Pareto Power Laws for Modeling Stock Price Changes

Summary

The document raises the proposal that stock price changes may be better described by a Pareto power-law tail than by a thin-tailed distribution. It states the exceedance form in which the probability of a change exceeding a threshold declines as a power of that threshold, and reports an approximate tail exponent for stocks attributed to the referenced paper. This motivates attention to rare, large price moves when modeling return distributions.

The author asks how to estimate such a distribution from historical prices and requests practical examples in common programming languages. No estimation procedure, data, scripts, fitted parameters, or comparative evidence are supplied in the document itself. The stated exponent is therefore a claim attributed to the paper, not a result demonstrated here. The discussion is limited to the tail behavior and does not explain how to choose a threshold, distinguish a Pareto tail from alternatives, account for changing market conditions, or use a fitted distribution in risk or trading decisions.

Key ideas

• A Pareto tail models exceedance probabilities with a power-law decay.
• The document attributes an approximate tail exponent for stocks to a cited paper.
• Historical price data could be used to estimate a return distribution, but no estimation steps are provided.
• Tail estimates require practical choices such as a threshold, which the document leaves unresolved.

Tags

# Example how to model stock price with Pareto distribution according to Mandelbrot and Taleb


# Example how to model stock price with Pareto distribution according to Mandelbrot and Taleb












There's a paper by B. Mandelbrot and N. Taleb Mild vs Wild Randomness that says that Pareto distributions is a better fit for modelling price changes.

```
P(X>x) = Kx^-α
where P(X>x) is the probability of exceeding a variable x
and α is the asymptotic power law exponent for x large enough

α ~ 3 for stocks
```

Is there a more detailed, practical example how it can be used? To estimate price distribution from historical prices? Ideally with some scripts in Python, R, Java etc.

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