Diagrams 1, shown below, is the surface of the Australian All Ordinaries Index from 1875 and is representative of the growth in the values inherent in the Australian Economy. The first of many of the characteristics of this surface is the beautifully consistent average rate of growth. This is demonstrated.

The All Ordinaries Index, monthly chart from 1875 to October 2009.  

Diagram 1

Diagram 1a

by the lower rising line on diagram 1a that defines perfectly the logarithmic trend in growth. The upper rising line on diagram 1a connects the high of 1889, the high in 1987 and the recent high in November 2007. It may be noticed that the upper line is rising slightly higher than the trend, suggesting that trend growth is slowly increasing over the years.

A study of this index and Australian history tends to provide a reality check to the minority group who, perhaps in good conscience, predict dire times ahead. The market has reacted and recovered from major events such as:

•  the collapse of the Australian Banking System 1889  to 1894
  
•  World War One with a substantial part of a generation lost
  
• The 1929 market collapse (not as severe as the US event)
• The Great Depression during the 1930’s
• World War Two when for a period it seemed possible that the axis powers could win.
• The constant threat of a nuclear holocaust at any time.
• The Korean War, the Malaysian emergency (courtesy of Indonesia) and Vietnam.
• The 1970 to 1974 collapse (!ustralia’s biggest) due to falling commodity prices and the OPEC oil crisis.
  
• An oil price crisis again in 1980 to 1982.
• Collapse in commodity prices post 1987.
• Iraq’s invasion of Kuwait and the first Gulf war in the early 1990’s.
• Asian currency crisis of October 1997 (10 year anniversary of the 1987 market collapse).
• The “Tech Wreck” bubble of 2000.
• The second Gulf war in 2003.
• The most recent collapse of the markets and the “Global Financial Crisis”.  

The surface referred to in diagram 1 could be described as very choppy with no apparent rhyme or reasoning to the market motion. In fact the surface could best be described as chaotic. That the market is a chaos model was determined by Benoit Mandelbrot in the mid 1990’s following the publication of “Fractals in Scaling and Finance”, 1997. Mandelbrot is considered the “father” of fractal geometry, the science relating to the order that exists in a chaotic structure.

Knowing the markets are a fractal model and applying this knowledge to a trading technique with more reliability, and a better win to loss ratio than existing trading strategies remains a work in progress. However, some broad brush techniques have emerged and two such behaviours of fractals are the subject of this paper; Fractal Intercepts and an example of an Expansion Model.

Diagram 2

A Fractal structure, in relation to the financial markets, refers to a structure that is composed of a form that is mathematically proportioned to ratios of irrational numbers. Irrational numbers are numbers that have no ending. That is, the numbers following the decimal point continue indefinitely. 

Each larger fractal (pattern) formed in the market consists of smaller fractals combining together to form the larger fractal.  Multiples of these larger fractals combine to form the next larger fractal pattern. This characteristic may be seen in Diagram 2, the Cauliflower, by observing that the structure of each floret is a number of smaller but similar florets.

(The sources of these photographs are unknown other than the belief that they date back to 2001. We wish to acknowledge the original source if known.)
 
The mathematics of irrational numbers are the reason that the seeds in a sunflower, the pine nuts in a pine cone and the florets in a cauliflower can spiral out both clockwise and anti-clockwise at the same time without ever colliding. This spiral is expanding at the rate of 1.618033989 and these digits continue, perhaps into eternity.

Diagram 3

We often read that it is not possible to forecast the future. A better expression would be that forecasting the future with reliability and accuracy may become possible through the development and application of Fractal Geometry. This assumes that Fractal Geometry develops into a set of physical laws. 

It is a reliable and accurate forecast to say that the sun will rise in the East and set in the West. Sunrise, Moon rise, the phase of the moon, the height and time in the future of the tides and seasons can all be forecast accurately because of the application of physical laws.  

The financial markets provide a massive and continuous stream of high quality statistical data that provides an excellent research tool in the quest to develop knowledge to allow better results in the trading environment.

Examples of applications in the financial Markets:

Fractal Intercepts

A fractal component in the financial markets is a complex multi faceted structure, which conforms to ratios of irrational numbers, with the dominant ratio being 1.618033989. The parameters of these structures are determined by the aggregate of the decisions made by a great many individuals that reflect each individual’s positive or negative view of the markets, the hope or wish of future market action and the extent or aggressiveness of that.

The decision making that affects the Australian markets occurs globally and encapsulates trading time frames from seconds through to many years. The stream of data, representing aggregate sentiment, once rising will tend to continue to rise until that aggregate positive sentiment is exhausted. The longer the time frame, the greater will be the magnitude of the move. The rise will be periodically characterised by down moves, representing the structure of smaller fractal components, but the move up will continue. See diagram 1

Looking back at the Cauliflower we see that the Cauliflower is comprised of florets that are miniature Cauliflowers, and the miniature florets are themselves comprised of even smaller florets. This raises the question, how big can a Cauliflower grow and why does it complete after a specific number of iterations? Transposing this question to the markets raises the same question, how big is big?

In the case of the Cauliflower, the supply of nutrients available to be drawn from the surrounding environment become beyond the capacity of the increasing demand of the exponentially expanding Cauliflower. At this time the Cauliflower seeds, and continues the replication process through the production of a greater number of new and smaller plants. 

In the case of the financial markets, after a set sequence of expanding fractal components, the financial Cauliflower matures, descends into reproduction mode and emerges into the beginning of a new financial Cauliflower. This new market action resumes the replication (fractal) process.

An insight into this process may be gained when viewing a monthly chart that rises continuously for a number of months. The smooth line shows no signs of chaotic behaviour, however switching to a weekly or daily chart will reveal obvious periods of chaos. The smaller the time frame the more evident will be the underlying fractal sequences.

When a market is rising, the rise is a line representing the limits of the forming fractal plane and is known as the initiator. When this line bends, resulting in a change of direction, the resultant plane is known as the generator. The initiator can be described as the order in the market (trending component) while the generator can be considered the chaos component (corrective component). The initiator is the driving or thrusting component, while the generator is the corrective or consolidation component of market action. 

If the initiator is rising, it is doing so because the underlying sentiment is positive. This underlying positive sentiment may be periodically undermined by negative sentiment events. 

Diagram 4

These smaller fractal components come together, add algebraically, and create a temporary level of negative sentiment sufficient to reverse the initiator line. These smaller fractal components of varying durations become unsynchronised, diminishing the negative affect, and this allows the initiator line to continue.

Eventually, these smaller fractals grow sufficiently in magnitude and duration such that the cumulative algebraic affect (of the different magnitude fractals) become synchronised and are sufficient to terminate the initiator and reverse the market direction through the formation of the generator. See Diagram 4

This is the equivalent of the Cauliflower going to seed. In simple terms, the fractal components of the generator form and enlarge faster than those of the bigger initiator. They do this because the duration of the formation grows progressively larger as does the magnitude. The relationship in size of the initiator and the generator are consistent with 1.618033989 when measured from the start and finish bifurcation points of a fractal of the same magnitude. The bifurcation points may not be the extreme points of the initiator and the generator.

An application of this principle may be seen if it is acknowledged that the peaks in the market represent the extent of positive sentiment (the price level at which no one wants to buy) while the lows represent the extent of negative sentiment (the price level at which no one wants to sell)

Consider the significance of the point at which these planes of negative and positive sentiment intersect. This is the price level at which no one wants to buy and simultaneously the price level at which no wants to sell. Does the market die or does the intersection identify a turn in the market, a point at which positive becomes negative and/or negative becomes positive?

Consider the chart of the All Ordinaries Index in Diagram 4. The lows in the market, point A and point B represent the extent of negative sentiment across the fractal segment while the horizontal plane across the 1994 high represents the extent of positive sentiment. 

It is essential in attempting this form of analysis that the pointers (A and B) are both the termination points of the same fractal segment. The intersection that occurs at point C is always an important point, and in this case was the rapid drop in the market on the tenth anniversary of the 1987 crash precipitated by the crash of the Thai Baht and the Malayan Ringgit. This currency crisis had Global ramifications at the time, even though the fractal is only a 1:3 on a magnitude scale where 1:1 is the concluding fractal. The 1 in this example indicates that the ‘intersect’ at point C started at the origin of the impending fractal structure.

Diagram 5

 
Diagram 5 shows a 2:3 magnitude fractal intersect with the termination point occurring at the time of the attack on the world trade centres.

Late in 1998 (just beyond point B) this chart was shown to a group of 140 attendees at an equity management course at a Coffs Harbour resort on the Central N.S.W. coast. After demonstrating the technique, the attendees were asked to “forecast” the next market event, and correctly predicted the final months of 2001. September 2001 was the month of the attack on the World Trade Centres in New York. Note the fall in the market terminating in the vicinity of the intercept. 

These intercepts, when drawn correctly, consistently signal a significant sentiment event including market terminations, the 1:1 event.

A higher magnitude fractal predicts the conclusion of what is a wave three in Elliott Wave terms. Fractal analysis reveals the genius of Elliott and Gann in developing rules and techniques that brought much order to a structure whose actual formation was completely unknown.

Diagram 6

 
The rules and techniques to determine the relative magnitude of the fractal structures under consideration are not a topic in this brief paper, however traders and analysts will be able to locate a sufficient number of resources to aid the trading experience.

Diagram 7

It may be seen that when point A and point B have formed it becomes possible to determine the likely zone that the market could retrace to. Several other techniques serve to validate this zone, and also to provide a useful indication as to the top of the market.

Expansion Model

Diagram 8

Diagram 8 provides a brief look at using the initiator ranges to predict the next segment range. The segment of interest following the completion of point E is the likely top at point F. The process to calculate point F is shown on the chart, however the sequence is dependent on other mathematical relationships in duration and magnitudes of the fractal components of the eventual structure. 

This technique requires that the correct bifurcation points are used. For example, the just completed A F G fractal commenced on the 17th November 1992 and not at the immediate low following the 1987 correction.

The bifurcation point represents the conclusion of chaos and the beginning of order in the time frame under examination. The underlying forms of fractal segments is quite complex and to date it has only been possible to map a small number. However, this small number is providing a view and and an understanding of the remarkable uniformity of group opinions and beliefs that are the market.