What we mean by ‘fractal markets’

When people say "Mandelbrot was right about markets", they usually mean something slightly different from what Mandelbrot actually showed. The popular reading is that crashes are more frequent than the bell curve predicts. That part is true, but it's not the most useful part.

The more useful part is this: the shape of the return distribution matters less for prediction and more for sizing.

The textbook picture

If you assume returns are normally distributed — the textbook assumption — you get neat answers to questions like:

• What's the chance of a 5σ move? (one in 1.7 million)
• How should I scale a position to lose at most 2% per trade?
• How quickly does my expected drawdown grow with leverage?

All three answers are wrong if returns aren't normal. The first answer is wrong by orders of magnitude. The second one over-allocates in calm markets and under-allocates in tail events. The third one hides the regime in which leverage actually breaks you.

The L-stable picture

Mandelbrot's preferred family — the L-stable distributions, of which the normal is the special case with α=2 — admits returns where:

• Variance can be infinite
• Higher moments don't exist
• Tail events arrive in clusters, not in isolation

In that world, no amount of "averaging more data" gives you a stable estimate of risk. The sample mean is unbiased; the sample variance isn't. The standard 95% confidence interval drawn around backtested performance is, mathematically, meaningless.

Why it matters more for sizing than for prediction

Here is the key practical point. Better return forecasts don't fix this problem. Better sizing rules do.

If your model has a 53% directional hit rate (which is good) and you size each trade as a fraction of your equity, the question that governs your long-run wealth isn't "is my prediction more accurate than yesterday?" — it's "what fraction is robust to the L-stable assumption?"

The full Kelly fraction is fragile under fat tails. Half-Kelly is the standard pragmatic adjustment. Even half-Kelly understates drawdown when tails are heavy.

In the working paper Mandelbrot–Kelly: from theory to position sizing, we walk through the correction we use in production at Evrydiki. The short version: the Kelly fraction has to be discounted by a function of the tail index α, not by a constant.

What we don't claim

We don't claim to predict crashes. We don't claim to know the true α for any given market. We don't claim our sizing is optimal in any formal sense — it is calibrated to be robust under a range of α values that are empirically plausible for the markets we trade.

The discipline isn't to outguess the market. It's to size such that no plausible tail breaks the firm.

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This is an Insider's Note — a short methodology piece. See research for the full archive and the underlying paper.