Much of the original motivation of fractal geometry was to familiarize people with the idea of exact scaling in geometry, so the statistical scaling in financial data would be more readily accepted. In that regard, we owe much of the development of the material in this (and several thousand other!) web pages to fractality in finance.
Mathematical models of financial markets have a long history: one of the earliest was proposed by Louis Bachelier in his 1900 thesis, Theorie de la speculation. Predating Einstein's work by several years, Bachelier's thesis included the first mathematical analysis of Brownian motion. The underlying mathematics now is well-understood, and this remains the foundation of most current models, including that of Black-Merton-Sholes. We shall argue Brownian motion is not a good model, but that leaves the question of why so many analysts use this approach. One answer involves how the data are plotted. Typical graphs are price vs time (or Log(price) vs time, for longer time records). One of the graphs below is a record of IBM stock from 1959 to 1996, the other is a similar record of Brownian motion. Is it obvious which is which?
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Instead of prices, below we plot the differences of successive prices. Certainly, the right graph corresponds to Brownian motion: the differences are independent and normally distributed. The left graph has very large jumps, including one over ten sample standard deviations from the sample mean, an event that should happen about one time in 1024 if the jumps were normally distributed. In addition, the varying thickness of the "central band" of jumps is a signature of correlations: large jumps tend to cluster together, small jumps tend to cluster together. When viewing the differences, the non-Brownian nature of IBM stock is revealed.
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After the fact, any Brownian model can be adjusted to fit features of the observed data. With enough fixes, the match can be made quite good. However, different data may require different fixes, and so far none has been effective for predicting future behavior. In some sense, the situation recalls Ptolemaic cosmology: a sufficient number of epicycles can match the observed motion of the planets to any accuracy desired. But this analogy is flawed, because Ptolemy's model could make successful predictions. On the other hand, the stock market may be more complicated than the solar system.
Another approach is motivated by the observed ubiquity of statistically self-affine scaling of stock data. Build the simplest system that produces these scalings, and see how many of the other observed features follow automatically. Because different distributions of large jumps and of correlations are observed, the system should have parameters, a few rather than many. Because it is designed giving no thought to representing the mechanics of the stock market, we call the system a cartoon instead of a model.
The cartoon is built with initiator the diagonal of the unit square, and generator a piecewise-linear curve with three linear segments, two up and one down.
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Here are the answers.
Other than confounding the experts, the point of this example is to show that an exceedingly simple cartoon can produce a convincing forgery of real data. This suggests that at least some features of the real market data may result from fairly simple aspects of market dynamics.
Another interesting direction made possible by this cartoon approach is the rescaling of time, by which the "global dependence" and "large jump" aspects of these systems can be disentangled. First, it is easy to see the equation
|dY1|D + |dY2|D + |dY3|D = 1
has a unique solution D. Taking the trading time generators to be dTi = |dYi|D, it is clear the Y-T curve is unifractal. Moreover, the large jumps of the Y-t graph have been absorbed into the time rescaling, so what remains is fractional Brownian motion (global dependence, no large jumps) in multifractal time. The t-values of high volatility are stretched into longer T-values; the t-values of low volatility are compressed into shorter T-values. Our students call this VCR time: compress the boring parts (fast-forward through the commercials) and stretch the interesting parts (slow-motion through those parts deserving careful study).
This is a potentially useful development, but the main problem lies in how to extract the T generators from data, when dYi and dti are not known.
The more immediate, practical message is that risk is much more pernicious than Brownian motion models suggest. Especially in an increasingly globalized market, reinsurance is an important consideration.