Stock Markets are Unpredictable, but Can be Exploitable

[摘要] In 1900, Bachelier [1] was the first to consider the variation of stock prices as a path of random walk. Subsequent researchersfound, however, the ups and downs of the stock indices are different from randomness except for very long time scales. Clearevidence of this non-randomness is that the distribution of returns is not Gaussian but universally has a fat tail [2]. Optimisticresearchers quickly attempted to utilize the information gathered from these returns to predict the market [3,4]. Others have usedagent-based models to generate series with return distributions similar to the real ones [5]. Here we have shown that the timesequences of some stock indices behave like the trajectory of a weakly-persistent random walk (WPRW). The trajectory of a WPRWis a crossover-fractal [6], a fractal which has one fractal dimension for short scales and another for large scales. We studied variouspieces of stock index series from ten different countries and found that some of them were crossover-fractals with the crossoversat about 20 minutes. Beyond that time the fractal dimension was close to 1.5, much like a simple random walk. Since the longrange variance of the WPRW is larger than that of a random walk, the stock index is harder to predict than a random walk. On theother hand, because of this persistent feature, one can exploit the market at shorter scales than 20 minutes and always profit byhigh-frequency transactions.

[关键词] Stock markets;Stock index;Crossover-fractal structure.