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How the Mathematics of Fractals Can Help Predict Stock Markets Shifts

In financial markets, two of the most common trading strategies used by investors are the momentum and mean reversion strategies. If a stock exhibits momentum (or trending behavior as shown in the figure below), its price on the current period is more likely to increase (decrease) if it has already increased (decreased) on the previous period.

When the return of a stock at time t depends in some way on the return at the previous time t-1, the returns are said to be autocorrelated. In the momentum regime, returns are positively correlated.

In contrast, the price of a mean-reverting stock fluctuates randomly around its historical mean and displays a tendency to revert to it. When there is mean reversion, if the price increased (decreased) in the current period, it is more likely to decrease (increase) in the next one.

A section of the time series of log returns of the Apple stock (adjusted closing price), shown below, is an example of mean-reverting behavior.

Note that, since the two regimes occur in different time frames (trending behavior usually occurs in larger timescales), they can, and often do, coexist.

In both regimes, the current price contains useful information about the future price. In fact, trading strategies can only generate profit if asset prices are either trending or mean-reverting since, otherwise, prices are following what is known as a random walk (see the animation below).

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