Chaos Theory in Trading by Paul Cottrell

It is possible to graph returns to determine the level of chaotic behavior in a financial system.  This is accomplished with graphing returns along the x-axis pertaining to t-1 and the y-axis with only t, whereby t is time.  By graphing returns in this fashion we are determining the previous return affects for the next period’s return.

The empirical evidence suggests that daily returns are not autocorrected from the previous day.  But on larger time scales trends do emerge—leading to autocorrelation of previous period returns with current returns.  Another way to determine the level of chaotic behavior in a time series is to calculate the fractal efficiency ratio.  A ratio equal to 1 is a pure trending system, whereas a ratio equal to zero is considered a pure chaotic system. 

Yet another method to determine the smoothness of the market is describing the market dynamics in terms of Mandelbrot markets.  A Mandelbrot market is a method to determine the level of trending in a time series.  This smoothness is defined by the value that H holds.

A mean reverting system has H < 0.5.  A pure Brownian motion system has H = 0.5.  For H > 0.5 a trending time series is present.  Higher H values are lumpier market surfaces.

Mandelbrot–time is another way to look at the strange behavior of financial markets Mandelbrot (2004).  There are two types of time: trading–time and clock–time.  In trading–time, the time is varying where the velocity depends on the speed of the price.  This can be expressed as the first derivative, whereby delta price is divided by delta time.  For an example of trading–time, during high volatility days the price action is high—leading to faster time within trade–time.  Lower volatility days have slower trade–time.  Many traders express their perceptions of time with the following terms: rapid price movement, or a slow trading day.  Clock–time is the standard time and is constant in velocity. 

Time is relative to level of price change, which can be used to help model discontinuous markets. To fill the gap in discontinuous markets one should build a Brownian motion bridge.  Mandelbrot–time can help frame volatility in terms of delta time.  This is similar in concept to the space–time bending with gravity, whereby trade–time bends with level of price action.