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Impressive profits can be accumulated just by staying with a position during a trend. We would all be millionaires if only we could identify the trend early in its onset. While the trends are obvious in retrospect, it's another matter altogether to identify the trend in the heat of battle. Not only that, there may not be a trend at all at the time we expect one.

If we make a reasonable mathematical model of the market we can examine it parametrically. The conclusions we draw from this model can help us establish our entry points and strategies for trading the trends. We will view the market as a random walk problem to create our model.

Random walk for the market

In the same way that water can only flow downstream, time cannot be reversed in trading. In addition, prices can only be higher or lower in the same way that the river can only bend to the right or left. These elements constrain the random walk problem to a special form that mathematicians call "drunkard's walk." In the simplest form of this walk, the "drunk" steps only into a square diagonally to the right or into a square diagonally to the left as he steps forward. He must make a new decision with each step. To make the decision random, he flips a coin to determine the direction he will take. Repeated many times, the overlay of paths that he follows will look like a smoke plume. The question of the drunkard's destination can be answered through a well-known partial differential equation called the Diffusion Equation. The density of the smoke particles in the plume is analogous to the probability of the drunkard's location. A multiple-exposure photograph of the drunkard's walk repeated over and over would show its randomness. This photograph would show the composite paths to have a uniform density, widening from the initial position. The uniform density would make the sum of the paths look like smoke plume.

Further, random walk does not necessarily mean chaos. A minor variation of the drunkard's walk problem is to allow the random coin-flip decision to control the change of direction rather than the direction itself— that is, the random variable becomes momentum instead of direction. The partial differential equation describing this condition is known as the Telegrapher's Equation. The equation describes electric waves along telegraph wires, among other subjects. You can picture the result as the drunk reeling back and forth. He overcorrects around a general direction trying to reach an objective. This formulation of the problem, expressed in terms of physics, accurately portrays the river and explains why the river meanders. In a multiple-exposure photograph the paths are still randomly distributed. Nevertheless, the cycles are apparent in the shorter case of a single path. By analogy, the market has short-term cycles when the appropriate conditions prevail.

If enough traders ask themselves whether the market will go up today, the random variable is direction. Thus, conditions are established for the solution of the Diffusion Equation. On the other hand, if enough traders ask themselves whether the trend will continue, the random variable now becomes momentum. You could then expect the conditions to be established for the solution of the Telegrapher's Equation. The market is ripe for short-term cycle activity.