Markets exhibit behaviour much like the wind: how turbulent and windy it was yesterday will determine how windy it is today, and how the markets will fair two months or two years from now. The idea that market prices are independent, that the changes of prices are the result of random coin flips according to the normal distribution is clearly wrong when you have a look at the daily changes of a financial index [ fig. 14 ]
fig. 14: The daily percentage changes of the Dow Jones industrial average from 1928 to 2006 compared to what MFT would have predicted.
Clearly MFT is a bit lacking in describing financial markets. Markets are much more like the randomness of the wind, than they are to the coin flips of the normal distribution. With this in mind Mandelbrot set about using fractal geometry and his ideas of wild randomness to create his financial model “Fractal Brownian Motion in multifractal Trading Time”. This model, while still being worked on, has proved a lot better at analysing and shock testing many stock portfolios against the chaotic nature of the markets. Fractals are clearly absolutely fascinating from the perspective of pure mathematics. You can generate n-ball volumes from hypercubes in all dimensions, and can see how they turn up in binomial coefficients as well as the huge amount of information conveyed in the M-set, and the way it determines the shape of a filled Julia set. We can say for sure that they are defiantly mathematical curiosities and a wonder of modern pure mathematics. However, they are defiantly not just that. The way they describe the intricacies of nature, from the fractal dimensions of the UK coastline to describing the pattern of a sea shell as well as their incredible power in describing the pattern of financial markets. Fractals are definitely not just mathematical play things; they are provide a framework to answer problems of corporations or the cotton industry to that of mapping the complex plane though an iterative function. Fractals are wonderful. Conclusion
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