Foreign exchange market fluctuations as random walk in demarcated complex plane

Abstract

We show that time-dependent fluctuations $\{\Delta x\}$ in foreign exchange rates are accurately described by a random walk in a complex plane that is demarcated into the gain (+) and loss (-) sectors. $\{\Delta x\}$ is the outcome of $N$ random steps from the origin and $|\Delta x|$ is the square of the Euclidean distance of the final $N$-th step position. Sign of $\{\Delta x(t)\}$ is set by the $N$-th step location in the plane. The model explains not only the exponential statistics of the probability density of $\{\Delta x\}$ for G7 markets but also its observed asymmetry, and power-law dependent broadening with increasing time delay.