We can see this clearly in [ fig 12 ]
In the first image our unit length is 200km. By using this length we get the coast of the UK to be 2400km. However, in the second image we use a unit length of 50km. This gives us a coastline of 3400km. What would happen if we make our ruler smaller and smaller, measuring more and more of the detailed intricacies of the coastline? Well the length increases at a rapid pace. By the time you are down to 1mm rulers, going into all the cracks and crevasses of the cliff face, you have a coastline of tens of thousands of kilometres. This is a basic fractal behaviour that
fig. 12: Coastline measuring
we can see in many fractals including the Koch curve mentioned before. Coastlines are not the only fractal appearances in nature, but the pattern of shells or lighting strikes, or the surface of clouds, all exhibit fractal behaviour. The human uses of fractal geometry however lie in two distinct places, in computer generated imaging and in financial modelling. Due to nature’s similarity to fractals, computers use fractal geometry and iterative processes to try and replicate nature. This works well for two reasons; fractal geometry is very good at describing and understanding nature, and computers are very good at iterative processes. To generate realistic looking landscapes, computers start with a basic shape, such as a triangle for a mountain, and apply a iterative stochastic algorithm to the shape. Stochastic means an unpredictable process or event. In generating fractal landscapes this means that you have several generators for your fractal, and at each iteration you randomly select what generator you use. This basic process can be used to generate realistic looking landscapes. In Finical models, fractal geometry has proved revolutionary. Up until the Russian financial crisis of 1998 many of the traders and portfolio managers of the world operated under the umbrella term of “Modern Financial Theory”. The issue with “Modern Financial Theory” or MFT is that it was based on some inaccurate assumptions. While all models by necessity distort reality in one way or another, the assumptions of the orthodox financial theory are ridiculous if viewed alone. The main assumption that MFT relied on was that the price changes of a market follow a Brownian motion. What this means is that each price change is independent from the last: that there is statistical stationarity of the price changes, and that price changes follow the normal distribution. The reality of all this is that life is a lot more complex, and viewing markets as a series of efficient, independent and normally distributed changes is misleading. Mandelbrot noticed that fractal geometry might prove invaluable to the area of financial modelling since market timelines, [ fig. 13 ], exhibit fractal behaviour. By looking at graph with its axes removed you can’t tell if you are looking at one hour, one day or a one-year period. These graphs are self- affine fractal patterns.
fig. 13: Market timelines display a price index over time
52
Made with FlippingBook - professional solution for displaying marketing and sales documents online