Recent studies have found that many financial market time series scale as f
ractals. Much of the methodology is explicitly derived from physics, so muc
h so that the field has been called econophysics. Since fractals have been
identified primarily in the physical sciences, there has been a tendency in
this literature to assume that the mechanisms leading to fractality in fin
ancial markets are analogous, The parallels between physics and economics c
annot be carried too far, however, since the structural equations used in e
conometric models do not posit the same causal mechanisms. The widely used
cascade model in physics does not operate in economics. For instance, excha
nge rates are not determined by multiplicative cascades, but rather by diff
erentials in rates of return and relative prices. Moreover, the property of
long-term scaling symmetries, which has been found in some physical proces
ses, runs counter to economic theory. Economic processes are turbulent at s
hort horizons, but theory states that at longer horizons they should conver
ge to an equilibrium state. Nevertheless, the equations normally used to pr
edict exchange rates do imply that currencies will show fractal properties,
at least at near-term horizons. These equations do not generate strong sca
ling symmetries, but rather only weaker scaling symmetries over shorter tim
e intervals. Empirical tests find that both exchange rates and the interest
rate differentials that cause them are fractal: they show non-integer dime
nsionality. The degree of fractality, measured by the codimension, diminish
es as a function of the time scale. At long horizons, financial markets mov
e toward a non-fractal state. The probability distribution also shifts as t
he time scale increases. (C) 2000 Elsevier Science B.V. All rights reserved
.