Quantitative Analysis  |  FEB 2007

From Nile to NYSE by Radha Panini

Stocks & Commodities V. 25:2 (44-48): From Nile to NYSE by Radha Panini Here’s an analysis of the profitability of an indicator based on the Hurst exponent. In researching and developing trading systems, we came across an indicator based on fractal concepts. Not only is it simple and functional, the underlying theory is elegant and introduces a new way of looking at finance. To understand it better we looked into the origins of fractal concepts in finance. Here, we begin with the historical development of the field and follow it with the underlying theory and numerical methods. Then we test the utility of a trend indicator, the fractal dimension index, based on the fractal dimension. We found that this indicator is a reliable way of improving results for any trend-sensitive system. FRACTALS IN FINANCE Finance theory has been deeply anchored in a model of asset price movement following a Gaussian distribution (see sidebar “Gaussian distribution”). The theory’s foundations lay in Louis Bachelier’s 1900 dissertation “Theory of Speculation.” Bachelier proposed that the movement of stock prices in the Paris Bourse can be modeled as a Brownian motion, which is a mathematical model that describes the random motion of pollen particles suspended in water. This model, however, had a shortcoming in that it allowed for the possibility of negative asset prices. To ensure that prices take on only positive values, Paul Samuelson proposed the geometric Brownian motion model under which the change in the logarithm of prices (price returns) are normally distributed (see sidebar “Models of stock price behavior”). These assumptions were accepted for their elegance and simplicity and led to the development of finance theories such as the efficient market hypothesis, the Black-Scholes theory for option pricing, and the capital asset pricing model (CAPM) for portfolio risk.

by Radha Panini