Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.
Benoit Mandelbrot
Clustering is a key feature or stylized fact that expresses itself in terms of the auto-correlated volatility or magnitude between time points when analyzing certain time series. With market data, large changes in price tend to be followed by more large changes in price, and vice versa.
This self-similarity can manifest in a decaying fashion such that the auto-correlation of increments remains high far into the future. This is known as long-range dependence (LRD). The self-similarity in LRD plays out over a longer period of time, such that auto-correlation decays by a power law rather than exponentially. A stationary process is said to have LRD if its autocorrelation function decays slowly in this characteristic way.
Fractional brownian motion is a statistically self-similar non-stationary random process that can model these auto-correlation effects. The history of this discovery goes back to Mandelbrot:
[Fractional Gaussian noise] was in itself controversial, because it gained the desirable and tractable property of stationarity at the price of introducing infinite-ranged temporal memory or LRD. LRD implies that in order to predict the next state of a system its whole past is needed. This is different from typical dynamical systems whose next state is determined just by the current state. Such systems are called Markovian. This property appeared to many to be inconsistent with the Markovian nature of the equations of motion.
Nature, A Dynamical Systems Explanation of the Hurst Effect..
The Hurst Effect
Recalling that a time series is just an ordered series of random values,
What happens when random values occurring either adjacently or far apart at various lags are correlated? In our multivariate case, this is autocovariance. And it turns out the following equation is used with the Hurst parameter H to compute the autocovariance at lags between lagged values x_1 and x_t.
A Hurst parameter above 0.5 indicates the system has long-range dependence, there may be a decaying, periodic or cyclical serial dependence. This often manifests as a trend due to auto-correlated increments. A value below 0.5 indicates that consecutive values have little or no auto-correlation, the time series is anti-persistent and noisy. A value of exactly 0.5 produces a Brownian motion random walk.
With this approach, synthetic data generating processes can also respect an important and unique property of stochastic processes and in the case of financial markets this is auto-correlation and the related but distinct concept of long-range dependence.
Volatility tends to heighten and cluster around significant events or due to certain natural cycles unfolding, and then slowly diffuses away. Very long running events (such as quantitative easing) seem to introduce LRD that moves the system quite far from equilibrium over an extended timeframe.
Notebook: https://github.com/regimelab/notebooks/blob/main/fractional_brownian_motion.ipynb
References & Further Reading
Fractional Brownian motions, Fractional Noises and Applications
M(andelbrot), Van Ness
Basic properties of the Multivariate Fractional Brownian Motion
Pierre-Olivier Amblard, Jean-François Coeurjolly, Frédéric Lavancier, Anne Philippe
A Dynamical Systems Explanation of the Hurst Effect and Atmospheric Low-Frequency Variability
Christian L. E. Franzke, Scott M. Osprey, Paolo Davini & Nicholas W. Watkins
Long Memory and Regime Switching
Francis X. Diebold, Atsushi Inuoe
& more in the README: https://github.com/regimelab/notebooks/blob/main/README.md#fbm--hurst-effectlong-memory