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From: Stephen Wolfram, A New Kind of Science
Notes for Chapter 8: Implications for Everyday Systems
Section: Financial systems
Page 1014

Randomness in markets. After the somewhat tricky process of correcting for overall trends, empirical price data from a wide range of markets seem to a first approximation to follow random walks and thus to exhibit Gaussian fluctuations, as noted by Louis Bachelier in 1900. However, particularly on ×cales less than a day, it has in the past decade become clear that, as suggested by Benoit Mandelbrot in the early 1960s, large price fluctuations are significantly more common than a Gaussian distribution would imply. Such an effect is easy to model with the approach used in the main text if different entities interact in clumps or herds - which can be forced if they are connected in a hierarchical network rather than just a line.

The observed standard deviation of a price - or essentially so-called volatility or beta - can be considered as a measure of the risk of fluctuations in that price. The Capital Asset Pricing Model proposed in the early 1960s suggested that average rates of price increases should be proportional to such variances. And the Black-Scholes model from 1973 implies that prices of suitably constructed options should depend in a sense only on such variances. Over the past decade various corrections to this model have been developed based on non-Gaussian distributions of prices.

Stephen Wolfram, A New Kind of Science (Wolfram Media, 2002), page 1014.
© 2002, Stephen Wolfram, LLC