A few selected generators yield so-called unifractal curves that exhibit the relatively tranquil picture of the market encompassed by modern portfolio theory. But tranquillity prevails only under extraordinarily special conditions that are satisfied only by these special generators. The assumptions behind this oversimplified model are one of the central mistakes of modern portfolio theory. It is much like a theory of sea waves that forbids their swells to exceed six feet.
The beauty of fractal geometry is that it makes possible a model general enough to reproduce the patterns that characterize portfolio theory’s placid markets as well as the tumultuous trading conditions of recent months. The just described method of creating a fractal price model can be altered to show how the activity of markets speeds up and slows down—the essence of volatility. This variability is the reason that the prefix “multi-” was added to the word “fractal.”
To create a multifractal from a unifractal, the key step is to lengthen or shorten the horizontal time axis so that the pieces of the generator are either stretched or squeezed. At the same time, the vertical price axis may remain untouched. In illustration 2, the first piece of the unifractal generator is progressively shortened, which also provides room to lengthen the second piece. After making these adjustments, the generators become multifractal (M1 to M4). Market activity speeds up in the interval of time represented by the first piece of the generator and slows in the interval that corresponds to the second piece (illustration 3).
Such an alteration to the generator can produce a full simulation of price fluctuations over a given period, using the process of interpolation described earlier. Each time the first piece of the generator is further shortened—and the process of successive interpolation is undertaken—it produces a chart that increasingly resembles the characteristics of volatile markets (illustration 4).
The unifractal (U) chart shown here (before any shortening) corresponds to the becalmed markets postulated in the portfolio theorists’ model. Proceeding down the stack (M1 to M4), each chart diverges further from that model, exhibiting the sharp, spiky price jumps and the persistently large movements that resemble recent trading. To make these models of volatile markets achieve the necessary realism, the three pieces of each generator were scrambled—a process not shown in the illustrations. It works as follows: imagine a die on which each side bears the image of one of the six permutations of the pieces of the generator. Before each interpolation, the die is thrown, and then the permutation that comes up is selected.
What should a corporate treasurer, currency trader or other market strategist conclude from all this? The discrepancies between the pictures painted by modern portfolio theory and the actual movement of prices are obvious. Prices do not vary continuously, and they oscillate wildly at all timescales. Volatility—far from a static entity to be ignored or easily compensated for—is at the very heart of what goes on in financial markets. In the past, money managers embraced the continuity and constrained price movements of modern portfolio theory because of the absence of strong alternatives. But a money manager need no longer accept the current financial models at face value.
Instead multifractals can be put to work to “stress-test” a portfolio. In this technique the rules underlying multifractals attempt to create the same patterns of variability as do the unknown rules that govern actual markets. Multifractals describe accurately the relation between the shape of the generator and the patterns of up-and-down swings of prices to be found on charts of real market data.