Abstract
It is now well established that the probability distribution of relative price changes of stock market aggregates has two prominent features. First, in its central region, the distribution closely resembles a Levy stable distribution with exponent α ≅ 1.4. Secondly, it has power-law tails with exponent v ≤ 4. Both these results follow from relatively low resolution analyses of the data. In this paper we present the results of a high-resolution analysis of a database consisting of 132,000 values of the S&P 500 index taken at 10 minute intervals. We find a third prominent feature, a delta function at the origin the amplitude of which shows power-law decay over time with an exponent c ≅ 2 / 3. We show that Continuous-Time Random-Walk (CTRW) theory can account for all three features, but predicts subdiffusion with a growth of the variance of the ln(price) as the cth power of time. We find instead superdiffusion with an exponent c ≅ 9/8 instead of 2/3. We conclude that CTRW theory must be extended to incorporate the effects of “Price Momentum”.
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