A Fractal Forecasting Model For Financial Time Series.pdf

Journal of Forecasting
J. Forecast. 23, 587–602 (2004)
Published online in Wiley InterScience (.com). DOI:
A Fractal Forecasting Model for
Financial Time Series
GORDON R. RICHARDS*
Sprint, Kansas, USA
ABSTRACT
Financial market time series exhibit high degrees of non-linear variability, and
frequently have fractal properties. When the fractal dimension of a time series
is non-integer, this is associated with two features: (1) inhomogeneity—
extreme fluctuations at irregular intervals, and (2) scaling symmetries—
proportionality relationships between fluctuations over different separation
distances. In multivariate systems such as financial markets, fractality is
stochastic rather than deterministic, and generally originates as a result of
multiplicative interactions. Volatility diffusion models with multiple stochastic
factors can generate fractal structures. In some cases, such as exchange rates,
the underlying structural equation also gives rise to fractality. Fractal princi-
ples can be used to develop forecasting algorithms. The forecasting method
that yields the best results here is the state transition-fitted residual scale ratio
(ST-FRSR) model. A state transition model is used to predict the conditional
probability of extreme events. Ratios of rates of change at proximate separa-
tion distances are used to parameterize the scaling symmetries. Forecasting
experiments are run using intraday exchange rate futures contracts measured
at 15-minute intervals. The overall forecast error is reduced on average by up
to 7% and in one instance by nearly a quarter. However, the forecast error
during the outlying events is reduced by 39% to 57%. The ST-FRSR reduces
the predictive error primarily by capturing extreme fluctuations more
accurately. Copyright © 2004 John Wiley & Sons, Ltd.
key words fractals; non-linear variability; state transitions; volatility diffu-
sions; financial markets; exchange r