………………………………... 7 3.1 Function of Chaos……………………………………………………. 10 3.2 Definition of Chaos………………………………………………….. 10 4. Technical Analysis of Financial Markets (TA)……………………… 11 4.1 Definition…………………………………………………………….. 11 4.2 Principles of Technical Analysis…………………………………….. 11 4.3 The Foundations of Technical Analysis……………………………… 11 4.3.1 Charles Dow Theory………………………………………… 11 4.3.2 Elliott Wave Theory…………………………………………. 11 4.3.2.1 Fibonacci Numbers…………………………………………. 12 4.3.2.1.1 Golden Ratio………………………………………………… 13 4.3.2.2 Basics of the Wave Theory…………………………………. 14 4.3.2.2.1 Fibonacci Numbers and Ratios in the Waves……………….. 16 4.3.2.2.2 Fibonacci in Percentage Retracements………………………. 17 4.3.2.2.3 Fibonacci in Time Targets…………………………………… 17 4.4 Some Critics of Technical Analysis…….……………………………. 20 5. Chaos Theoretical Analysis of Financial Markets (CA)……………... 21 5.1 System approach of CA to the analysis of Financial Markets……….. 21 5.2 Price Distributions…………………………………………………… 22 5.2.1 Bachelier Method……………………………………………………. 22 5.2.1.1 Historical Volatility………………………………………………….. 23 5.2.1 2 Square Root of Time…………………………………………………. 24 5.3 Hausdorff (Fractal) Dimension………………………………………. 24 5.3.1 Sierpinski Carpet…………………………………………………….. 25 5.3.2 Fractal Time Scale……………………………………………………. 26 5.3.3 Koch Curve…………………………………………………………… 27 5.4 Leptokurtic Distributions……………………………………………… 28 5.4.1 Probability Densities………………………………………………….. 28 5.4.1.1 Location and Scale……………………………………………………. 30 5.5 Symmetric Stable Distributions and the Golden Mean Law ………. 31 5.6 Measuring Sensitivity………………………………………………… 31 5.6.1 How to Find the Prediction Window…………………………………. 32 5.7 To Summarise………………………………………………………… 34 Summary and Conclusions……………………………………………………. 35 List of Figures…………………………………………………………………. 36 Appendix 1 Dow Jones and his Theory………………………………….. i Appendix 2 Benoit Mandelbrot and his Contribution in the Analysis of the Fractal Geometry of Nature……………………………… vi Appendix 3 Definition of a Dynamical System…………………………. x Appendix 4 Fibonacci and the Power of the Golden Mean …………….. xi Appendix 5 N.R. Elliott and his Wave Theory………………………….xxiii Appendix 6 Brownian Motion and Bachelier Process…………………… xl Appendix 7 Lyapunov Exponent ………………………………………..xliii Appendix 8 Hurst Exponent…………………………………………….…liii References

Introduction The essay aims at giving a general theoretical and philosophical view on the development of financial markets. The purpose of the essay is not to list the existing tools of Technical Analysis, like types of charts, nor is it a handbook of their interpretation, like analysis of patterns and trends. The essay gives a philosophical picture of the movement of the market towards globalisation as part of the general natural process in the view of the theory of chaos and fractals. The main part of the paper consists of 5 chapters. The first chapter gives a general concept of markets as a dynamical social organism. It briefly outlines the history of trade and points the tendency to globalisation. The development of markets shows random highs and lows, which produce a general impression of unpredictability. That leads us logically to the notion of chaos, which is being studied from the philosophical point of view in Chapter 2, while Chapter 3 gives a brief explanation of the modern Theory of Chaos. Chapter 4 contains the description of the Technical approach to the analysis of markets. It deals with the definition of Technical Analysis, its principles and its foundations. A big part of the chapter is devoted to the Elliott Wave Theory as an important contribution into the Technical analysis of Financial Markets, which served basis for thousands of modern constructions used for prediction purposes. The Fibonacci relations in market movements as well as its fractal geometry are given special attention, as an indicator of the existence of an inner structure in the system. The chapter also mentions some criticism of the methods of Technical Analysis. Chapter 5 is devoted to the modern Chaos Theoretical analysis of Financial Markets. It introduces a number of mathematical notions needed to understand the methods and points out, that with the Chaos Theory the Fibonacci golden ratio can also be traced in the markets. The conclusion offers the idea to view the development of the markets towards globalisation as a natural process of social economic development, following the natural law and having a tendency to the Golden Mean. The 8 Appendices to the paper elaborate some of the aspects, touched upon in the main part, for the purpose of giving a more complete picture of some important issues that haven’t been given enough attention in the main body of the essay.

Chapter 1. What is Markets? A Brief History of Trade An historical perspective is probably the most common way to begin a description of any kind of human activity[68]. History is the backbone of all disciplines and its study is important in understanding human behaviour. In our case, the sheer task of covering the entire trading history of homo economicus is at the least, bewildering. Was the first commercial transaction struck some time around 10000 BC? We will probably never know. The earliest writings were pictograms inscribed on clay tablets, they were lists of goods. These ancient ledgers were used to keep a detailed account of daily or periodic activities. They date back to 2000 BC and belong to the Mesopotamian civilization. Phoenicians were probably the first to embody the quintessence of a merchant: a person living by trade, profiting from wares which others had produced. It was exactly this activity that made them sail the entire Mediterranean, go beyond the Herculean Pillars and find new sources of copper and tin in Cornwall. During their journeys they founded several trading stations, scattered along the Mediterranean coasts. And that was around 1000 BC. Actually wherever a civilization sprang up, it was accompanied by trading activities. Ancient Egypt, Minoan civilisation, Crete, Classical Greece, Roman Empire, Byzantium, Asian civilisations in the ancient times, which sprang up and faded away to find a rebirth in the middle ages in other forms and with another names, and on a different level of their trading activities, where trade became closely connected with shipping and a little later with finance. Italian cities, like Genoa, Venice, Florence and Pisa were rivals in establishing economic control over Crete and Cyprus. They were fighting for trade supremacy. Trade and military power went hand in hand, the latter securing uninhibited operation for the former, while the former financed the latter. Hanse merchants from the Baltic region built up a league. It was strong not only due to the trading skills of its merchants, but also due to the much needed financial skills required to organize shipping expeditions and to create the soundness and reliability that the Hanse tradesmen were renown for. Later colonial powers like Portugal, Spain, the Netherlands, took over international trade. Most of all, the Dutch era saw the creation of institutions that allowed and sustained the existence of trading empires. From the “freedom on the seas” to contract law, those new institutions were a tantamount to the prosperity of private enterprise. Contrary to the state run empires like Portugal and Spain, the Dutch, and later the English, empire was based on individuals and companies. That was probably the starting point in history of trade, when on more or less a global scale free trade started to emerge. For modern neo-classical economists, this state of economic affairs meant the emergence of Adam Smith’s “invisible hand” which could manage human activity, rendering state intervention needless. This era brought about a substantial decrease in information costs, increased capital mobility, risk diversification, and a legal framework that formalised contracts and ensured their enforceability. On the 31st December 1600 a new company was incorporated in London, that would become a symbol of global English predominance over the following 2 centuries. Its name was “East India Company” and would join numerous other companies that were being set up to take advantage of promising trade opportunities around the world. Although it would be difficult to put a time stamp on the beginning of modern global economics, it can approximately be placed in the second half of the 19th century. Europe, the

Americas and the Far East were all entangled in a global economic web, at the heart of which lay trade[68]. The data on market analysis is only available since the end of the last century, when the first modern globalisation trend appeared, with the boom in transportation technologies. The 50 years before the First World War saw large cross-border flows of goods, capital and people. That period of globalisation, like the present one, was driven by reductions in trade barriers and by sharp falls in transport costs, thanks to the development of railways and steamships. It was then that Charles Dow created his first industrial and railway indices. The present surge of globalisation is in a way a resumption of that previous trend. That earlier attempt at globalisation ended abruptly with the First World War, after which the world moved into a period of fierce trade protectionism and tight restrictions on capital movement. During the early 30-s America sharply increased its tariffs, and other countries retaliated, making the Great Depression even greater. The volume of world trade fell sharply. International capital flows virtually dried up in the inter-war period as governments imposed capital controls to try to insulate their economies from the impact of a global slump. Capital controls were maintained after the Second World War, as the victors decided to keep their exchange rates fixed – an arrangement known as the Bretton Woods system, after the American town in which it was approved. But the big economic powers also agreed that reducing trade barriers was vital to recovery. They set up the General Agreement on Tariffs and Trade (GATT), which organised a series of negotiations that gradually reduced import tariffs. GATT was replaced by the World Trade Organisation (WTO) in 1995. Trade flourished. In the early 1970s the Bretton Woods system collapsed and currencies were allowed to “float” against one another at whatever rates the market set. This signalled the rebirth of the global capital market. The boom in communication and information technologies supports the present globalisation trend. With the costs of communication and computing falling rapidly, the natural barriers of time and space that separate national markets have been falling too. The second driving force is liberalisation. As a result of both the WTO negotiations and unilateral decision, almost all countries lowered barriers to foreign trade. Most countries have welcomed international capital as well. Could the trend towards globalisation be reversed again? New technology and new types of financial instruments make it tricky for governments to impose effective capital controls. The growth of multinational firms that can switch production from one country to another would also make it harder to erect effective trade barriers. New technology also creates distribution channels that protectionist governments will find hard to block. Another reason to suppose that globalisation is irreversible is that free trade is built upon firmer institutional foundations now. Withdrawal from the WTO would not be done lightly. However, even now the available data is not global. Financial and commodities exchange indices are only tools to use the liquid markets, they don’t give the global picture, because a substantial part of modern world is still closed with economic and political barriers. The goal of the process of globalisation is to secure stability of the market, with the potential to ensure steady growth of productivity and living standards everywhere, by means of a better

division of labour and free shift of capital to whatever place offers most productive investment opportunities. The history of homo economicus shows fractal geometry of the development of the system. Local trading activities, started with the first civilisations, found its replicas on smaller and larger scales, growing consistently into the global scale which we evidence now. Looking back, we may percept the whole picture as random: civilisations springing up here or there, at different times, with different characteristics, however, the choice of trading centres was never random, it was always very rational and can be explained in every case. Modern markets are a complex social organism, embracing a great number of interacting components, incorporating not only demand and supply, but almost all human actions, reactions, fashion trends, moods, intentions, traditions, happenings, information and its interpretation, etc. Markets are sort of an index of social economic development. Such markets are called “Efficient Markets”, meaning that actual prices reflect all the information that effects them. In financial markets all the interacting components effect the development of the organism, and the result of the action produces a general impression of chaos.

Chapter 2. Philosophy of Chaos Chaos has a bad name in some ways. It was chaos that caused the Trojan War. Eris, goddess of chaos, upset at not being invited to the wedding of Peleus and Thetis, showed up anyway and rolled a golden apple marked "kalliste" ("for the prettiest one") among the guests. Each of the goddesses Hera, Athena, and Aphrodite claimed the golden apple as her own. Zeus appointed Paris, son of Priam, king of Troy, judge of the beauty contest. Paris gave the apple to Aphrodite, and Hera and Athena went off fuming to plot the destruction of Troy[12]. While the Greeks had a specific goddess dedicated to Chaos, early religions gave chaos an even more fundamental role. In the Babylonian New Year festival, Marduk separated Tiamat, the dragon of chaos, from the forces of law and order. This primal division is seen in all early religions. Traditional New Year festivals returned symbolically to primordial chaos through a deliberate disruption of civilized life. One shut down the temples, extinguished fires, had orgies and otherwise broke social norms. Then a purification period followed, whereby the dragon of chaos was overthrown, and life went back to normal. Around us in the world today we see the age-old battle between order and chaos. In the international sphere, the old order of communism has collapsed. In its place is a chaotic matrix of competing, breakaway states, wanting not only political freedom and at least a semi- market economy, but also their own money supplies and nuclear weapons, and in some cases a society with a single race, religion, or culture. We also have proclamations of a New World Order, on the other hand, accompanied by the outbreak of sporadic wars. In fact, wherever we look, central command is losing control. Even in the sphere of the human mind we have increasing attention paid to cases of multiple personality, or as some would say, mental chaos. The most recent theories see human identity and the human ego as a network of cooperative subsystems, rather than a single entity. (Examples of the viewpoint are found in Robert Ornstein, Multimind[55] and Michael Gazzanaga, The Social Brain[9]). An average person, educated or not, is not comfortable with chaos. Faced with chaos, people begin talking about the fall of Rome, the end of time. Faced with chaos people begin to deny its existence, and present the alternative explanation that what appears as chaos is a hidden agenda of historical or prophetic forces that lie behind the apparent disorder. They begin talking about the "laws of history" or proclaiming that "God has a hidden plan". Well, is it really the case there is a hidden plan, or does the goddess Eris have a non- hidden non-plan? People are so uncomfortable with chaos, in fact, that Newtonian science as interpreted by Laplace and others saw the underlying reality of the world as deterministic. If you knew the initial conditions you could predict the future far in advance. Then came quantum mechanics, with uncertainty and indeterminism, which even Einstein refused to accept, saying: "God doesn't play dice." Philosophically, Einstein was afraid of the threatened return of chaos, preferring to believe for every effect there was a cause. R.N. Elliott (See Appendix 3) wrote in his “Wave Principle”[61]: “No truth meets more general acceptance than that the universe is ruled by law. Without law, it is self-evident there would be chaos, and where chaos is, nothing is.... Very extensive

research in connection with... human activities indicates that practically all developments which result from our social-economic processes follow a law that causes them to repeat themselves in similar and constantly recurring serials of waves or impulses of definite number and pattern... The stock market illustrates the wave impulse common to social-economic activity... It has its law, just as is true of other things throughout the universe.” (August 31st, 1938). He based his theory on that presumption. A consequence of this was the notion that if you could control the cause, you could control the effect. However, you can't see the future precisely because you don't really know what's causing it. The myth of causality denies the role of Eris. At the beginning of this century, the mathematician Henri Poincaré, who was studying planetary motion, began to get an inkling of the basic problem: "It may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible" (1903) [58]. It needed the 20th century to come with the development of computer technologies to allow people put order to chaos. Now when we say “chaos” we mean a deterministic non-linear dynamical system sensitive to initial conditions. Chaos theory is about explaining apparent disorder in a very ordered way.

Chapter 3. Chaos Theory Since the 1960s science has been undergoing an intellectual revolution that may be as significant as the development of quantum mechanics. It is now widely understood that deterministic is not the same as predictable. When was chaos first discovered? The first true experimenter in chaos was a meteorologist, named Edward Lorenz. In 1960, he was working on the problem of weather prediction. He had a computer set up, with a set of twelve equations to model the weather. It didn't predict the weather itself, however this computer program did theoretically predict what the weather might be. One day in 1961, he wanted to see a particular sequence again. To save time, he started in the middle of the sequence, instead of the beginning. He entered the number off his printout and left to let it run. When he came back an hour later, the sequence had evolved differently. Instead of the same pattern as before, it diverged from the pattern, ending up wildly different from the original. Eventually he figured out what happened. The computer stored the numbers to six decimal places in its memory. To save paper, he only had it print out three decimal places. In the original sequence, the number was .506127, and he had only typed the first three digits, .506. This effect came to be known as the butterfly effect. The amount of difference in the starting points of the two curves is so small that it is comparable to a butterfly flapping its wings. The flapping of a single butterfly's wing today produces a tiny change in the state of the atmosphere. Over a period of time, what the atmosphere actually does diverges from what it would have done. So, in a month's time, a tornado that would have devastated the Indonesian coast doesn't happen. Or maybe one that wasn't going to happen, does [67]. The “butterfly effect”, common to chaos theory, is also known as sensitive dependence on initial conditions. Just a small change in the initial conditions can drastically change the long-term behaviour of a system. (This phenomenon is given a detailed explanation in Appendix 7). From the idea of the “Butterfly effect”, Lorenz stated that it is impossible to predict the weather accurately. However, this discovery led Lorenz on to other aspects of what eventually came to be known as chaos theory. Lorenz started to look for a simpler system that had sensitive dependence on initial conditions. His first discovery had twelve equations, and he wanted a much more simple version that still had this attribute. He simplified the system to only three equations. The equations for this system also seemed to give rise to entirely random behaviour. However, when he graphed it, a surprising thing happened. The output always stayed on a curve, a double spiral. There were only two kinds of order previously known: a steady state, in which the variables never change, and periodic behaviour, in which the system goes into a loop, repeating itself indefinitely. Lorenz's equations are definitely ordered - they always followed a spiral. They never settled down to a single point, but since they never repeated the same thing, they weren't periodic either. He called the image he got when he graphed the equations the Lorenz attractor.

Here it is. Notice how the line loops around, never intersecting itself. Figure 1. Lorenz Attractor The system keeps looping around two general areas, as though it were drawn to them. The points from where a system feels compelled to go in a certain direction are called the basin of attraction. The place it goes to is called the attractor. Such attractors as above are called strange attractors. Chaotic systems are non-linear and follow trajectories that end up on non-intersecting loops called strange attractors. Classical systems of equations from physics were linear. Linear means that outputs are proportional to inputs. For example, here is a simple linear equation from the capital-asset pricing model used in corporate finance: E(R) = α + β E(Rm). It says the expected return on a stock, E(R), is proportional to the return on the market, E(Rm). The input is E(Rm). You multiply it by β ("beta"), then add α ("alpha") to the result—to get the output E(R). This defines a linear equation.

Equations which cannot be obtained by multiplying isolated variables (not raised to any power except the first) by constants, and adding them together, are non-linear. The equation y = x2 is non-linear, for example. Strange attractors have fractal patterns. Fractals are mathematical structures, which on an ever-smaller scale infinitely repeat themselves. Loistl and Betz are writing, that strange attractors, despite their complexity, have a high degree of order According to these authors, the geometrical structure of an attractor gives a detailed picture of the dynamics of a system. The understanding of the system’s dynamics should help predict its future development[47]. Such patterns are traced out by self-organizing systems. Names other than strange attractor may be used in different areas of science. In biology (or socio-biology) one refers to collective patterns of animal (or social) behaviour. In Jungian psychology, such patterns may be called archetypes. An “efficient market” is a self-organizing system by definition. The main feature of chaos is that simple deterministic systems can generate what appears to be random behaviour. It’s not hard to imagine that if a system is complicated and hence governed by complicated mathematical equations, then its behaviour might be complicated and unpredictable. What has come as a surprise to most scientists is that even very simple systems, described by simple equations, can have chaotic solutions. However, everything is not chaotic. After all, we can make accurate predictions of eclipses and many other things. An even more curious fact is that the same system can behave either predictably or chaotically, depending on small changes in a single term of the equations that describe the system. Chaotic processes are not random; they follow rules, but even simple rules can produce extreme complexity. This blend of simplicity and unpredictability also occurs in music and art. A piece of music that consists of random notes or of an endless repetition of the same sequence of notes would be either disastrously discordant or unbearably boring. Likewise, a work of art produced by throwing paint at a canvas from a distance or by endlessly replicating a pattern, as in wallpaper, is unlikely to have aesthetic appeal. Nature is full of visual objects, such as clouds and trees and mountains, as well as sounds, like the cacophony of excited birds, that have both structure and variety. The mathematics of chaos provides the tools for creating and describing such objects and sounds. Chaos theory reconciles our intuitive sense of free will with the deterministic laws of nature. However, it has an even deeper philosophical ramification. Not only do we have freedom to control our actions, but also the sensitivity to initial conditions implies that even our smallest act can drastically alter the course of history, for better or for worse. Like the butterfly flapping its wings, the results of our behaviour are amplified with each day that passes, eventually producing a completely different world than would have existed in our absence! 3.1. Function of Chaos Why chaos? Does it have a physical or biological function? The answer is yes. One role of chaos is the prevention of entrainment.

A chaotic world economic system is desirable in itself. It prevents the development of an international business cycle, whereby many national economies enter downturns simultaneously. Otherwise national business cycles may become harmonized so that many economies go into recession at the same time. "A chaotic system with a strange attractor can actually dissipate disturbance much more rapidly. Such systems are highly initial-condition sensitive, so it might seem that they cannot dissipate disturbance at all. But if the system possesses a strange attractor which makes all the trajectories acceptable from the functional point of view, the initial-condition sensitivity provides the most effective mechanism for dissipating disturbance”[2]. 3.2. Definition of Chaos To conclude the chapter, let’s give a precise mathematical definition to the term “Chaos”: Chaos is a deterministic non-linear dynamical system characterized by three hallmarks: • Sensitivity to initial conditions • Fractal structure • Mixing behaviour

Chapter 4. Technical Analysis of Financial Markets Technical analysis of financial markets was born as an attempt to put order to excessive information on price fluctuations. 4.1 Definition “Technical analysis is the study of market action, primarily through the use of charts, for the purpose of forecasting future price trends.” [54] 4.2 Principles of Technical Analysis There are three premises on which technical approach is based: 1. Market action discounts everything. This statement is a cornerstone of technical analysis. It means that anything that can possibly effect the price – fundamentally, politically, psychologically, or otherwise –is actually reflected in the price of that market. 2. Prices move in trends. The concept of trend is absolutely essential to the technical approach. There is a corollary to this premise – a trend in motion is more likely to continue than to reverse. 3. History repeats itself The key to understanding the future lies in a study of the past. 4.3 The Foundations of Technical Analysis 4.3.1 Dow Theory The most famous work done in technical analysis was begun by Charles H. Dow. Over 100 years ago he founded a newspaper called, "The Wall Street Journal." Back in 1884 he made up an average of the daily closing prices of eleven important stocks and began to record the fluctuations of this average. Charles Dow believed that everything known about the future business of a company was already priced into its stock. Therefore, he felt this tool was the best predictor there could be of future economic activity. And, by choosing the most important stocks in the United States to compose his index, he could determine where the economy of the country was headed. Mr. Dow's work led to what is commonly referred to as the Dow Theory. The Dow Theory, which started to include investigations into the trends and cycles of the stock market, created the foundation for what are now the thousands of types of technical analysis approaches practiced in the market today. More information about Charles Dow and his theory is given in Appendix 1. 4.3.2 Elliott Wave Theory

An important step in the development of Dow’s foundations was the Elliott Wave principle, discovered in the late 1920s by Ralph Nelson Elliott. In part Elliott based his work on the Dow Theory, but Elliott discovered the fractal nature of market action. Thus Elliott was able to analyse markets in greater depth, identifying the specific characteristics of “wave patterns” and making detailed market predictions based on the patterns he had identified. The patterns that Elliott discovered are built the following way: An impulsive wave, which goes with the main trend, always shows five waves in its pattern. On a smaller scale, within each of the impulsive waves of the before mentioned impulse, again five waves will be found. In this smaller pattern, the same pattern repeats itself ad infinitum. A little later, in the 1960s Benoit Mandelbrot, an applied mathematician, proved the existence of fractals in his book "The Fractal Geometry of Nature" [52]. He recognized the fractal structure in numerous objects and life forms (See Appendix 2). In the 70s the Wave Principle gained popularity through the work of Frost and Prechter. They published a legendary book on the Elliott Wave “The Wave Principle” [61], wherein they predicted, in the middle of the crisis of the 70s, the great bull market of the 1980s. Not only did they correctly forecast the bull market but Robert R. Prechter also predicted the crash of 1987 in time and pinpointed the high exactly. Apart from these patterns, R.N. Elliott was the first who encountered (with the help of his publisher Charles Collins) Fibonacci ratios in financial markets. 4.3.2.1 Fibonacci Numbers The Fibonacci sequence of numbers is a sequence in which each number is the sum of the previous two: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, … This mathematical sequence appeared 1202 A.D. in the book Liber Abaci, written by the Italian mathematician Leonardo da Pisa, who was popularly known as Fibonacci (son of Bonacci). The Fibonacci sequence represents the path of a dynamical system. The Fibonacci dynamical system looks like this: F(n+2) = F(n+1) + F(n). We shall see later in the course of this paper, that financial markets are dynamical systems. Appendix 3 explains the notion of a dynamical system. 4.3.2.1.1 The Golden Ratio What is very special and important about this sequence is what is known as golden ratio, that evolves from the Fibonacci sequence the following way: If we take the ratio of two successive numbers in Fibonacci's series, (1, 1, 2, 3, 5, 8, 13, ..) and divide each by the number before it, we will find the following series of numbers: 1 /1 = 1, 2/1 = 2, 3/2 = 1.5, 5/3 = 1.66..., 8/5 = 1.6, 3/8 = 1.625, 21 /13 = 1.61538...

It is easier to see what is happening if we plot the ratios on a graph: Fig.2 Ratio of Successive Fibonacci Terms[32] The ratio is settling down to a particular value, which we call the golden ratio or the golden number. φ can also be expressed as a linear ratio: , where the ratio of AC/CB equals to the ratio of CB/AB. If CB = 1 (as our unit of measurement), then AC / 1 = 1 / (1+AC). Multiplying the equation out we get AC2 + AC – 1 = 0. This gives two solutions: AC = [- 1 + 50.5] / 2 = 0.618033…, and AC = [- 1 - 50.5] / 2 = -1.618033… The first, positive solution (AC = 0.618033…) is called the golden mean. Using CB = 1 as our scale of measurement, then AC, the golden mean, is the solution to the ratio AC / CB = CB / (AC+CB). The golden mean, or its reciprocal equivalent, is found throughout the natural world. Numerous books have been devoted to the subject. Appendix 4 gives some examples of this phenomenon. These same ratios are found in financial markets, and we trace them in Elliott waves.

The ratios of alternate numbers approach 2.618 or its reverse, 0.382. For example, 13/34=0.382; 34/13=2.615. These values are commonly found in Elliott waves too. The next chapter will outline the basic wave theory and show how Fibonacci sequence is used there. (The details of the Elliott theory and Fibonacci retracements are given in Appendix 5 for reference.) In Chapter 5 we shall see, how the Chaos theoretical method is approaching Fibonacci ratio. 4.3.2.2 Basics of the Wave Theory According to physical law: "Every action creates an equal and opposite reaction". The same goes for the financial markets. A price movement up or down must be followed by a contrary movement, as the saying goes: "What goes up must come down"(and vice versa). Price movements can be divided into trends on the one hand and corrections or sideways movements on the other hand. Trends show the main direction of prices, while corrections move against the trend. In Elliott terminology these are called Impulsive waves and Corrective waves. The Impulse wave formation has five distinct price movements, three in the direction of the trend (1,3,and 5) and two against the trend (2 and 4). Fig. 3 Impulse Wave Formation Obviously the three waves in the direction of the trend are impulses and therefore these waves also have five waves. The waves against the trend are corrections and are composed of three waves. Fig.4 Corrective Wave Formation The corrective wave formation normally has three, in some cases five or more distinct price movements, two in the direction of the main correction ( A and C) and one against it (B). Wave 2 and 4 in the above picture are corrections. These waves have the following structure:

Fig.5 Structure of Corrective Waves Note that these waves A and C go in the direction of the shorter term trend, and therefore are impulsive and composed of five waves, which is shown in the picture above. An impulse wave formation followed by a corrective wave, form an Elliott wave degree, consisting of trend and counter trend. Although the patterns pictured above are bullish, the same applies for bear markets, where the main trend is down. The following example shows the difference between a trend (impulse wave) and a correction (sideways price movement with overlapping waves). It also shows that larger trends consist of (a lot of) smaller trends and corrections, but the result is always the same. Fig.6 Trends and Corrections Very important in understanding the Elliott Wave Principle is the basic concept that wave structures of the largest degree are composed of smaller sub waves, which are in turn composed of even smaller sub waves, and so on, which all have more or less the same structure (impulsive or corrective) like the larger wave they belong to. Elliott distinguished nine wave degrees ranging from two centuries to hourly:

Wave degree Trend Correction Grand Supercycle Supercycle Cycle Primary Intermediate Minor Minute Minuette Sub minuette Fig.7 List of Wave Degrees This indicates that you can trade the investment horizon, which is most suited for you, from very aggressive intra day trading to longer term investing. The same rules and patterns apply over and over again. [26,27]. 4.3.2.2.1 Fibonacci numbers and ratios in the waves The established wave theory says: “The wave counts of the impulsive and corrective patterns (5 + 3 = 8 total) are Fibonacci numbers, and breaking down wave patterns into their respective sub waves produces Fibonacci numbers indefinitely. Since Fibonacci ratios manifest themselves in the proportions of one wave to another, waves are often related to each other by the ratios of 2.618, 1.618, 1, 0.618 and 0.382. If, for example, a wave 1 or A of any degree (or time frame) has been completed, retracements of 0.382, 0.50 and 0.618 for wave 2 or B can be projected. Most of the time the third wave is the strongest, so often you will find that wave 3 is approximately 1.618 times wave 1. Wave 4 normally shows a retracement, which is less than wave 2, like 0.382. If wave three is the longest wave, the relationship between wave 5 and three often is 0.618. Also wave 5 equals wave 1 most of the time. The same relations can be found between A and C waves. Normally C equals A or is 1.618 times the length of A. Waves can be combined to find support and resistance zones. For example the net price movement of wave 1 and 3 times 0.618 creates another interesting target for wave 5.” [25,26]

It looks like the Elliott’s vision of the rhythm of the markets follows the Fibonacci rule. A detailed outline of correction patterns and Fibonacci proportional retracements in them is given in Appendix 5. 4.3.2.2.2 Fibonacci in Percentage Retracements The preceding ratios help to determine price objectives in both impulse and corrective waves. Another way to determine price objectives is by the use of percentage retracements. The most commonly used numbers in retracement analysis are 61.8%(usually rounded off to 62%), 38% and 50%. In a strong trend a minimum retracement is usually around 38%. In a weaker trend a maximum percentage retracement is usually 62%. See Figures 8.1 and 8.2. (Figures quoted from Murphy John J.: “Technical Analysis of the Futures Markets”[54]. Fibonacci ratios approach 0.618 only after the first four numbers. The first three ratios are 1/1 (100%), ½ (50%) and 2/3 (66%). So the famous 50% retracement is actually a Fibonacci ratio, as is the two-thirds retracement. A complete retracement (100%) of a previous bull or bear market also should mark an important support or resistance area. 4.3.2.2.3 Fibonacci Time Targets Fibonacci relationships exist even in the aspect of time in wave analysis. Fibonacci time targets are found by counting forward from significant tops and bottoms. On a daily chart, the analyst counts forward the number of trading days from an important turning point with the expectation that future tops or bottoms will occur on Fibonacci days – that is, on the 13th, 21st, 34th, 55th or 89th trading day in the future. The same technique can be used on weekly, monthly, or even yearly charts. On the weekly chart, the analyst picks a significant top or bottom and looks for weekly time targets that fall on Fibonacci numbers. (See Figures 9.1 and 9.2) [54]. The ideal situation occurs when wave form, ratio analysis and time targets come together. Suppose that the study of waves revealed that a fifth wave has been completed, that it has gone 1.618 times the distance from the bottom of wave 1 to the top of wave 3, and that the time from the beginning of the trend has been 13 weeks from a previous low and 34 weeks from a previous top. Suppose further that the fifth wave has lasted 21 days. That would indicate, that an important top was near. A study of price charts in both stocks and futures markets reveals a number of Fibonacci time relationships. Part of the problem, however, is the variety of possible relationships. Fibonacci time targets can be taken from top to top, top to bottom, bottom to bottom, and bottom to top. These relationships can always be found after the fact. It’s not always clear, which of the possible relationships are relevant to the current trend.[54] That’s the way classical technical analysis has been treating the price charts.

4.4 Some Critics of TA George Soros, a famous investor, who made a fortune on market speculations, criticised the theory of “efficient markets”, the cornerstone of technical analysis. In his book “Le döfi de l’argent”[66] he writes, that economy attempts to follow physics. Classical economists tried to follow Isaak Newton, however, they forgot that Newton lost his own capital in a speculative “South Sea bubble” that took place at his time in England. Soros compares social-economical studies with alchemy that flourished in the Middle Ages. Technical Analysis is an empirically developed system of methods, which don’t have any theoretical background to support their successful application. The application criteria are not defined. Strictly speaking, chart analysis as a method doesn’t fulfil the criteria of scientific approach. That doesn’t mean however, that its application must be unsuccessful. The application of fractals and Fibonacci numbers though marked the point when Technical Analysis of Financial Markets (TA), founded by N. Dow, gave birth to a new branch within itself, named later Chaos- Theoretical Analysis (CA). We shall see in the next Chapter, how CA came to the same ratio by estimating the time scale rate determining the distribution of price variables. The chaos theoretical approach, however, is based on a purely mathematical background, and it’ll take some more effort to understand its methods.

Chapter 5. Chaos Theoretical Approach (CA) Why is it Chaos Theory, that has found its way to market analysis? What are the characteristics of financial markets that allow its application? The non-linear statistical analysis forms the basis for a possible application of the chaos theory for modelling and prognosis of financial markets. Various analysis methods and the results available from it allow us draw essential conclusions: - Market dynamics is not linear. This was proved by the correlation calculations. This is important to know, because chaotic systems are non-linear. - Prices do not have a normal distribution. That was proved by the two-dimensional KS test[69], where real market price movements were compared to normal-distributed pseudo random price movements with the same average values and the same standard deviation. - Market dynamics is not random. This is confirmed by several tests: the Chi2 test, the conditional entropy method and the conditional probability test, developed by Savit and Green[69]. The “return maps” method also shows clear differences between real market data and random numbers. For most markets this is also proved by the fractal dimensions they show[69]. - Markets conform with the 3 properties characterizing chaotic systems: o Sensitivity to initial conditions o Fractal nature o Mixing behaviour Those are the key arguments in favour of the application of the chaos theory to the analysis of financial markets. 5.1 System approach of CA to the analysis of Financial Markets CA consists of quite a number of methods, handling non-linear dynamical systems (See Appendix 2). The methods are applied systematically: 1 First it is checked, if a system is chaotic, or random. That is done with the help of Hurst exponent. (Hurst exponent is explained in Appendix 8), or with the “return maps”(RM) method (see www.arbitrage-trading.com). 2 Its complexity is stated (the number of variables is set). 3 Then the state position of the system is presented in the form of an attractor, which gives the information about the dynamics of the system. The graphic is at least two- dimensional. 4 This helps reconstruct the model of the system. 5 The further development of the system can be predicted with the help of the attractor itself, or a model is built on its basis. 6 If there is enough data to make up a statistical forecast, it is possible to calculate the degree of probability that the forecast is correct. The CA methods have gone through multiple critical tests and were proved reliable. They have been proved not only on mathematical and physical model systems. CA has proved itself reliable in solving real world problems in physical, medical and social spheres. In future it can completely replace TA in market forecast[69]. The methods are however quite complex, and not easy to use for a non-specialist.

The major notions of chaos theory were introduced in Chapter 3. Here we shall concentrate on its methods applied to the analysis of financial markets. For that purpose a few important mathematical notions need to be introduced. 5.2. Price Distributions In stock prices, interest rates, or commodity prices – the principles are the same. Here the assumption is that the systems generating the prices are nondeterministic (stochastic, random) – but that doesn’t prevent there being a hidden form, hidden order, in the shape of probability distributions. In the early 60-s Benoit Mandelbrot stepped up with an idea of the probabilities involved in price distributions (See Appendix 2). Before that the price distributions were considered to be “normal”, or Gaussian. When you deal with prices, you first take logs, and then look at the changes between the logs of prices [65]. The changes between these log prices are what are often referred to as the price distribution. They may, for example, form a Bell-shaped curve around a mean of zero. In that case, the changes between logs would have a normal (Gaussian) distribution, with a mean of zero, and some standard deviation. (The actual prices themselves would have a lognormal distribution. But that’s not what is meant by "non-normal" in most economic contexts, because the usual reference is to changes in the logs of prices, and not to the actual prices themselves.) 5.2.1 Bachelier Method Below is an example of price treatment method, developed by Bachelier (Bachelier process and Brownian motion are described in Appendix 6). These are data of NYSE Composite Average in May 1999 (quoted from “Chaos and Fractals in Financial Markets” by J.O. Grabbe [38]). The calendar date in the first column of the table; the NYSE Composite Average, S(t), in the second column; the log of S(t) in the third column; the change in log prices, x(t) = log S(t) – log S(t-1) in the fourth column; and x(t)2 in the last column. The sum of the variables in the last column is given at the bottom of the table. Date S(t) log S(t) x(t) x(t)2 May 14 638.45 6.459043 May 17 636.92 6.456644 -.002399 .000005755 May 18 634.19 6.452348 -.004296 .000018456 May 19 639.54 6.460749 .008401 .000070577 May 20 639.42 6.460561 -.000188 .000000035 May 21 636.87 6.456565 -.003996 .000015968

May 24 626.05 6.439430 -.017135 .000293608 May 25 617.34 6.425420 -.014010 .000196280 May 26 624.84 6.437495 .012075 .000145806 May 27 614.02 6.420027 -.017468 .000305131 May 28 622.26 6.433358 .013331 .000177716 sum of all x(t)2 = .001229332 Fig.10 NYSE Composite Average in May 1999. On this basis historic volatility is measured. The variables x(t), which are the one-trading-day changes in log prices, are x(t) = log S(t) – log S(t-1). 5.2.1.1 Historical Volatility Bachelier thinks these should have a normal distribution. A normal distribution has a location parameter m and a scale parameter c (which is also called the "standard deviation" in the context of the normal distribution, and “historical volatility”). The location parameter m is close to zero. In fact, it is not quite zero. Essentially there is a drift in the movement of the stock index S(t), given by the difference between the interest rate (such as the broker-dealer loan rate) and the dividend yield on stocks in the average. But this is negligible over eleven trading days (which gives us ten values for x(t)). So Bachelier just assumes m is zero. To estimate c we add up the terms in the right-hand column in the table and get the value of .001229332. And there are 10 observations. So we have variance = c2 = .001229332/10 = .0001229332. Taking the square root of this, we have standard deviation = c = (.0001229332)0.5 = .0110875. In the financial markets, the scale parameter c is often called "volatility". The way we calculated the scale c is called "historical volatility," because we used actual historical data to estimate c. In the options markets, there is another measure of volatility, called "implied volatility." Implied volatility is found by back-solving an option value (using a valuation formula) for the volatility, c, that gives the current option price. Hence this volatility, which pertains to the future (specifically, to the future life of the option) is implied by the price at which the option is traded.

5.2.1.2 The Square Root of Time Among other things, Bachelier observed that the probability intervals into which prices fall seemed to increased or decreased with the square root of time (T0.5). This was a key insight. "Probability interval" means a given probability for a range of prices. For example, prices might fall within a certain price range with 65 percent probability over a time period of one year. But over two years, the same price range that will occur with 65 percent probability will be larger than for one year. How much larger? Bachelier said the change in the price range was proportional to the square root of time. Since the current price on May 28, from the table, is 622.26, this interval becomes: (622.26 exp(– .0110875 T0.5), 622.26 exp(.0110875 T0.5) ). This expression for the probability interval tells us the probability distribution over the next T days. Let P be the current price. After a time T, the prices will (with a given probability) fall in the range (P –a T0.5, P + a T0.5), for some constant a. For example, if T represents one year (T=1), then the last equation simplifies to (P –a, P + a), for some constant a. The price variation over two years (T=2) would be a T0.5 = a(2)0.5 = 1.4142 a or 1.4142 times the variation over one year. By contrast, the variation over a half-year (T=0.5) would be a T0.5 = a(0.5) 0.5 = .7071 a or about 71 percent of the variation over a full year. That is, after 0.5 years, the price (with a given probability) would be in the range (P –.7071a , P + .7071a ). Here the constant a will be different for different types of prices: a may be bigger for silver prices than for gold prices, for example. So, in financial markets "time" is fractal. Time does not always move with the rhythms of a pendulum. Sometimes time is less than that. In the Bachelier process the log of probability moves according to a T0.5. 5.3 Hausdorff (Fractal) Dimension Next, it’s very important to understand the what is called Hausdorff dimension, because it is always used in calculation of symmetric stable distributions [65], which are fractal in nature. We are used to 1, 2 and 3 dimensional objects. However, the Hausdorff dimension is a

variable, and it may take any value in between. To understand how it works, let’s consider the Sierpinski carpet fractal below. 5.3.1 Sierpinski Carpet Fig.11 Sierpinski Carpet Notice that it has a solid blue square in the centre, with 8 additional smaller squares around the centre one. 1 2 3 8 centre square 4 7 6 5 Each of the 8 smaller squares looks just like the original square. Multiply each side of a smaller square by 3 (increasing the area by 3 x 3 = 9), and you get the original square. Or,

doing the reverse, divide each side of the original large square by 3, and you end up with one of the 8 smaller squares. At a scale factor of 3, all the squares look the same (leaving aside the discarded centre square). You get 8 copies of the original square at a scale factor of 3. This defines a fractal dimension of log 8 / log 3 = 1.8927. Each of the smaller squares can also be divided up the same way: a centre blue square surrounded by 8 even smaller squares. So the original 8 small squares can be divided into a total of 64 even smaller squares—each of which will look like the original big square if you multiply its sides by 9. So the fractal dimension is log 64 / log 9 = 1.8927. In a fractal this process goes on forever. Meanwhile, we have just defined a fractal (or Hausdorff ) dimension. If the number of small squares is N at a scale factor of r, then these two numbers are related by the fractal dimension D: N = rD . Or, taking logs, we have D = log N / log r. The same things keep appearing when we scale by r, because the object we are dealing with has a fractal dimension of D. The different solution trajectories of chaotic equations form strange attractors. If similar patterns appear in the strange attractor at different scales, governed by some multiplier or scale factor r, they are said to be fractal. They have a fractal dimension D, governed by the relationship N = rD. Chaos equations like the one here generate fractal patterns. Benoit Mandelbrot defined a fractal as an object whose Hausdorff dimension is different from its topological dimension. 5.3.2 Fractal Time Scale Bachelier observed that if the time interval was multiplied by 4, the probability interval only increased by 2. In other words, at a scale of r = 4, the number N of similar probability units was N = 2. So the Hausdorff dimension for time was: D = log N/ log r = log 2/ log 4 = 0.5. In going from Bachelier to Mandelbrot, then, the innovation is not in the observation that time is fractal: that was Bachelier’s contribution. Instead the question is: What is the correct fractal dimension for time in speculative markets? Is the Hausdorff dimension really D = 0.5, or does it take other values? To answer this question we will first need to have a look at another important fractal named Koch curve:

5.3.3 Koch Curve We take a line segment. For future reference, let’s say its length L is L = 1. Now we divide it into three parts (each of length 1/3), and remove the middle third. But we replace the middle third with two line segments (each of length 1/3), which can be thought of as the other two sides of an equilateral triangle. This is stage two (b) of the construction in the graphic below: Fig.12 Koch Curve At this point we have 4 smaller segments, each of length 1/3, so the total length is 4(1/3) = 4/3. Next we repeat this process for each of the 4 smaller line segments. This is stage three (c) in the graphic above. This gives us 16 even smaller line segments, each of length 1/9. So the total length is now 16/9 or (4/3)2. At the n-th stage the length is (4/3)n, so as n goes to infinity, so does the length L of the curve. The final result "at infinity" is called a Koch curve. At each of its points it has a sharp angle. However, the Koch curve is continuous, because we can imagine taking a pencil and tracing its (infinite) length from one end to the other. So, from the topological point of view, the Koch curve has a dimension of one, just like the original line. Or, as a topologist would put it, we can deform (stretch) the original line segment into a Koch curve without tearing or breaking the original line at any point, so the result is still a "line", and has a topological dimension T = 1. To calculate a Hausdorff dimension, we note that at each stage of the construction, we replace each line segment with N = 4 segments, after dividing the original line segment by a scale factor r = 3. So its Hausdorff dimension D = log 4/log 3 = 1.2618… A very interesting observation is, that financial variables are symmetric stable distributions with standard deviation (volatility, Hausdorff dimension) between the values of Hausdorff dimension of Sierpinski carpet (α = log8/log3= 1.8927….) and Koch curve (α = log4/log3 = 1.2618….). Moreover, Orlin Grabbe in his “Chaos and Fractals in Financial Markets” writes, that the empirical evidence shows, that the Hausdorff dimension of some symmetric stable distributions encountered in financial markets is approximately α = 1.618033…. – that is the Fibonacci golden mean[38].

5.4 Leptokurtic Distributions Another important aspect to consider is the notion of Cauchy distribution, as an example of leptokurtic distributions, because it’s just the leptokurtic distributions, and not Gaussian (normal) price distribution, that function in financial markets. 5.4.1 Probability Densities (partially quoted from [38]) The normal or Gaussian distribution distributes the probability across the real line (from minus infinity to plus infinity) using the density function: f(x) = [1/(2π )0.5] exp(-x2/2) , - ∞ < x < ∞ Here f(x) creates a bell-shaped curve (x is on the horizontal line, and f(x) is the blue curve above it): Fig.13 Gaussian Probability Distribution [38] The probability is distributed between the horizontal line and the curve. The curve f(x) is called the probability density. So we can calculate the probability density for each value of x using the function f(x). Here are some values: x f(x) -3 .0044 -2 .0540 -1 .2420 -.75 .3011 -.50 .3521 -.25 .3867 0 .3989 .25 .3867 .50 .3521 .75 .3011 1 .2420 2 .0540 3 .0044

At the centre value of x = 0, the probability density is highest, and has a value of f(x) = .3989. Around 0, the probability density is spread out symmetrically in each direction. The probability that x lies between a and b, a < x < b, is just the area under the curve measured from a to b, as indicated by the red portion in the graphic below, where a = -1 and b = +1: Fig. 13.1 Elaboration of Probability Distributions Here is a picture of F(b) when b = 0: Fig. 13.2 Elaboration of Probability Distributions For any value x, F(x) is the cumulative probability function. It represents the total probability up to (and including) point x. It represents the probability of all values smaller than (or equal to) x. Now. Here is a different function for spreading probability, called the Cauchy density: g(x) = 1/[π (1 + x2)], - ∞ < x < ∞ Here is a picture of the resulting Cauchy curve: Fig.14 Cauchy curve It is symmetric like the normal distribution, but is relatively more concentrated around the centre, and taller in the tails than the normal distribution. We can see this more clearly by looking at the values for g(x):

x g(x) -3 .0318 -2 .0637 -1 .1592 -.75 .2037 -.50 .2546 -.25 .2996 0 .3183 .25 .2996 .50 .2546 .75 .2037 1 .1592 2 .0637 3 .0318 At every value of x, the Cauchy density is lower than the normal density, until we get out into the extreme tails, such as 2 or 3 (+ or -). Note that at –3, for example, the probability density of the Cauchy distribution is g(-3) = .0318, while for the normal distribution, the value is f(-3) = .0044. There is more than 7 times as much probability for this extreme value with the Cauchy distribution than there is with the normal distribution! (The calculation is .0318/.0044 = 7.2.) Relative to the normal, the Cauchy distribution is fat-tailed[38]. There are other distributions that have more probability in the tails than the normal, and also more probability at the peak (in this case, around 0). But since the total probability must add up to 1, there is less probability in the intermediate ranges. Such distributions are called leptokurtic. Leptokurtic distributions have more probability both in the tails and in the centre than does the normal distribution, and are to be found in all asset markets—in foreign exchange, shares of stock, interest rates, and commodity prices [6]. 5.4.1.1 Location and Scale So far, as we have looked at the normal and the Cauchy densities, we have seen they are centred around zero. However, since the density is defined for all values of x, - ∞ < x < ∞ , the centre can be elsewhere. To move the centre from zero to a location m, we write the normal probability density as: f(x) = [1/(2π )0.5] exp(-(x-m)2/2) , - ∞ < x < ∞. Here is a picture of the normal distribution after the location has been moved from m = 0 (the blue curve) to m = 2 (the red curve):

Fig.15 Normal Distribution with a moved location Here is a picture of the normal distribution for different values of scale c: Fig.16 Normal Distribution for Different Standard Deviation Values The blue curve represents c = 1, while the peaked red curves has c < 1, and the flattened red curve has c > 1. For the Cauchy density, the corresponding alteration to include a location parameter m and scale parameter c is: π g(x) = 1/[cπ (1 + ((x-m)/c)2)], - ∞ < x < ∞ Operations with location and scale are well defined, whether or not the mean or the variance exist. Most of the probability distributions in finance lie somewhere between the normal and the Cauchy. These two distributions form the "boundaries", so to speak, of our main area of interest. Just as the Sierpinski carpet has a Hausdorff dimension that is a fraction which is greater than its topological dimension of 1, but less than its Euclidean dimension of 2, so do the probability distributions in which we are chiefly interested have a dimension that is greater than the Cauchy dimension of 1, but less than the normal dimension of 2 [38]. 5.5. Symmetric Stable Distributions and the Golden Mean Law Symmetric stable distributions are a type of probability distribution that are fractal in nature: a sum of n independent copies of a symmetric stable distribution is related to each copy by a scale factor n1/ α , where α is the Hausdorff dimension of the given symmetric stable distribution [65]. In the case of the normal or Gaussian distribution, the Hausdorff dimension α = 2, which is equivalent to the dimension of a plane. A Bachelier process is governed by a T1/α = T1/2 law.

In the case of the Cauchy distribution the Hausdorff dimension α = 1, which is equivalent to the dimension of a line. A Cauchy process would be governed by a T1/α = T1/1 = T law. In general, 0 < α <=2. This means that between the Cauchy and the Normal are all sorts of interesting distributions, including ones having the same Hausdorff dimension as a Sierpinski carpet (α = log 8/ log 3 = 1.8927….) or Koch curve (α = log 4/ log 3 = 1.2618….). Interestingly, however, many financial variables are symmetric stable distributions with an α parameter that hovers around the value of the golden mean, 1.618033. Orlin Grabbe investigated that in application to daily changes in the dollar/deutschemark exchange rate for the first six years following the breakdown of the Bretton Woods Agreement of fixed exchange rates in 1973. [11] (The time period was July 1973 to June 1979.) In his “Chaos And Fractals in Financial Markets” he writes: “The value of α was calculated using maximum likelihood techniques. The value I found was α = 1.62 with a margin of error of plus or minus .04. You can’t get much closer than that to α = h = 1.618033… This implies that these market variables follow a time scale law of T1/α = T1/h = Tg = T0.618033... That is, these variables following a T-to-the-golden-mean power law, by contrast to the Bachelier’s assumption, whereby it follows a T-to-the-one-half power law” [38]. Grabbe writes further: “In this and other financial asset markets, it would seem that time scales not according to the commonly assumed square-root-of-T law, but rather to a Tg law.” The research that has been done in this area suggests however, that what Grabbe calls α can take values from 1.2 to 2. For more detailed information read “Chaos and Order in the Capital Markets” by J. Edgar Peters [57]. So we can suggest, that the golden mean value in this context is the tendency. 5.6. Measuring Sensitivity An important aspect of the chaos theoretical approach is the estimation of reliability of forecast. This is achieved with the help of the Lyapunov exponent, which is used to measure the extent to which errors are amplified. (Appendix 7 explains the Lyapunov exponent concept).

5.7. To Summarise The Chapter shows, that CA is a reliable tool for market analysis. It helps understand, how far it is possible to predict market behaviour in every specific case, and estimate the possibility of prediction error. CA has a variety of useful methods, mentioned or described in the course of the chapter, like return maps analysis, - which shows if a certain behaviour has a random or a chaotic character, and hence if it’s predictable or not. As soon as typical chaotic features are detected, Fractal Dimension technique is applied. By iterating price distribution equations it reconstructs the model of price movement in the form of a strange attractor, which shows exactly the character of the system. Paragraph 5.1 gives an overview of the chaos theoretical methods applied for the analysis of financial markets. Paragraphs 5.2 and 5.4 explain, what are price distributions. On the basis of the data used in price distribution equations strange attractors can be drawn with the help of the Fractal Dimension technique, which is explained in Paragraph 5.3. Paragraph 5.6 shows the possibility of measuring prediction error. For prediction purposes the best technique is the application of Lyapunov exponent to the attractor (Appendix 7 gives a more detailed explanation of the technique), to show, how far the prediction is possible, and at which areas of the system it is most reliable. An interesting observation has been given special attention in the Chapter (Paragraphs 5.3- 5.5): the variables of equations, showing price movement, i.e. timescale and fractal dimension – both seem to have a tendency to the Golden Mean.

Summary and Conclusions The essay has given an overview of the methods applied to analyse modern financial markets as a self-regulating system. Chart analysis techniques as empirically based methods, as well as Chaos theoretical approach as scientifically grounded methods have been studied and compared. We have seen the power of the „golden mean“, discovered mathematically by Fibonacci. It is valid for major natural processes and systems. It is shown that the golden mean is present in the organism of the markets too. Taken the development of markets is fractal, the logical assumption would be that what is called globalisation is a fractal process too. The ideal goal of globalisation is to achieve a smooth economical growth on the planet. If we assume that the economical growth tends to follow the time-to-the-golden-mean-power law, and the growth itself tends to the golden mean ratio, it would be interesting to see, where we stand in the process of economical development, and how many more steps we need to follow, till we reach the stable golden mean growth ratio, or is it extrapolated to infinity. The speed of our life grows exponentially with time, from century to century, from decade to decade, from year to year. Who knows, the acceleration might also follow the Fibonacci numbers plotted on the time axis at equal distances, or the golden mean ratio. Likewise the markets, as an indicator of human social development, however chaotic, seem to follow the law of nature.

List of Figures Figure 1. Lorenz Attractor p. 8 Figure 2. Ratio of Successive Fibonacci Terms p.13 Figure 3. Impulse Wave Formation p.14 Figure 4. Corrective Wave Formation p.14 Figure 5. Structure of Corrective Waves p.15 Figure 6. Trends and Corrections p.15 Figure 7. List of Wave Degrees p.16 Figures 8.1 and 8.2 Examples of Fibonacci Percentage Retracements p.18 Figures 9.1 and 9.2 Examples of Fibonacci Time Targets p.19 Figure 10. NYSE Composite Average in May 1999 p.23 Figure 11. Sierpinski Carpet p.25 Figure 12. Koch Curve p.27 Figure 13. Gaussian Probability Distribution p.28 Figures 13.1 and 13.2 Elaboration of Probability Distributions p.29 Figure 14. Cauchy Curve p.29 Figure 15. Normal Distribution with a moved location p.30 Figure 16. Normal Distribution for Different Standard Deviation Values p.30 Figure 17a Quarterly Changes in German Stock Market (3-dimentional) p.32 Figure 17b Prediction Error of Quarterly Changes in German Stock Market (3-dimentional) p.32 Figure 18a Quarterly Changes in German Stock Market (contour) p.33 Figure 18b Prediction Error of Quarterly Changes in German Stock Market (contour) p.33

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Appendix 1. Charles Dow and his Theory* CHARLES DOW - FATHER OF THE DOW THEORY Charles Dow was born in Sterling, Connecticut November 6, 1851. He died in 1902. His father was a farmer. Dow's father died when he was six years old. Dow was a newspaperman his entire adult life. In 1872 he went to work for 'The Springfield Daily Republican' in Springfield, Massachusetts. Dow covered the city beat for the Republican. Dow remained at The Springfield Republican until 1875. He then left for Providence, Rhode Island and 'The Providence Star'. There he was the night editor. The Providence Star went out of business in 1877; and Dow moved to The Providence Journal. Dow was a reporter for the Journal from 1877 to 1879. Through five articles published by The Journal and Dow, Dow became known as a sort of local historian. He traced the histories of the steamship and local life in and around Providence. In 1879, Dow visited Leadville, Colorado. He did this at the request of the Journal when carbonates were discovered in Leadville. At the time, Leadville was the most famous mining town in the country. The Journal felt Dow was the best equipped to write about them. It was in Leadville that Dow met prominent men in the financial community. It was also in Leadville that Dow discovered that in financial journalism he could attain importance and usefulness far beyond that which he could expect in the course of ordinary journalism. It was in Leadville that Dow began to focus on the financial aspects of journalism. He reported extensively on the financial aspect of the mining boom. In 1880, Dow moved to New York City. He went to Wall Street and found a job reporting on mining stocks. He soon became known as a reliable reporter who was capable of expert financial analysis. He could also be trusted with confidential information. At some point around 1880, Dow went to work for the Kiernan News Agency. Edward D. Jones was also a reporter at Kiernan. Coincidentally, Jones was also a fellow worker of Dow's in Providence. In November of 1882, Dow and Jones left Kiernan and formed Dow Jones & Company. Jones stayed with the company until 9 January 1899. He died in Providence in 1920. The business of Dow Jones & Company was delivering 'flimsies' or 'slips' to financial institutions. Both Dow and Jones collected the news and messenger boys delivered the news. In the evening, Dow and Jones prepared the news for the next day. Initially, the first office of Dow Jones & Company was located at 15 Wall Street (right next door to the entrance of the New York Stock Exchange). It was the back room to a soda fountain. By 1884, still located behind the soda fountain, the company had grown in size. Dow and Jones no longer went out and

collected the news themselves. There were now newsgatherers, manifold writers, and 'lively boys' who delivered the news. Every employee of the company solicited subscriptions to the news service and were paid a commission on each sale. And every employee would report news they came across whether this was their job or not. In 1883, Dow Jones & Company began printing a news sheet containing the news of the day. This was the precursor to 'The Wall Street Journal'. From December 1885 through April 1891, Dow was a member of the New York Stock Exchange. He was listed as a partner in Goodbody, Glynn, & Dow from December 24, 1885 through April 30, 1891. This may have been a partnership of convenience. Because members had to be American citizens, Goodbody, an Irish immigrant, could not qualify as a member. Dow executed orders on the floor of the exchange for the firm. In 1891, when Goodbody became a naturalized American citizen, Dow withdrew from the exchange. The Wall Street Journal was first published on 8 July 1889. Dow was the editor. Jones took care of the deskwork of the firm. Interestingly, it was not until 1892 that Dow Jones & Company got their first telephone. It was as editor of The Wall Street Journal that Dow was able to publish his observations on finance and investment. [33] DOW THEORY Overview In 1897, Charles Dow developed two broad market averages. The "Industrial Average" included 12 blue-chip stocks and the "Rail Average" was comprised of 20 railroad enterprises. These are now known as the Dow Jones Industrial Average and the Dow Jones Transportation Average. The Dow Theory resulted from a series of articles published by Charles Dow in The Wall Street Journal between 1900 and 1902. The Dow Theory is the common ancestor to most principles of modern technical analysis. Interestingly, the Theory itself originally focused on using general stock market trends as a barometer for general business conditions. It was not originally intended to forecast stock prices. However, subsequent work has focused almost exclusively on this use of the Theory. Interpretation The Dow Theory comprises six assumptions: 1. The Averages Discount Everything. An individual stock's price reflects everything that is known about the security. As new information arrives, market participants quickly disseminate the information and the price adjusts accordingly. Likewise, the market averages discount and reflect everything known by all stock market participants. 2. The Market Is Comprised of Three Trends. At any given time in the stock market, three forces are in effect: the Primary trend, Secondary trends, and Minor trends.

The Primary trend can either be a bullish (rising) market or a bearish (falling) market. The Primary trend usually lasts more than one year and may last for several years. If the market is making successive higher-highs and higher-lows the primary trend is up. If the market is making successive lower-highs and lower-lows, the primary trend is down. Secondary trends are intermediate, corrective reactions to the Primary trend. These reactions typically last from one to three months and retrace from one-third to two-thirds of the previous Secondary trend. The following chart shows a Primary trend (Line "A") and two Secondary trends ("B" and "C"). Minor trends are short-term movements lasting from one day to three weeks. Secondary trends are typically comprised of a number of Minor trends. The Dow Theory holds that, since stock prices over the short-term are subject to some degree of manipulation (Primary and Secondary trends are not), Minor trends are unimportant and can be misleading. 3. Primary Trends Have Three Phases. The Dow Theory says that the First phase is made up of aggressive buying by informed investors in anticipation of economic recovery and long-term growth. The general feeling among most investors during this phase is one of "gloom and doom" and "disgust." The informed investors, realizing that a turnaround is inevitable, aggressively buy from these distressed sellers. The Second phase is characterized by increasing corporate earnings and improved economic conditions. Investors will begin to accumulate stock as conditions improve. The Third phase is characterized by record corporate earnings and peak economic conditions. The general public (having had enough time to forget about their last "scathing") now feels comfortable participating in the stock market--fully convinced that the stock market is headed for the moon. They now buy even more stock, creating a buying frenzy. It is during this phase that those few investors who did the aggressive buying during the First phase begin to liquidate their holdings in anticipation of a downturn. The following chart of the Dow Industrials illustrates these three phases during the years leading up to the October 1987 crash.

In anticipation of a recovery from the recession, informed investors began to accumulate stock during the First phase (box "A"). A steady stream of improved earnings reports came in during the Second phase (box "B"), causing more investors to buy stock. Euphoria set in during the Third phase (box "C"), as the general public began to aggressively buy stock. 4. The Averages Must Confirm Each Other. The Industrials and Transports must confirm each other in order for a valid change of trend to occur. Both averages must extend beyond their previous secondary peak (or trough) in order for a change of trend to be confirmed. The following chart shows the Dow Industrials and the Dow Transports at the beginning of the bull market in 1982. Confirmation of the change in trend occurred when both averages rose above their previous secondary peak. 5. The Volume Confirms the Trend. The Dow Theory focuses primarily on price action. Volume is only used to confirm uncertain situations.

Volume should expand in the direction of the primary trend. If the primary trend is down, volume should increase during market declines. If the primary trend is up, volume should increase during market advances. The following chart shows expanding volume during an up trend, confirming the primary trend. 6. A Trend Remains Intact Until It Gives a Definite Reversal Signal. An up-trend is defined by a series of higher-highs and higher-lows. In order for an up-trend to reverse, prices must have at least one lower high and one lower low (the reverse is true of a downtrend). When a reversal in the primary trend is signaled by both the Industrials and Transports, the odds of the new trend continuing are at their greatest. However, the longer a trend continues, the odds of the trend remaining intact become progressively smaller. The following chart shows how the Dow Industrials registered a higher high (point "A") and a higher low (point "B") which identified a reversal of the down trend (line "C"). * The text and charts of this Appendix were taken from [22], [33] and [30].

Appendix 2. Benoit Mandelbrot and his Contribution in the Analysis of the Fractal Geometry of Nature* Benoit Mandelbrot was largely responsible for the present interest in fractal geometry. He showed how fractals can occur in many different places in both mathematics and elsewhere in nature. Mandelbrot was born in Poland in 1924 into a family with a very academic tradition. His father, however, made his living buying and selling clothes while his mother was a doctor. As a young boy, Mandelbrot was introduced to mathematics by his two uncles. Mandelbrot's family emigrated to France in 1936 and his uncle Szolem Mandelbrojt, who was Professor of Mathematics at the Collège de France, took responsibility for his education. In fact the influence of Szolem Mandelbrojt was both positive and negative since he was a great admirer of Hardy and Hardy's philosophy of mathematics. This brought a reaction from Mandelbrot against pure mathematics, although as Mandelbrot himself says, he now understands how Hardy's deep felt pacifism made him fear that applied mathematics, in the wrong hands, might be used for evil in time of war. Mandelbrot attended the Lycée Rolin in Paris up to the start of World War II, when his family moved to Tulle in central France. This was a time of extraordinary difficulty for Mandelbrot who feared for his life on many occasions. In [1] the effect of these years on his education was emphasised:- The war, the constant threat of poverty and the need to survive kept him away from school and college and despite what he recognises as "marvellous" secondary school teachers he was largely self taught. Mandelbrot now attributed much of his success to this unconventional education. It allowed him to think in ways that might be hard for someone who, through a conventional education, is strongly encouraged to think in standard ways. It also allowed him to develop a highly geometrical approach to mathematics, and his remarkable geometric intuition and vision began to give him unique insights into mathematical problems. After a very successful performance in the entrance examinations of the Ecole Polytechnique, Mandelbrot began his studies there in 1944.

After completing his studies at the Ecole Polytechnique, Mandelbrot went to the United States where he visited the California Institute of Technology. After a Ph.D. granted by the University of Paris, he went to the Institute for Advanced Study in Princeton where he was sponsored by John von Neumann. Mandelbrot returned to France in 1955 and worked at the Centre National de la Recherche Scientific. He married Aliette Kagan during this period back in France and Geneva, but he did not stay there too long before returning to the United States. Clark gave the reasons for his unhappiness with the style of mathematics in France at this time [2]: Still deeply concerned with the more exotic forms of statistical mechanics and mathematical linguistics and full of non standard creative ideas he found the huge dominance of the French foundational school of Bourbaki not to his scientific tastes and in 1958 he left for the United States permanently and began his long standing and most fruitful collaboration with IBM as an IBM Fellow at their world renowned laboratories in Yorktown Heights in New York State. IBM presented Mandelbrot with an environment which allowed him to explore a wide variety of different ideas. He has spoken of how this freedom at IBM to choose the directions that he wanted to take in his research presented him with an opportunity which no university post could have given him. After retiring from IBM, he found similar opportunities at Yale University, where he is presently Sterling Professor of Mathematical Sciences. In 1945 Mandelbrot's uncle had introduced him to Julia's important 1918 paper claiming that it was a masterpiece and a potential source of interesting problems, but Mandelbrot did not like it. Indeed he reacted rather badly against suggestions posed by his uncle sice he felt that his whole attitude to mathematics was so different from that of his uncle. Instead Mandelbrot chose his own very different course which, however, brought him back to Julia's paper in the 1970s after a path through many different sciences which some characterise as highly individualistic or nomadic. In fact the decision by Mandelbrot to make contributions to many different branches of science was a very deliberate one taken at a young age. It is remarkable how he was able to fulfil this ambition with such remarkable success in so many areas. With the aid of computer graphics, Mandelbrot who then worked at IBM's Watson Research Center, was able to show how Julia's work is a source of some of the most beautiful fractals known today. To do this he had to develop not only new mathematical ideas, but also he had to develop some of the first computer programs to print graphics. The Mandelbrot set is a connected set of points in the complex plane. Pick a point Z0 in the complex plane. Calculate: Z1 = Z02 + Z0 Z2 = Z12 + Z0 Z3 = Z22 + Z0 ... If the sequence Z0, Z1, Z2, Z3, ... remains within a distance of 2 of the origin forever, then the point Z0 is said to be in the Mandelbrot set. If the sequence diverges from the origin, then the point is not in the set. The Mandelbrot fractal looks like this:

His work was first put elaborated in his book Les objets fractals, forn, hasard et dimension (1975) and more fully in The fractal geometry of nature in 1982. On 23 June 1999 Mandelbrot received the Honorary Degree of Doctor of Science from the University of St Andrews. At the ceremony Peter Clark gave an address [2] in which he put Mandelbrot's achievements into perspective. We quote from that address:- ... at the close of a century where the notion of human progress intellectual, political and moral is seen perhaps to be at best ambiguous and equivocal there is one area of human activity at least where the idea of, and achievement of, real progress is unambiguous and pellucidly clear. That is mathematics. In 1900 in a famous address to the International Congress of mathematicians in Paris David Hilbert listed some 25 open problems of outstanding significance. Many of those problems have been definitively solved, or shown to be insoluble, culminating as we all know most recently in the mid-nineties with the discovery of the proof of Fermat's Last Theorem. The first of Hilbert's problems concerned a thicket of issues about the nature of the continuum or the real line, a major concern of 19th and indeed of 20th century analysis. The problem was both one of geometry concerning the nature of the line thought of as built up of points and of arithmetic thought of as the theory of the real numbers. The integration of those two fields was one of the great achievements of Richard Dedekind and George Cantor, the latter of whom we [St Andrews University] were intelligent enough to honour in 1911. Now lurking about so to speak in the undergrowth of that achievement lay certain very extraordinary geometric objects indeed. To all at the time, they seemed strange, indeed rather pathological monsters. Odd indeed they were, there were curves - one dimensional lines in effect - which filled two dimensional spaces, there were curves which were well behaved, that is nice and continuous but which had no slope at any point (Not just some points, ANY points) and they went by strange names, the Peano Space filling curve, the Sierpinski gasket, the Koch curve, the Cantor Ternary set. Despite their pathological qualities, their extraordinary complexity, especially when viewed in greater and greater detail, they were often very simple to describe in the sense that the rules which generated them were absurdly simple to state. So odd were these objects that mathematicians set about barring these monsters and they were set aside as too strange to be of interest. That is until our honorary graduand created out of them an entirely new science, the theory of fractal geometry: it was his insight and vision which saw in those objects and the many new ones he discovered, some of which now bear his name, not mathematical curiosities, but signposts to a new mathematical universe, a new geometry with as much system and generality as that of Euclid and a new physical science.

As well as IBM Fellow at the Watson Research Center Mandelbrot was Professor of the Practice of Mathematics at Harvard University. He also held appointments as Professor of Engineering at Yale, of Professor of Mathematics at the Ecole Polytechnique, of Professor of Economics at Harvard, and of Professor of Physiology at the Einstein College of Medicine. Mandelbrot's excursions into so many different branches of science was, as we mention above, no accident but a very deliberate decision on his part. It was, however, the fact that fractals were so widely found which in many cases provided the route into other areas [3]:- I should not ... give the impression that we have here before us a mathematician alone. Let me explain why. The first of his great insights was the discovery that the extraordinarily complex almost pathological structures, which had been long ignored, exhibited certain universal characteristics requiring a new theory of dimension to treat them adequately which he had generalised from earlier work of Hausdorff and Besicovitch but the second great insight was that the fractal property so discovered, the general theory of which he had provided, was present almost universally in Nature. What he saw was that the overwhelming smoothness paradigm with which mathematical physics had attempted to describe Nature was radically flawed and incomplete. Fractals and pre-fractals once noticed were everywhere. They occur in physics in the description of the extraordinarily complex behaviour of some simple physical systems like the forced pendulum and in the hugely complex behaviour of turbulence and phase transition. They occur as the foundations of what is now known as chaotic systems. They occur in economics with the behaviour of prices and as Poincaré had suspected but never proved in the behaviour of the Bourse or our own Stock exchange in London. They occur in physiology in the growth of mammalian cells. Believe it or not ... they occur in gardens. Note closely and you will see a difference between the flower heads of broccoli and cauliflower, a difference which can be exactly characterised in fractal theory. Mandelbrot has received numerous honours and prizes in recognition of his remarkable achievements. For example, in 1985 Mandelbrot was awarded the 'Barnard Medal for Meritorious Service to Science'. The following year he received the Franklin Medal. In 1987 he was honoured with the Alexander von Humboldt Prize, receiving the Steinmetz Medal in 1988 and many more awards including the Nevada Medal in 1991 and the Wolf prize for physics in 1993. *The materials for this Appendix were taken from the following sources, which are not listed in the main list of references: Article of J J O'Connor and E F Robertson in http://www-groups.dcs.st- and.ac.uk/~history/Mathematicians/Mandelbrot.html, as well as 1. D J Albers and G L Alexanderson (eds.), Mathematical People: Profiles and Interviews (Boston, 1985), 205-226. 2. P Clark, Presentation of Professor Benoit Mandelbrot for the Honorary Degree of Doctor of Science (St Andrews, 23 June 1999). 3. B Mandelbrot, Comment j'ai decouvert les fractales, La Recherche (1986), 420-424.

Appendix 3 Definition of a Dynamical System A simple and vivid explanation of a dynamical system is given by J.O. Grabbe in his work “Chaos and Fractals in Financial Markets”[38]: “Johnny grows 2 inches a year. This system explains how Johnny’s height changes over time. Let x(n) be Johnny’s height this year. Let his height next year be written as x(n+1). Then we can write the dynamical system in the form of an equation as: x(n+1) = x(n) + 2. If we plug Johnny’s current height of x(n) = 38 inches in the right side of the equation, we get Johnny’s height next year, x(n+1) = 40 inches: x(n+1) = x(n) + 2 = 38 + 2 = 40. Going from the right side of the equation to the left is called iteration. We can iterate the equation again by plugging Johnny’s new height of 40 inches into the right side of the equation (that is, let x(n)=40), and we get x(n+1) = 42. If we iterate the equation 3 times, we get Johnny’s height in 3 years, namely 44 inches, starting from a height of 38 inches). This is a deterministic dynamical system. If we wanted to make it nondeterministic (stochastic), we could let the model be: Johnny grows 2 inches a year, more or less, and write the equation as: x(n+1) = x(n) + 2 + e where e is a small error term (small relative to 2), and represents a drawing from some probability distribution. The original equation, x(n+1) = x(n) + 2, is linear. Linear means you either add variables or constants or multiply variables by constants. The equation z(n+1) = z(n) + 5 y(n) –2 x(n) is linear, for example. But if you multiply variables together, or raise them to a power other than one, the equation (system) is non-linear.”

Appendix 4. Fibonacci and the Power of the Golden Mean * Fibonacci (c.1175 - c.1240) One of the best modern sources of information about Fibonacci is the following article: A.F. Horadam, "Eight hundred years young," (1975) 123-134. Department of Mathematics, University of New England, which is partially quoted in this Appendix. "...considering both the originality and power of his methods, and the importance of his results, we are abundantly justified in ranking Leonardo of Pisa as the greatest genius in the field of number theory who appeared between the time of Diophantus [4th century A.D.] and that of Fermat" [17th century] [13]. 1. The world of Fibonacci. During the twelfth and thirteenth centuries, many far-reaching changes in the social, political and intellectual lives of people and nations were taking place. Europe had emerged from the period of barbarian invasions and disruption known as the Dark Ages. Improved techniques in farming had led to greater food production, population growth and commercial expansion which were to pave the way for the industrial, scientific and technological progress of later centuries. Contacts with Eastern civilizations were made by the Crusaders, by curious travellers and by merchants eager for trading opportunities. By the end of the twelfth century, the struggle between the Papacy and the Holy Roman Empire had left many Italian cities independent republics. Having consolidated their military victories, many of these cities embarked on substantial trading enterprises, and some established centres for higher learning. In particular, ships from Genoa and Venice, laden with cargo for and from distant lands, helped to extend the maritime dominions of these cities which became the capitals of small empires. Among these important and remarkable republics was the small but powerful walled city-state of Pisa which played a major role in the commercial revolution which was transforming Europe. A description of the many facets of the bustling life of medieval Pisa (which is thought to have had a population of about 10,000) may be found in Gies and Gies [4]. "Its citizens are brave", observed Benjamin of Tudela, a Spanish Jew, "and they have neither king nor prince to whom they owe obedience".

Into this world of change and cross-fertilisation of Christian and Moslem cultures, Fibonacci, a man for all seasons, was born in Pisa. Unknown to most people is a statue in the Giardino Scotto erected by the citizens of Pisa to its most famous citizen, Fibonacci, the most outstanding Western mathematician of the Middle Ages and a man very much in advance of his time. In this progressive and energetic city, Fibonacci was born in about 1175, i.e., about 800 years ago, though no one knows for certain the exact date of his birth. 2. Fibonacci’s Life Fibonacci's father is mentioned by name by a contemporary writer as Gulielmus (William). Not much is known about the father except that he was a state official associated with the new mercantile class which had emerged from the commercial revolution. [4]. All that we do know about Fibonacci is contained in a few sentences about himself in the 1228 edition of his famous Liber Abbaci (sometimes spelt Liber Abaci). The translation of these passages, along with the original Latin, is given by Grimm [5] as follows: “After my father's appointment by his homeland as state official in the customs house of Bugia for the Pisan merchants who thronged to it, he took charge; and in view of its future usefulness and convenience, had me in my boyhood come to him and there wanted me to devote myself to and be instructed in the study of calculation for some days. There, following my introduction, as a consequence of marvellous instruction in the art, to the nine digits of the Hindus, the knowledge of the art very much appealed to me before all others, and for it I realized that all its aspects were studied in Egypt, Syria, Greece, Sicily, and Provence, with their varying methods; and at these places thereafter, while on business. I pursued my study in depth and learned the give-and-take of disputation. But all this even, and the algorism, as well as the art of Pythagoras I considered as almost a mistake in respect to the method of the Hindus. Therefore, embracing more stringently that method of the Hindus, and taking stricter pains in its study, while adding certain things from my own understanding and inserting also certain things from the niceties of Euclid's geometric art. I have striven to compose this book in its entirety as understandably as I could, dividing it into fifteen chapters. Almost everything which I have introduced I have displayed with exact proof, in order that those further seeking this knowledge, with its pre-eminent method, might be instructed, and further, in order that the Latin people might not be discovered to be without it, as they have been up to now. If I have perchance omitted anything more or less proper or necessary, I beg indulgence, since there is no one who is blameless and utterly provident in all things.” From this tantalizingly brief glimpse of Fibonacci's life, we gain some insight into his personality and the mathematical quality of his mind. Quite apart from the intellectual curiosity and excitement which these passages reveal, they leave us with a respect and warmth of feeling for the modest humility of the man. "Leonardo's humility graces his genius", says Grimm [5]. No one knows when Fibonacci died nor under what circumstances his death occurred. Within a few decades of his death, Pisa was disastrously defeated by Genoa in the grim naval battle of Meloria and the decline of his native city had irreversibly begun. The proud inscription on Pisa's "golden" sea gate, "Love justice, ye rulers of the earth!" became a nostalgic memory of past grandeur.

3. The Importance of Fibonacci From Fibonacci's commercial problems we can learn much about the society in which he lived. For instance, the sociologist and economist can discover that pepper was a very important item of merchandise transported by Pisan ships, and that the Pisan colony in Constantinople traded extensively with Egypt. Further evidence is also gleaned about the relative values of money coined in the mints of different cities, and about the problem of alloying of coins to be minted. After Fibonacci's death, his influence languished for many centuries and indeed Mathematics made no real progress for 300 years. What is the legacy we have inherited from Fibonacci? Hindu-Arabic numerals were obviously relevant to the expanding commerce-oriented society of his day. Of greater importance was the long-range impact on Science and Mathematics of the new system of numeration which he publicised. The quality of his mind is evident in the techniques he used to demolish the more challenging problems he encountered. Yet few latter-day mathematicians recognised his brilliance, possibly because of the forbidding barrier placed in their way by his untranslated Latin, coupled with his relative obscurity. Apart from the statue to his memory in the Giardino Scotto, Fibonacci's name is perpetuated in two streets, the quayside Lungarno Fibonacci in Pisa and the Via Fibonacci in Florence. Few people know of these minor memorials. Strangely, Fibonacci is best remembered by the sequence which bears his name but which, ironically, he treated only lightly. Modern mathematicians, in naming an Association, a Journal, and a Bibliographical and Research Centre after Fibonacci, ensure that his name will not easily be forgotten. 4. Fibonacci's works. The mathematical writings of Fibonacci known to us and their dates are [2]: Liber Abbaci (The Book of Calculation), 1202 (1228); Practica Geometriae (The Practice of Geometry), 1220; Liber Quadratorum (The Book of Square Numbers), 1225; Flos (The Flower), 1225; Letter to Master Theodore (cf. §3). 5. Liber Abbaci: Hindu-Arabic numerals. In 1202, Fibonacci's hand-written account, Liber Abbaci, of his new mathematical experiences arising from the contacts he made on his Mediterranean travels was completed in Pisa. "Its publication", say Gies and Gies [4], "was a landmark in both the history of the Middle Ages and the history of mathematics". In this book he introduces to Western Europe in a clear, comprehensive and independent way, the new Hindu-Arabic numerals which had been successfully used in the Middle East and the Orient. McClenon [13] states that the Liber Abbaci was the "greatest arithmetic of the middle

ages and the first one to show by examples from every field the great superiority of the Hindu-Arabic numeral system over the Roman system exemplified by Boethius" (the arithmetic most generally taught throughout Europe before the thirteenth century). Fibonacci's Liber Abbaci begins (page 2 of [2]) with the simplest but profound statement: The nine Indian figures are: 987654321 With these nine figures, and with the sign 0 ... any number may be written, as is demonstrated below. In the Liber Abbaci, Fibonacci explains the nature of the Hindu-Arabic numerals, their use in calculations with integers and fractions, and their applications to practical commercial problems relating to weights and measures, bartering, interest and money changing. In the remainder of his book, Fibonacci is more concerned with theoretical, rather than practical, problems, e.g. Series and Proportions, the Extraction of Square and Cube Roots, the use of the Rule of False Position, the Method of Casting out Nines (for checking calculations), and other techniques discovered by the Hindu and Arab mathematicians. Finally, he deals with geometry and algebra. For ease in calculation with the new numerals, Fibonacci provides tables for the four arithmetical operations. While discussing the merits of the new system of numeration, Fibonacci also makes reference to the then current Roman numerals for comparison, and to the hexadecimal system of the Babylonians. Not everyone, however, accepted the usefulness of these Oriental symbols. For example, in 1299, Florentine merchants issued an ordinance prohibiting the use of Hindu- Arabic numerals because it was too easy for an unscrupulous person to write a sum of money, say 1000 lire, as 1999 lire. Such a fraudulent alteration was impossible in Roman numerals. To give something of the flavour of Fibonacci's work we quote [4] the following theoretical and commercial problems: PROBLEM 1 (A Voyage). A certain man doing business in Lucca doubled his money there, and then spent 12 denarii. Thereupon, leaving he went to Florence; there he also doubled his money, and spent 12 denarii. Returning to Pisa, he there doubled his money and spent 12 denarii, nothing remaining. How much did he have in the beginning? PROBLEM 2 (An Inheritance). A man whose end was approaching summoned his sons and said: "Divide my money as I shall prescribe." To his eldest son, he said, "You are to have 1 bezant and a seventh of what is left." To his second son he said, "Take 2 bezants and a seventh of what remains." To the third son, "You are to take 3 bezants and a seventh of what is left." Thus he gave each son 1 bezant more than the previous son and a seventh of what remained, and to the last son all that was left. After following their father's instructions with care, the sons found that they had shared their inheritance equally. How many sons were there, and how large was the estate? 5a. Liber Abbaci: the Fibonacci sequence. In Chapter 12 of the Liber Abbaci (pages 283-4 of [8]) Fibonacci states the problem which unvolves the famous sequence, with which his name is irrevocably linked (Quot paria coniculorum in uno anno ex uno paro germinentur): A certain man put a pair of rabbits in a place surrounded by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

Since Fibonacci's time his sequence (which was not important to him) has generated nearly as many research papers as it has hypothetical rabbits! For the history of the work on the sequence until about 1920 we might consult Dickson [4]. Since the foundation of the Fibonacci Association in San José, California, in the 1960's, and the production of the Fibonacci Quarterly there has been an orgy of creativity. For information on these fascinating aspects of Mathematics, see Hoggatt [7], Jarden [11], Lucas [12] and Vorobev [15]. Readily accessible material on the Fibonacci numbers may be found in past issues of The Australian Mathematics Teacher in articles by Guest [6], Horadam [8]-[10] and MacDonald [14]. 6. The Power of the Golden Mean Fibonacci numbers appear in (idealised) rabbit, cow and bee populations, and in the arrangements of petals round a flower, leaves round branches and seeds on seed-heads and pinecones, in everyday fruit and vegetables, in shells and in crystals. They are also found in architecture, art and music. We will only give a couple of vivid illustrations how we trace the golden mean in nature, architecture, art and music and try to see why it is so powerful. The golden ratio 1.618034 is also called the golden section or the golden mean or just the golden number. It is often represented by a Greek letter Phi . The closely related value which we write as phi with a small "p" is just the decimal part of Phi, namely 0.618034. Phi in Nature Why does Phi appear in nature? The answer lies in packing - the best arrangement of objects to minimise wasted space. The arrangement of leaves is the same as for seeds and petals. All are placed at 0.618034.. leaves, (seeds, petals) per turn. In terms of degrees this is 0·618034 of 360° which is 222·492...°. If there are Phi (1.618...) leaves per turn (or, equivalently, phi=0.618... turns per leaf), then we have the best packing so that each leaf gets the maximum exposure to light, casting the least shadow on the others. This also gives the best possible area exposed to falling rain so the rain is directed back along the leaf and down the stem to the roots. For flowers or petals, it gives the best possible exposure to insects to attract them for pollination.

The whole of the plant seems to produce its leaves, flowerhead petals and then seeds based upon the golden number. Phyllotaxis is the study of the ordered position of leaves on a stem. The leaves on this plant are staggered in a spiral pattern to permit optimum exposure to sunlight. If we apply the Golden Ratio to a circle we can see how it is that this plant exhibits Fibonacci qualities. By dividing a circle into Golden proportions, where the ratio of the arc length is equal to the Golden Ratio, we find the angle of the arcs to be 137.5 degrees. In fact, this is the angle at which adjacent leaves are positioned around the stem. This phenomenon is observed in many types of plants. Similarly, sunflowers have a Golden Spiral seed arrangement. This provides a biological advantage because it maximizes the number of seeds that can be packed into a seed head.

Fibonacci in living organisms We can make another picture showing the Fibonacci numbers 1,1,2,3,5,8,13,21,.. if we start with two small squares of size 1 next to each other. On top of both of these draw a square of size 2 (=1+1). We can now draw a new square - touching both a unit square and the latest square of side 2 - so having sides 3 units long; and then another touching both the 2-square and the 3-square (which has sides of 5 units). We can continue adding squares around the picture, each new square having a side which is as long as the sum of the latest two square's sides. This set of rectangles whose sides are two successive Fibonacci numbers in length and which are composed of squares with sides which are Fibonacci numbers, are called Fibonacci Rectangles. The diagram shows that we can draw a spiral by putting together quarter circles, one in each new square. This is a spiral (the Fibonacci Spiral). A similar curve to this occurs in nature as the shape of a snail shell or some seashells: Whereas the Fibonacci Rectangles spiral increases in size by a factor of Phi (1.618..) in a quarter of a turn (i.e. a point a further quarter of a turn round the curve is 1.618... times as far from the centre, and this applies to all points on the curve), the Nautilus spiral curve takes a whole turn before points move a factor of 1.618... from the centre. Starfish If a regular pentagon is drawn and diagonals are added, a five-sided star or pentagram is formed. Where the sides of the pentagon are one unit in length, the ratio between the

diagonals and the sides is Phi, or the Golden Ratio. This five-point symmetry with Golden proportions is found in starfish: Fibonacci in Anatomy. Humans exhibit Fibonacci characteristics, too. The Golden Ratio is seen in the proportions in the sections of a finger. The cochlea of the inner ear forms a Golden Spiral: Looking at DaVinci's Vitruvian Man, we can see that the entire human form may be broken down into a series of Golden Ratios:

Fibonacci in Art: The Golden Rectangle The Golden Ratio may be applied to the sides of a rectangle to form what is known as a Golden Rectangle. It is believed that the most visually pleasing dimensions are found in a rectangle whose length: width ratio is equal to Phi w l Where l / w = Phi = 1.618... Golden Rectangle First constructed by Pythagoras in the 6th Century BC. It is not surprising that the Golden Rectangle, given its aesthetically pleasing proportions, has a ubiquitous presence in art. It can be found in art and architecture of ancient Greece and Rome, in works of the Renaissance period, through to modern art of the 20th Century. The Golden Rectangles present in pictures below are quite obvious:

The Parthenon, Greece Leonardo DaVinci, Mona Lisa

Piet Mondrian, Place de la Concorde Fibonacci in Music Fibonacci sequence is used in music too. One calls the "golden sequence" simple sequence of 3 intervals in major: minor 6 (V-III), perfect 5 (V-II) and major 3 (I-III), which can be found in very many popular compositions in all styles. The most known example in classics is the beginning of Joseph Haydn's Finale from the Symphony No.103 Es-dur. The sections of a violin are constructed using the Golden Ratio.

The Golden Ratio can also be found in many musical compositions. For example, the familiar 5 bar motto in Beethoven's 5th Symphony appears at the beginning and end of the piece, but it is also reiterated at measure 377. There are 610 measures in total; this means that the second repetition of the motto divides the piece at its Golden Section. Other composers such as Debussy, Bach and Mozart also incorporated the Fibonacci numbers and Golden Ratio into their works, either consciously or unconsciously. It appears that the golden mean tendency is a powerful law of nature. In the seeming randomness of the natural world, we can find many instances of mathematical order involving the Fibonacci numbers themselves and the closely related "Golden" elements. Illustrations are taken from [32]

Appendix 5. Ralph Nelson Elliott and his Wave Theory 1. Short Biography of Ralph Nelson Elliott* Ralph Nelson Elliott was of those rare breeds, a true scholar in the practical world of finance. Financial analyst Hamilton Bolton accurately described the enormity of Elliott's feat when he said that "he developed his principle into a rational method of stock market analysis on a scale never before attempted." Brilliant and persistent, Elliott reached his ultimate achievement late in life by a circuitous route that included fortune in the disguise of disaster. Elliott was born on July 28, 1871 in Marysville, Kansas, and later moved to San Antonio, Texas. Around 1896, he entered the accounting field, and for twenty-five years held executive positions primarily with railroad companies in Mexico and Central America. By rescuing numerous companies from financial difficulty, Elliott earned a reputation as an expert business organizer. Finally, in early 1920, he moved to New York City. Elliott’s specialty made him the perfect choice for one of the U.S. government’s international projects. In 1924, the U.S. State Department chose him to become the Chief Accountant for Nicaragua, which was under the control of the U.S. Marines at the time. In February 1925, Elliott began applying his experience in corporate reorganization to reorganizing the finances of an entire country. When the U.S. extricated itself from Nicaragua, Elliott moved to Guatemala City to assume another corporate executive position: general auditor of the International Railway of Central America. During this time, Elliott wrote two books: Tea Room and Cafeteria Management, published in August 1926 by Little, Brown & Company, and The Future of Latin America, an analysis of the economic and social problems of Latin America and a proposal for creating economic stability and lasting prosperity in the region. With one book sold and the second one under consideration, Elliott decided to return to the United States to set up an independent management consulting business. It was around this time that he began to feel the symptoms of an alimentary tract illness caused by the organism amoeba histolytica that he contracted Central America. Elliott's reputation, built upon a distinguished career, his new book, and a long list of references, was soaring. Book reviews were favourable, he was in demand as a speaker, and his consulting business was beginning to grow. Just when Elliott's future appeared its brightest, however, his illness suddenly worsened. By 1929, it had developed into a debilitating case of pernicious anaemia, leaving him bedridden. The adventurous and

productive R.N. Elliott was forced into an unwanted retirement at the age of 58. Several times over the next five years, he came extremely close to death. Elliott needed something to occupy his mind while recuperating between the worst attacks of his illness. It was around this time that he turned his full attention to studying the behaviour of the stock market. Investigating the possibility of form in the marketplace, Elliott examined yearly, monthly, weekly, daily, hourly and half-hourly charts of the various indexes covering 75 years of stock market behaviour. In doing so, he was fulfilling a mission that he had enunciated for all responsible men in his manuscript on Latin America: "There is a reason for everything, and it is [one's] duty to try to discover it." In May 1934, two months after his final brush with death, Elliott's observations of stock market behaviour began coming together into a general set of principles that applied to all degrees of wave movement in the stock price averages. Today's scientific term for a large part of Elliott's observation about markets is that they are "fractal," coming under the umbrella of chaos science, although he went further in actually describing the component patterns and how they linked together. The former expert organizer of businesses had uncovered, through meticulous study, the organizational principle behind the movement of markets. As he got more proficient in the application of his principles and corrected initial errors in their formulation, their accuracy began to amaze him. By November 1934, R.N. Elliott's confidence in his ideas had developed to the point that he decided to present them to at least one member of the financial community: Charles J. Collins of Investment Counsel, Inc. in Detroit. Collins had traditionally put off the numerous correspondents who offered him systems for beating the market by asking them to forecast the market for a while, assuming that any truly worthwhile system would stand out when applied in current time. Not surprisingly, the vast majority of these systems proved to be dismal failures. Elliott's principle, however, was another story. The Dow Jones Averages had been declining throughout early 1935, and Elliott had pinpointed hourly turns by telegram with a fair degree of accuracy. In the second week of February, the Dow Jones Rail Average, as Elliott had previously predicted, broke below its 1934 low of 33.19. Advisors were turning negative and memories of the 1929-32 crash were immediately rekindled as bearish pronouncements about the future course of the economy proliferated. The Dow Industrials had fallen about 11% and were approaching the 96 level while the Rails (a more important average then) had fallen 50% from their 1933 peak to the 27 level. On Wednesday, March 13, 1935, just after the close of trading, with the Dow Jones averages finishing near the lows for the day, Elliott sent a telegram to Collins and flatly stated the following: "NOTWITHSTANDING BEARISH (DOW) IMPLICATIONS ALL AVERAGES ARE MAKING FINAL BOTTOM." Collins read the telegram on the morning of the next day, Thursday, March 14, 1935, the day of the closing low for the Dow Industrials that year. The day prior to the telegram, Tuesday, March 12, marked the 1935 closing low for the Dow Jones Rails. The thirteen-month “correction” was over, and the market immediately turned to the upside. Two months later, as the market continued on its upward climb, Collins, "impressed by [Elliott's] dogmatism and accuracy," agreed to collaborate on a book on the Wave Principle

suitable for public distribution. The Wave Principle was published on August 31, 1938. The first chapter makes the following statements: “No truth meets more general acceptance than that the universe is ruled by law. Without law, it is self-evident there would be chaos, and where chaos is, nothing is.... Very extensive research in connection with... human activities indicates that practically all developments which result from our social-economic processes follow a law that causes them to repeat themselves in similar and constantly recurring serials of waves or impulses of definite number and pattern... The stock market illustrates the wave impulse common to social-economic activity... It has its law, just as is true of other things throughout the universe.” Within weeks after the publication of his groundbreaking book, Elliott packed up his belongings and moved to an apartment hotel in Columbia Heights, Brooklyn, a short subway stop from Manhattan's financial district. On November 10, he published the first in a long series of Interpretive Letters that analysed and forecasted the path of the stock market. Ralph Elliott was finally back in the saddle, and as independently in business as he had planned eleven years before. In early 1939, Elliott was commissioned to write twelve articles on the Wave Principle for Financial World magazine. These articles established Elliott's reputation with the investment community, and led to his publishing a series of "Educational Bulletins." One of these was a groundbreaking work that lifted the Wave Principle from a comprehensive catalogue of the market's behavioural patterns to a broad theory of collective human behaviour that was new to the fields of economics and sociology. By the early 1940s, Elliott had fully developed his concept that the ebb and flow of human emotions and activities follow a natural progression governed by laws of nature. He tied the patterns of collective human behaviour to the Fibonacci, or "golden" ratio, a mathematical phenomenon known for millennia by mathematicians, scientists, artists, architects and philosophers as one of nature's ubiquitous laws of form and progress. Elliott then put together what he considered his definitive work, Nature's Law -- The Secret of the Universe. This rather grandly titled monograph, which Elliott published at age 75, includes almost every thought he had concerning the theory of the Wave Principle. The book was published June 10, 1946, and the reported 1000 copies sold out quickly to various members of the New York financial community. Less than two years before his death, Elliott had finally made his mark upon history. As a result of Elliott’s pioneering research, today, thousands of institutional portfolio managers, traders and private investors use the Wave Principle in their investment decision- making. Ralph Elliott undoubtedly would be gratified to see it. _________________________ This part was excerpted from a detailed 64-page biography in R.N. Elliott’s Masterworks, edited by R. Prechter (New Classics Library, 1994).

2. Classic Elliott Wave patterns* We have shown the basics of Elliott theory in the main part of this essay. Here we shall elaborate on the patterns, which he discovered. Explaining the following descriptions, on the left you will find a picture of a bull market, on the right - of a bear market. The pattern section depicts the structure, while the description gives additional information. The pattern should follow the rules and guidelines, which can also be derived from the picture. Furthermore the section, in which wave explains in which wave, as a part of a larger wave degree, the patterns normally occur. Last but not least the pattern must have an internal structure as described. This is very important to determine which pattern you are dealing with. I. Trends a. Impulse Pattern Description Impulses are always composed of five waves, labelled 1,2,3,4,5. Waves 1, 3 and 5 are themselves each impulsive pattern and are approximately equal in length. Waves 2 and 4 on the contrary are always corrective patterns. Rules and guidelines - Wave 2 cannot be longer in price than wave 1, and it must not go beyond the origin of wave1. - Wave 3 is never the shortest when compared to waves 1 and 5. - Wave 4 cannot overlap wave 1, except in diagonal triangles and sometimes in wave 1 or A waves, but never in a third wave. In most cases there should not be an overlap between wave 1 and A. As a guideline the third wave shows the greatest momentum, except when the fifth is the extended wave. - Wave 5 must exceed the end of wave 3. As a guideline the internal wave structure should show alternation, which means different kind of corrective structures in wave 2 and 4. In which wave Impulse patterns occur in waves 1, 3, 5 and in waves A and C of a correction (this correction could be a wave 2, 4 or a wave B, D, E or wave X).

Internal structure It is composed of five waves. The internal structure of these waves is 5-3-5-3-5. Note that the mentioned 3s are corrective waves, which should be composed of 5 waves in a corrective triangle. b. Extension Pattern Description By definition an extension occurs in an impulsive wave, where waves 1, 3 or 5 can be extended, being much longer than the other waves. It is quite common that one of these waves will extend, which is normally the third wave. The two other waves then tend to equal each other. Rules and guidelines - It is composed of 5, 9, 13 or 17 waves. - Wave 2 cannot be longer in price length than wave 1, so it should not go beyond the origin of wave 1. - Wave 3 is never the shortest when compared to waves 1 and 5. - Wave 4 cannot overlap wave 1. - Wave 5 exceeds the end of wave 3. - The extended wave normally shows the highest acceleration. In which wave Extensions occur in waves 1, 3, 5, and in A and C waves, when compared to each other. Internal structure As a minimum it is composed of 9 waves, though 13 or 17 waves could occur. So the minimal internal structure of the 9 waves is 5-3-5-3-5-3-5-3-5. Note that the 3s mentioned are corrective waves, which could be composed of 5 waves in the case of a corrective triangle. c. Diagonal triangle type 1 Pattern

Description Diagonals are sort of impulsive patterns, which normally occur in terminal waves like a fifth or a C wave. Don’t confuse them with corrective triangles. Diagonals are relatively rare phenomena for large wave degrees, but they do occur often in lower wave degrees on intra-day charts. Usually Diagonal triangles are followed by a violent change in market direction. Rules and guidelines - It is composed of 5 waves. - Waves 4 and 1 do overlap. - Wave 4 can’t go beyond the origin of wave 3. - Wave 3 cannot be the shortest wave. - Internally all waves of the diagonal have a corrective wave structure. - Wave 1 is the longest wave and wave 5 the shortest. - The channel lines of Diagonals must converge. As a guideline the internal wave structure should show alternation, which means different kind of corrective structures. In which wave Diagonal triangles type 1 occur in waves 5, C and sometimes in wave 1. Internal structure The internal structure of the five waves is 3-3-3-3-3. c. Diagonal triangle type 2 Pattern Description Diagonal type 2 is a sort of impulsive pattern, which normally occurs in the first or A wave. The main difference with the Diagonal Triangle type 1 is the fact that waves 1, 3 and 5 have an internal structure of five waves instead of three. Diagonals are relatively rare phenomena for large wave degrees, but they do occur often in lower wave degrees in intra day charts. These Diagonal triangles are not followed by a violent

change in market direction, because it is not the end of a trend, except when it occurs in a fifth or a C wave. Rules and guidelines - It is composed of 5 waves. - Wave 4 and 1 do overlap. - Wave 4 can’t go beyond the origin of wave 3. - Wave 3 cannot be the shortest wave. - Internally waves 1, 3 and 5 have an impulsive wave structure. - Wave 1 is the longest wave and wave 5 the shortest. As a guideline the internal wave structure should show alternation, which means that wave 2 and 4 show a different kind of corrective structure. In which wave Diagonal triangles type 2 occur in waves 1 and A. Internal structure The five waves of the diagonal type 2 show an internal structure of 5-3-5-3-5. d. Failure or Truncated 5th Pattern Description A failure is an impulsive pattern in which the fifth wave does not exceed the third wave. Fifth waves, which travel only slightly beyond the top of wave 3, can also be classified as a kind of failure. It indicates that the trend is weak and that the market will show acceleration in the opposite direction. Rules and guidelines - Wave 2 cannot be longer in price distance than wave 1, so it should not go beyond the origin of wave 1. - Wave 3 is never the shortest when compared to waves 1 and 5. Wave 4 cannot overlap wave 1, except for diagonal triangles and sometimes in waves - 1 or A, but never in a third wave. There should not be overlap between wave 1 and A. - Wave 5 fails to go beyond the end of wave 3. As a guideline the third wave shows the greatest momentum. As a guideline the internal wave structure should show alternation, which means different kinds of corrective structures. In which wave A failure can only occur in a fifth wave or a C wave, but normally not in the fifth wave of wave 3. Internal structure It must be composed of five waves.

II. Corrections a. Zigzag Pattern Description A Zigzag is the most common corrective structure, which starts a sharp reversal. Often it looks like an impulsive wave, because of the acceleration it shows. A zigzag can extend itself into a double or triple zigzag, although this is not very common, because it lacks alternation (the same two patterns follow each other). Zigzag can only be the first part of a corrective structure. Rules and guidelines - It is composed of 3 waves. - Waves A and C are impulses, wave B is corrective. - The B wave retraces no more then 61.8% of A. - The C wave must go beyond the end of A. - The C wave normally is at least equal to A. In which wave Most of the time it happens in A, X or 2. Also quite common in B waves as a part of a Flat, (part of) Triangles and sometimes in 4. Internal structure A single Zigzag is composed of 3 waves, a double of 7 waves separated by an X wave in the middle, a triple of 11 waves separated by two X waves (see pictures below). The internal structure of the 3 waves is 5-3-5 in a single Zigzag, 5-3-5-3-5-3-5 in a double.

Example of a Double Zigzag b. Flat Pattern Description Flats are very common forms of corrective patterns, which generally show a sideways direction. Waves A and B of the Flat are both corrective patterns. Wave C on the contrary is an impulsive pattern. Normally wave C will not go beyond the end of wave A. Rules and guidelines - It is composed of 3 waves. - Wave C is an impulse, wave A and B are corrective. - Wave B retraces more then 61.8% of A. - Wave B often shows a complete retracement to the end of the previous impulse wave. - Wave C shouldn’t go beyond the end of A. - Normally wave C is at least equal to A. In which wave It occurs mostly in B waves, though also quite common in 4 and 2. Internal structure As mentioned before a Flat consists of 3 waves. The internal structure of these waves is 3-3-5. Both waves A and B normally are Zigzags. c. Expanded Flat or Irregular Flat Pattern

Description This is a common special type of Flat. Here the B wave is extended and goes beyond the (orthodox) end of the previous impulsive wave. The strength of the B wave shows that the market wants to go in the direction of B. Often a strong acceleration will take place, which starts a third wave or an extended fifth. If the C wave is much longer then A, the strength will be less. Rules and guidelines - It is composed of 3 waves. - Wave C is an impulse; waves A and B are corrective. - Wave B retraces beyond the end of the previous impulse, which is the start of wave A. - The C wave normally is much longer then A. In which wave This corrective pattern can happen in 2, 4, B and X. If it happens in 2 and C is relatively short, normally acceleration in the third will take place. Internal structure It is composed of five waves, which have an internal structure of 3-3-5. c. Triangles Contracting Triangle: Pattern Description A triangle is a corrective pattern, which can contract or expand. Furthermore it can ascend or descend. It is composed of five waves; each of them has a corrective nature. Rules and guidelines - It is composed of 5 waves.

- Wave 4 and 1 do overlap. - Wave 4 can’t go beyond the origin of wave 3. - Wave 3 cannot be the shortest wave. - Internally all waves of the diagonal have a corrective wave structure. - In a contracting Triangle, wave 1 is the longest wave and wave 5 the shortest. In an expanding Triangle, wave 1 is the shortest and wave 5 the longest. - Triangles normally have a wedged shape, which follows from the previous. As a guideline the internal wave structure should show alternation. In which wave Triangles occur only in waves B, X and 4. Never in wave 2 or A. Internal structure It is composed of five waves, of which the internal structure is 3-3-3-3-3. Expanding Triangle: Ascending Triangle: This is a triangle, which slopes upwards. This pattern has been implemented in the Modern Rules. Descending Triangle: This is a triangle, which slopes downwards. This pattern has been implemented in the Modern Rules. Running Triangle: This is a triangle where the B wave exceeds the origin of wave A. d. Combinations Many kinds of combinations are possible. Below a rather complex example has been depicted.

Pattern Description A Combination combines several types of corrections. These corrections are labelled as WXY and WXYXZ if it is even more complex. It starts for example with a Zigzag (wave W), then an intermediate X wave, then a Flat (wave Y) and so on. A so-called double or triple three is also a Combination, but this pattern combines Flats separated by X waves. Rules and guidelines - All types of corrective patterns can combine to form a bigger corrective pattern. - The rules and guidelines, as mentioned for other corrective patterns apply. - A triangle in a Combination should normally occur at the end. - Corrective patterns in a Combination normally show alternation. In which wave Generally a Combination occurs mostly in B, X and 4, it is less common in A and rare in 2. Internal structure For example a Zigzag, followed by a Flat, followed by a Triangle has the following internal structure: 5-3-5(Zigzag)-5-3-5(X)-3-3-5(Flat)-3-3-3-3-3(Triangle). e. Running Flat Pattern Description The Running correction is a rare special form of a failure. This pattern is a kind of Flat, with an elongated B wave and a very small C wave. According to theory wave C should be so short that it doesn’t get to the price territory of wave A.

Instead of a running correction this could in theory be an extension in an impulsive wave, where the wave has subdivided in two (or more) 1,2 combinations. If the B is a clear three wave, then it is a Running correction, otherwise an extension. Rules and guidelines - The B wave must be composed of three waves. - The C wave must be composed of five waves. - Wave C must be very short and normally will not reach the price territory of A. - Wave C must not retrace more than 100% of wave B but more than 60% of wave A. In which wave Most of the time it should occur in wave 2 or B. Internal structure It is a three-wave structure. The internal structure is 3-3-5. X wave Description An X wave is an intermediate wave in a more complex correction. This wave is always corrective and can take many forms like a Zigzag, Double Zigzag, Flat, Expanded Flat, combination and a triangle. 3. Modern Elliott Wave Patterns On the basis of classical Elliott patterns a number of Modern Rules have been developed, that are mostly hybrid patterns derived from the known patterns that have existed from the beginning. Some of them are described below: I. Trends a. Impulse 2 Pattern Description An Impulse 2 is an uncommon pattern that resembles a normal impulse considerably. Rules and guidelines The same rules and guidelines apply as with a normal impulse except for the following: - Wave 4 is allowed to retrace between 51.5% and 62%, without penetrating the region of wave 1. As a guideline, wave 4 very often is a Zigzag.

In which wave Impulse 2 patterns mostly occur in waves 1,A or C, never in a wave 3. Internal structure It is composed of five waves. The internal structure of these waves is 5-3-5-3-5. Note that the mentioned 3s are corrective waves, which could be composed of 5 waves in a corrective triangle. II. Corrections a. ZigzagFlat Pattern Description It is a common pattern that is exactly the same as a Zigzag, except for the fact that the B wave is allowed to retrace more than 61.8% of wave A. b. Running Zigzag Pattern Description Apart from contracting Triangles, a failure in a corrective pattern happens when the C wave is shorter than wave A and fails to go beyond the end of A. This mostly happens in Running Flats and or in Zigzags. It indicates strength in the direction of the main trend.

Rules and guidelines The rules as mentioned with other corrective patterns apply. Wave C fails to go beyond the end of wave A. In which wave Failures can occur in a C wave of wave 2, in a C or E wave of wave 4, in a C wave of wave B or X. c. Failed Flat Pattern Description This pattern is exactly the same as a Flat, except for the fact that wave C does not reach the end of wave A and therefore is shorter than wave B. d. Running Flat (modern) Pattern Description This pattern is exactly the same as a Running Flat, except for the fact that it must retrace more than 60%, if not we consider it to be a normal Running Flat. This distinction is necessary, because normally a Running Flat is rare. But if it retraces more than 60% and still fails to

reach the end of wave A, it suddenly becomes much more probable the pattern will occur. In which case it will get a much higher score. e. Ascending and descending Triangles Description These are mentioned under the Triangles description in the Classic patterns section. Basically these patterns are the same as common contracting triangles, except for the fact that ascending and descending triangles slope up or down. 4. Channelling Channelling is an important tool not only to determine which sub waves belong together, but also to project targets for the next wave up. Channels are parallel lines, which more or less contain the complete price movement of a wave. Although the trend lines of a Triangle are not parallel lines, they will also be considered as a channel. Underneath you see an example of a channel in an impulsive wave and all channels in a corrective wave. Note that all patterns in the section "Patterns" show their channels. The picture of the corrective structure labelled A,B,C shows clearly how channels indicate which waves should be grouped together. Waves of the same degree can be recognized by drawing channels. Especially this is the case for Impulse (5) wave structures, Zigzags and Triangles. If these waves do not equate properly, you have a strong indication to search for an alternative count. How to draw channels and how to project targets using channels? Targets for wave 3 or C Channels are drawn as soon as waves 1 and 2 are finished. Connect the origin of wave 1, which has been labelled as zero, and the end of wave 2. Then draw a parallel line from the top of wave 1. The parallel line serves as an absolute minimum target for the 3rd wave under development. If the 3rd wave can’t break through the upper line or fails to reach it, that means that it’s probably a C wave. If C is finished, a trend line can be drawn, connecting wave A and the end of wave C to get a target for wave E. Wave E almost never stops at the trend line precisely, it either never reaches the trend line or it overshoots the trend line fast and temporarily.

Targets in a Double Zigzag Drawing a channel is very useful to separate Double Zigzags from impulsive waves, which is difficult since both have impulsive characteristics. Double Zigzags tend to fit a channel almost perfectly, while in an impulsive wave the third wave clearly breaks out of the channel. That’s what modern Elliotticians have noticed in the classical waves. 4. Fibonacci proportional relationships in the waves. If we summarize the Fibonacci proportional relationships in the waves, it’ll look like this: Targets for wave 1 The first wave, a new impulsive price movement, tends to stop at the base of the previous correction, which normally is the B wave. This often coincides with a 38.2% or a 61.8% retracement of the previous correction. Targets for wave 2 Wave 2 retraces at least 38.2% but mostly 61.8% or more of wave 1. It often stops at sub wave 4 and more often at sub wave 2 of previous wave 1. Targets for wave 3 Wave 3 is at least equal to wave 1, except for a Triangle (see patterns in Appendix B). If wave 3 is the longest wave it will tend to be at least161% of wave 1. Targets for wave 4 Wave 4 retraces at least 23% of wave 3 but more often reaches a 38.2% retracement. It normally reaches the territory of sub wave 4 of the previous 3rd wave. In very strong markets wave 4 should only retrace 14% of wave 3. Targets for wave 5 Wave 5 normally is equal to wave 1, or travels a distance of 61.8% of the length of wave 1. It could also have the same relationships to wave 3 or it could travel 61.8% of the net length of

wave 1 and 3 together. If wave 5 is the extended wave it mostly will be 161.8% of wave 3 or 161.8% of the net length of wave 1 and 3 together. Targets for wave A After a Triangle in a fifth wave, wave A retraces to wave 2 of the Triangle of previous wave 5. When wave A is part of a Triangle, B or 4 it often retraces 38.2% of the complete previous 5 wave (so not only the fifth of the fifth) into the territory of the previous 4th wave. In a Zigzag it often retraces 61.8% of the fifth wave. Targets for wave B In a Zigzag, wave B mostly retraces 38.2% or 61.8% of wave A. Targets for wave C Wave C has a length of at least 61.8% of wave A. It could be shorter in which case it normally is a failure, which foretells an acceleration in the opposite direction. Generally wave C is equal to wave A or travels a distance of 161.8% of wave A. Wave C often reaches 161.8% of the length of wave A in an Expanded Flat. In a contracting Triangle wave C often is 61.8% of wave A. * Quoted from [25-27]

Appendix 6. Brownian Motion and Bachelier’s Process* In 1827 an English botanist, Robert Brown got his hands on some new technology: a microscope "made for me by Mr. Dolland, . . . of which the three lenses that I have generally used, are of a 40th, 60th, and 70th of an inch focus."[1] Right away, Brown noticed how pollen grains suspended in water jiggled around in a furious, but random, fashion. What was going on was a puzzle. Were these tiny bits of organic matter somehow alive? Robert Brown himself said he didn’t think the movement had anything to do with tiny currents in the water, nor was it produced by evaporation. He explained his observations in the following terms: "That extremely minute particles of solid matter, whether obtained from organic or inorganic substances, when suspended in pure water, or in some other aqueous fluids, exhibit motions for which I am unable to account, and from which their irregularity and seeming independence resemble in a remarkable degree the less rapid motions of some of the simplest animalcules of infusions. That the smallest moving particles observed, and which I have termed Active Molecules, appear to be spherical, or nearly so, and to be between 1/20,000dth and 1/30,000dth of an inch in diameter; and that other particles of considerably greater and various size, and either of similar or of very different figure, also present analogous motions in like circumstances. "I have formerly stated my belief that these motions of the particles neither arose from currents in the fluid containing them, nor depended on that intestine motion which may be supposed to accompany its evaporation."[1] Brown noted that others before him had made similar observations in special cases. For example, a Dr. James Drummond had observed this fishy, erratic motion in fish eyes: "In 1814 Dr. James Drummond, of Belfast, published in the 7th Volume of the Transactions of the Royal Society of Edinburgh, a valuable Paper, entitled ‘On certain Appearances observed in the Dissection of the Eyes of Fishes.’ "In this Essay, which I regret I was entirely unacquainted with when I printed the account of my Observations, the author gives an account of the very remarkable motions of the spicula which form the silvery part of the choroid coat of the eyes of fishes."[1] Today, we know that this motion, called Brownian motion in honour of Robert Brown, was due to random fluctuations in the number of water molecules bombarding the pollen grains from different directions. Experiments showed that particles moved further in a given time interval if you raised the temperature, or reduced the size of a particle, or reduced the viscosity of the fluid. In 1905, in a celebrated treatise entitled The Theory of the Brownian Movement [5], Albert Einstein developed a mathematical description which explained Brownian motion in terms of particle size, fluid viscosity, and temperature. Later, in 1923, Norbert Wiener gave a mathematically rigorous description of what is now referred to as a "stochastic process." Since that time,

Brownian motion has been called a Wiener process, as well as a "diffusion process", a "random walk", and so on. But Einstein wasn’t the first to give a mathematical description of Brownian motion. That honour belonged to a French graduate student Louis Bachelier. In1900 in Paris he presented his doctoral thesis, entitled Théorie de la spéculation. What interested Bachelier were not pollen grains and fish eyes. Instead, he wanted to know why the prices of stocks and bonds jiggled around on the Paris bourse. He was particularly interested in bonds known as rentes sur l’état— perpetual bonds issued by the French government. What were the laws of this jiggle? He thought the answer lay in the prices being bombarded by small bits of news. More about the implication of fractional brownian motion can be learned from “Stochastic Processes: From Physics to Finance” by Wolfgang Paul and Jorg Baschnagel. The authors give a very clear introduction to the concepts underlying stable distributions. In the main part of the essay we have mentioned the key insight of Bachelier - that the probability intervals into which prices fall seemed to increased or decreased with the square- root of time (T0.5). The range of prices for a given probability, then, depends on the constant a, and on the square root of time (T0.5). However, he didn’t take into consideration another variable, that is the current price level P. In finance always logarithms of prices are taken. This is for many reasons. Most changes in most economic variables are proportional to their current level. For example, it is plausible to think that the variation in gold prices is proportional to the level of gold prices: $800 dollar gold varies in greater increments than does gold at $260. The change in price, ∆P, as a proportion of the current price P, can be written as: ∆P/P . But this is approximately the same as the change in the log of the price: ∆P/P ≈ ∆ (log P) . The range of prices for a given probability, then, depends on the constant a, and on the square root of time (T0.5), as well as the current price level P. The difference in the two approaches is that if price increments (∆P) are independent, and have a finite variance, then the price P has a normal (Gaussian distribution). But if increments in the log of the price (∆ log P) are independent, and have a finite variance, then the price P has a lognormal distribution. Here is a picture of a normal or Gaussian distribution:

The left-hand tail never becomes zero. No matter where we centre the distribution (place the mean), there is always positive probability of negative numbers. Here is a picture of a lognormal distribution: The left-hand tail of a lognormal distribution becomes zero at zero. No matter where we centre the distribution (place the mean), there is zero probability of negative numbers. A lognormal distribution assigns zero probability to negative prices. But a normal distribution assigns positive probability to negative prices. *The information and charts for this Appendix is partially quoted from [38].

Appendix 7. Lyapunov Exponent* Alexander M. Lyapunov was born in Yaroslavl, Russia, in 1857, and died in Odessa, Russia, in 1918. He did important work on differential equations, potential theory, stability of systems and probability theory. His work concentrated on the stability of equilibrium and motion of a mechanical system and the stability of a uniformly rotating fluid. He devised important methods of approximation. Lyapunov's methods, introduced by him in 1899, provide ways of determining the stability of sets of ordinary differential equations. In Chaos theory a quantity called the "Lyapunov Exponent" is used to measure the extent to which ever-smaller "infinitesimal" errors are amplified. In other words, the Lyapunov Exponent measures how sensitive a system is to errors. Growth of small errors As we have mentioned in Chapter 3 of this essay, chaotic systems are very sensitive to initial conditions. Suppose we have the following simple system (called a logistic equation) with a single variable, appearing as input, x(n), and output, x(n+1): x(n+1) = 4 x(n)[1-x(n)] (another expression for the same is f(x) = 4x(1-x) ) The graph of this function is just an upside-down parabola passing through the (x,y) pairs (0,0), (1/2,1), (1,0), as shown in the following figure: The number 4 in the definition of the function is there so that the graph of the function fits neatly into the unit box; in other words, if we apply the function to any number in the unit interval 0 ≤ x ≤ 1, then we get another number in the unit interval.

This is a very well known deterministic equation, a famous one in the study of non-linear dynamics, for amongst other things it can be used to demonstrate that a very simple deterministic expression can result with what otherwise appears to be random or stochastic process, when in fact it is predictable. The input is x(n). The output is x(n+1). The system is non-linear, because if you multiply out the right hand side of the equation, there is an x(n)2 term. So the output is not proportional to the input. Let x(n) = .75. The output is 4 (.75) [1- .75] = .75. That is, x(n+1) = .75. If this were an equation describing the price behaviour of a market, the market would be in equilibrium, because today’s price (.75) would generate the same price tomorrow. If x(n) and x(n+1) were expectations, they would be self-fulfilling. Given today's price of x(n) = .75, tomorrow's price will be x(n+1) = .75. The value .75 is called a fixed point of the equation, because using it as an input returns it as an output. It stays fixed, and doesn't get transformed into a new number. But, suppose the market starts out at x(0) = .7499. The output is 4 (.7499) [1-.7499] = .7502 = x(1). Now using the previous day's output x(1) = .7502 as the next input, we get as the new output: 4 (.7502) [1-.7502] = .7496 = x(2). And so on. Going from one set of inputs to an output is called iteration. Then, in the next iteration, the new output value is used as the input value, to get another output value. The first 100 iterations of the logistic equation, starting with x(0) = .7499, are shown in the table below. Finally, we repeat the entire process, using as our first input x(0) = .74999. These results are also shown in the Table below. Each set of solution paths—x(n), x(n+1), x(n+2), etc.—are called trajectories. The table shows three different trajectories for three different starting values of x(0). The sequence of numbers x(0), x(1), x(2)… is called the orbit of the initial condition x(0). Look at iteration number 20. If you started with x(0) = .75, you have x(20) = .75. But if you started with x(0) = .7499, you get x(20) = .359844. Finally, if you started with x(0) = .74999, you get x(20) = .995773. Clearly a small change in the initial starting value causes a large change in the outcome after a few steps. The equation is very sensitive to initial conditions. Table : First One Hundred Iterations of the Equation x(n+1) = 4 x(n) [1- x(n)] with Different Values of x(0). x(0): .75000 .74990 .74999 1 .7500000 .750200 .750020 Iteration 2 .7500000 .749600 .749960

3 .7500000 .750800 .750080 29 .7500000 .211328 .898598 4 .7500000 .748398 .749840 30 .7500000 .666675 .364478 5 .7500000 .753193 .750320 31 .7500000 .888878 .926535 6 .7500000 .743573 .749360 32 .7500000 .395096 .272271 7 .7500000 .762688 .751279 33 .7500000 .955981 .792558 8 .7500000 .723980 .747436 34 .7500000 .168326 .657640 9 .7500000 .799332 .755102 35 .7500000 .559969 .900599 10 .7500000 .641601 .739691 36 .7500000 .985615 .358082 11 .7500000 .919796 .770193 37 .7500000 .056712 .919437 12 .7500000 .295084 .707984 38 .7500000 .213985 .296289 13 .7500000 .832038 .826971 39 .7500000 .672781 .834008 14 .7500000 .559002 .572360 40 .7500000 .880587 .553754 15 .7500000 .986075 .979056 41 .7500000 .420613 .988442 16 .7500000 .054924 .082020 42 .7500000 .974791 .045698 17 .7500000 .207628 .301170 43 .7500000 .098295 .174440 18 .7500000 .658075 .841867 44 .7500000 .354534 .576042 19 .7500000 .900049 .532507 45 .7500000 .915358 .976870 20 .7500000 .359844 .995773 46 .7500000 .309910 .090379 21 .7500000 .921426 .016836 47 .7500000 .855464 .328843 22 .7500000 .289602 .066210 48 .7500000 .494582 .882822 23 .7500000 .822930 .247305 49 .7500000 .999883 .413790 24 .7500000 .582864 .744581 50 .7500000 .000470 .970272 25 .7500000 .972534 .760720 51 .7500000 .001877 .115378 26 .7500000 .106845 .728099 52 .7500000 .007495 .408264 27 .7500000 .381716 .791883 53 .7500000 .029756 .966338 28 .7500000 .944036 .659218 54 .7500000 .115484 .130115 xcix

55 .7500000 .408589 .452740 81 .7500000 .950432 .526042 56 .7500000 .966576 .991066 82 .7500000 .188442 .997287 57 .7500000 .129226 .035417 83 .7500000 .611727 .010822 58 .7500000 .450106 .136649 84 .7500000 .950068 .042818 59 .7500000 .990042 .471905 85 .7500000 .189755 .163938 60 .7500000 .039434 .996843 86 .7500000 .614991 .548250 61 .7500000 .151515 .012589 87 .7500000 .947108 .990688 62 .7500000 .514232 .049723 88 .7500000 .200378 .036901 63 .7500000 .999190 .189001 89 .7500000 .640906 .142159 64 .7500000 .003238 .613120 90 .7500000 .920582 .487798 65 .7500000 .012911 .948816 91 .7500000 .292444 .999404 66 .7500000 .050976 .194258 92 .7500000 .827682 .002381 67 .7500000 .193508 .626087 93 .7500000 .570498 .009500 68 .7500000 .624252 .936409 94 .7500000 .980120 .037638 69 .7500000 .938246 .238190 95 .7500000 .077939 .144886 70 .7500000 .231761 .725821 96 .7500000 .287457 .495576 71 .7500000 .712191 .796019 97 .7500000 .819301 .999922 72 .7500000 .819899 .649491 98 .7500000 .592186 .000313 73 .7500000 .590658 .910609 99 .7500000 .966007 .001252 74 .7500000 .967125 .325600 100 .7500000 .131350 .005003 75 .7500000 .127178 .878338 76 .7500000 .444014 .427440 77 .7500000 .987462 .978940 78 .7500000 .049522 .082465 79 .7500000 .188278 .302657 80 .7500000 .611319 .844223 c

Measuring Sensitivity: The Lyapunov Exponent The Lyapunov exponent λ is a measure of the exponential rate of divergence of neighbouring trajectories. We saw that a small change in the initial conditions of the logistic equation (Table above) resulted in widely divergent trajectories after a few iterations. How fast these trajectories diverge is a measure of our ability to forecast. For a few iterations, the three trajectories of the Table above look pretty much the same. This suggests that short-term prediction may be possible. A prediction of "x(n+1) = .75", based solely on the first trajectory, starting at x(0) = .75, will serve reasonably well for the other two trajectories also, at least for the first few iterations. But, by iteration 20, the values of x(n+1) are quite different among the three trajectories. This suggests that long-term prediction is impossible. So let's think about the short term. How short is it? How fast do trajectories diverge due to small observational errors, small shocks, or other small differences? That’s what the Lyapunov exponent tells us. Let ε denote the error in our initial observation, or the difference in two initial conditions. In Table 1, it could represent the difference between .75 and .7499, or between .75 and .74999. Let R be a distance (plus or minus) around a reference trajectory, and suppose we ask the question: how quickly does a second trajectory which includes the error ε get outside the range R? The answer is a function of the number of steps n, and the Lyapunov exponent λ , according to the following equation (where "exp" means the exponential ε = 2.7182818…, the basis of the natural logarithms): R = ε • exp(λ n). λ The following diagrams illustrate this phenomenon: first, we show the orbit of an initial point in blue. We plot the points x[k] (for k=0,1,2 etc.) against the step-number k: this is called a "time-series". 101

Orbit of an initial point In the next picture, we compare this with the orbit of another close-by point, shown in red. In other words, we add a small error and look at what happens to the orbit: Orbits of two initially close-by points Notice that for the first few steps (on the left-hand side of the diagram), the orbits remain very close together and it seems that the small error hasn't made much difference. However, after just a few steps (toward the right-hand side of the diagram) the orbits begin to look completely different. The error has grown until the orbits are far apart. The next picture shows the size of the error (i.e. the difference between the two orbits) at each step: 102

Error between the orbits Notice how the error starts off very small, but after a few steps it has grown as large as the orbits themselves. This is like the noise in a radio transmission becoming as large as the signal itself and swamping it completely. Order and Chaos in the same system Even some very simple systems are capable of showing both orderly and chaotic behaviour. These systems may make transitions (called "bifurcations") from one type of behaviour to another, as some parameter is varied, rather like tuning-in to different radio stations by turning the dial on a radio. For example, remember our function f(x) = 4x(1−x) whose graph was a parabola through (x,y)=(0,0), (1/2,1), and (1,0). We can add a parameter p to let us move the height of the parabola up and down: fp(x) = (4p)x(1−x) For a fixed number p (with magnitude between 0 and 1) the graph of fp against x is a parabola still passing through (0,0) and (1,0), but now having its "top" (turning point) at (1/2,p), as shown in the following picture: 103

Graph of function with a parameter p Parameter p is like the radio tuning dial: adjusting the value of p moves the top of the parabola up and down smoothly, but as we will soon see the behaviour of the function can change suddenly. When tuning a radio, you sometimes happen to hit the right wavelength and hear a different radio station. In the same way, when the value of p (the tuning dial) passes certain special values, the behaviour of orbits can change suddenly. This is the idea behind the Lyapunov exponent: we take a general function f and use this same formula to see how much small errors tend to be magnified. We take more and more iterates (larger and larger n) and calculate the corresponding values for the factor at which errors are magnified. Plotting the stability One idea to help us understand the way that the behaviour changes (as we change the parameter) is to plot the value of the Lyapunov exponent (for some chosen initial point x[0]) against the parameter p. This will let us estimate how the stability of our system changes as p changes. The following diagram shows a plot of the calculated Lyapunov exponent (using the starting point x[0]=1/2) against the parameter p. 104

Lyapunov exponent vs. parameter p Notice the large negative spikes pointing downward. In reality, these spikes in the graph go off to minus infinity, showing that the behaviour of the function is incredibly stable at those parameter values. If Lyapunov exponent, which is the logarithm of the error-multiplier, is very large and negative, then the error-multiplier is very small, which indicates that errors tend to vanish away and the orbit of the system is super-stable. Between the spikes, the graph touches the p-axis (so that the Lyapunov exponent is zero): this indicates that these parameters give an orbit, which is on the edge of stable and unstable behaviour. It is here that sudden changes (bifurcations) in the behaviour of orbits take place. Notice that the graph stays below zero until it reaches a point near the right hand side of the plot: this is where sensitivity first appears. As the graph rises above the axis, this means that the Lyapunov exponent is positive so that errors tend to be magnified. Even so, in this right- hand region there are still spikes where super-stable orbits appear. There is much complicated structure in this diagram and even a simple one-parameter family of functions (like our parabola functions) still holds mysteries for mathematical research. The different solution trajectories of chaotic equations form patterns called strange attractors. If similar patterns appear in the strange attractor at different scales (larger or smaller, governed by some multiplier or scale factor r, as we saw previously), they are said to be fractal. They have a fractal dimension D, governed by the relationship N = rD. Chaos equations like the one here (namely, the logistic equation) generate fractal patterns. We will not give the comprehensive mathematical explanation of how the exponent is calculated. One can read about it in [21].We we shall only underline, that the system diverges because the Lyapunov exponent is positive. If it were the case the Lyapunov exponent were negative, λ < 0, then exp(λ n) would get smaller with each step. So it must be the case that λ > 0 for the system to be chaotic. Note also that the particular logistic equation, x(n+1) = 4 x(n) [1-x(n)], which we used in our Table, is a simple equation with only one variable, namely x(n). So it has only one Lyapunov exponent. In general, a system with M variables may have as many as M Lyapunov 105

exponents. In that case, an attractor is chaotic if at least one of its Lyapunov exponents is positive. Lyapunov Exponent Pictures Lyapunov exponent can be presented in the form of pictures. Again, I shall not explain here, how the pictures are drawn (Refer to [21] to see the technique of drawing fractal images), I shall only quote a few examples and explain how to understand them. Like most fractal images, Lyapunov Exponent fractals are produced by iterating functions and observing the chaotic behaviour that may result. The sharply-defined curves which appear in these pictures correspond to very large negative Lyapunov exponents (meaning that the system is very stable, in fact “super”-stable, and errors tend to vanish away). The background regions indicate positive exponents (where the system is sensitive, i.e. small errors tend to grow, and may be chaotic). The presence of regions appear to “cross over” each other reveals that there are two or more different types of behaviour, that are “competing” with each other (in fact these are competing 106

attractors) – different behaviours “win” the competition for different values of input parameters, so we see what looks like different shapes crossing over each other. There are also repeating patterns and progressions of shapes (fractals), revealing lots of structure, that would be difficult to guess just from looking at the simple equations used to generate pictures. By visualising such structures and seeking to understand them, we can make progress in understanding the behaviour of more complicated models of systems in the real world, as well as in financial markets, an example of which is given in Chapter 5 of this essay. The beauty of fractal images is an invitation to try and understand such patterns. There is any number of sites on the World Wide Web dedicated to galleries of computer-generated fractal images. Pictures based on Lyapunov Exponent fractals, such as the one pictured above, are some of the most striking and unusual. _______________ * The materials for this Appendix were taken from the article by Andy Burbanks in [21], as well as from the article by J.O.Grabbe in [38] 107

Appendix 8. Hurst Exponent* Harold Edwin Hurst was a poor Leicester boy who worked his way into Oxford, and later became a British "civil servant" in Cairo in 1906. He got interested in the Nile. For centuries, perhaps millennia, the yearly flooding of the Nile was the basis of agriculture which supported much of known civilization. The annual overflowing of the river deposited rich top soil from the Ethiopian Highland along the river banks. The water and silt were distributed by irrigation, and the staple crops of wheat, barley, and flax were planted. Hurst looked at 800 years of records and noticed that there was a tendency for a good flood year to be followed by another good flood year, and for a bad (low) flood year to be followed by another bad flood year. That is, there appeared to be non-random runs of good or bad years. Later Mandelbrot and Wallis [51] used the term Joseph effect to refer to any persistent phenomenon like this (alluding to the seven years of Egyptian plenty followed by the seven years of Egyptian famine in the biblical story of Joseph). Of course, even if the yearly flows were independent, there still could be runs of good or bad years. So Hurst calculated a variable which is now called a Hurst exponent (H). The expectation was that H = ½ if the yearly flood levels were independent of each other. Calculating the Hurst Exponent J. Grabbe gives a specific example of Hurst exponent calculation which illustrates the general rule. Suppose there are 99 yearly observations of the height h of the mid-September Nile water level at Aswan: h(1), h(2), . . ., h(99). Calculate a location m and a scale c for h. If we assume in general that h has a finite variance, then m is simply the sample mean of the 99 observations, while c is the standard deviation. The first thing is to remove any trend, any tendency over the century for h to rise or fall as a long-run phenomena. So we subtract m from each of the observations h, getting a new series x that has mean zero: x(1) = h(1) - m, x(2) = h(2) - m, … x(99) = h(99) - m . The set of x’s are a set of variables with mean zero. Positive x’s represent those years when the river level is above average, while negative x’s represent those years when the river level is below average. Next we form partial sums of these random variables, each partial sum Y(n) being the sum of all the years prior to year n: Y(1) = x(1), Y(2) = x(1) + x(2), 108

. . . Y(n) = x(1) + x(2) + . . . + x(n), . . Y(99) = x(1) + x(2) + x(3) + . . . + x(99). Since the Y’s are a sum of mean-zero random variables x, they will be positive if they have a preponderance of positive x’s and negative if they have a preponderance of negative x’s. In general, the set of Y’s {Y(1), Y(2), …. , Y(99)} will have a maximum and a minimum: max Y and min Y, respectively. The difference between these two is called the range R: R = max Y - min Y If we adjust R by the scale parameter c, we get the rescaled range: rescaled range = R/c . Now, the probability theorist William Feller [7] had proven that if a series of random variables like the x’s 1) had finite variance, and 2) were independent, then the rescaled range formed over n observations would be equal to: R/c = k n 1/2 where k is a constant (in particular, k = (p /2)1/2 ) . That is, the rescaled range would increase much like the partial sums of independent variables (with finite variance), namely, the partial sums would increase by a factor of n1/2. In particular, for n = 99 in our hypothetical data, the result would be: R/c = k 991/2 . he found that in general the rescaled range was governed by a power law R/c = k nH where the Hurst exponent H was greater than ½ (Hurst found H @ .7) This implied that succeeding x’s were not independent of each other: x(t) had some persistent effect on x(t+1). This was what Hurst had observed in the data, and his calculation showed H to be above ½ [45]. To summarize, for reference, for the Hurst exponent H: • H = ½ : the flood level deviations from the mean are independent, random; the x’s are independent and correspond to a random walk; • ½ < H <=1: the flood level deviations are persistent - high flood levels tend to be followed by high flood levels, and low flood levels by low flood levels; x(t+1) tends to deviate from the mean the same way x(t) did; the probability that x(t+1) deviates from the mean in the same direction as x(t) increases as H approaches 1; • 0<=H< 1/2: the flood level deviations are anti-persistent - the x’s are mean-reverting; high flood levels have a tendency to be followed by low flood levels, and vice-versa; the probability that x(t+1) deviates from the mean in the opposite direction from x(t) increases as H approaches 0. The role of the Hurst exponent is to inform us whether the yearly flood deviations are independent or persistent. It is a measure of the way in which a data series varies in time. In financial markets it can be applied to predict trends. * The materials for this Appendix were quoted from [38]. 109

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