Economic fluctuations and anomalous diffusion

We quantify the relation between trading activity — measured by the number of transactions $N_{\Delta t}$—and the price change $G_{\Delta t}$ for a given stock, over a time interval $[t,t+\Delta t].$ To this end, we analyze a database documenting every transaction for 1000 U.S. stocks for the two-year period 1994–1995. We find that price movements are equivalent to a complex variant of classic diffusion, where the diffusion constant fluctuates drastically in time. We relate the analog for stock price fluctuations of the diffusion constant—known in economics as the volatility—to two microscopic quantities: (i) the number of transactions $N_{\Delta t}$ in $\Delta t,$ which is the analog of the number of collisions and (ii) the variance $W_{\Delta t}^2$ of the price changes for all transactions in $\Delta t,$ which is the analog of the local mean square displacement between collisions. Our results are consistent with the interpretation that the power-law tails of ${P(G}_{\Delta t})$ are due to ${P(W}_{\Delta t}),$ and the long-range correlations in $|G_{\Delta t}|$ are due to $N_{\Delta t}.$ DOI: http://dx.doi.org/10.1103/PhysRevE.62.R3023 © 2000 The American Physical Society