The multifractal model of asset returns composes a price process with a nonuniform trading-time process, combining discontinuity-like wild variation and clustered activity rather than merely fitting a single heavy-tailed return distribution.

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Price is a function of trading time, which in turn is a function of clock time. Again, the two steps in the model combine to produce a baby, far different from either parent. The final product has wild price fluctuations, the big jumps and fat tails we saw in cotton and many other non-normal price charts. It has the volatility that clusters here and there. Periods of big price changes group together, interspersed by intervals of more sedate variation. The tell-tale marks of long memory and persistence. It shows scaling. The moments, a term for the basic statistical characteristics of the price series, follow a familiar scaling pattern that is now captured in the function f of alpha. In fact, you can fashion an entire spectrum of price charts, some very wild, some very dependent, some partaking of both, just as we did earlier with the simplistic cartoons. Research so far suggests the model is accurate.

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And exactly how my current and best mathematical simulations of the market work. Their engine is a multifractal process. It takes normal clock time, deforms it into a unique form of trading time, and then generates a price chart from it all. Or it can go in the other direction. It can start with a normal price chart, and break it down into its two primitive components. One process that deforms time, and another that generates a price. To what end? To make a lab bench model of the market that we can use to assess risk, analyze investments, or guard against ruin. The key to it is multifractals, a subtle and beautiful topic. Recall the definition of a fractal, a pattern or object whose parts echo the whole, only scaled down. By contrast, a multifractal has more than one scaling ratio in the same object. Some parts of the object shrink quickly, others slowly. Put it another way, a fractal is like an object defined to be shown in black and white.