live logic
What It Is
Power-law tails describe risk that refuses to disappear.
Market returns are not well described by a clean bell curve. Extreme moves arrive far more often than Gaussian assumptions imply, and the tail probability decays as a power law: P(X > x) ~ x^-alpha.
Here, X is the return or loss random variable, x is the threshold level, and alpha is the tail exponent. When alpha is near 3, variance can exist while skewness and kurtosis diverge. In practice, that means rare events are not decorative edge cases; they are part of the model.
Trading Edge
Estimate alpha, then size for the tail.
The Hill estimator focuses on the largest observations and estimates how heavy the tail is. If you price options under Gaussian assumptions, out-of-the-money puts can become catastrophically underpriced.
Estimate alpha with the Hill estimator.
Price OTM options with power-law tails.
Size positions for 4 sigma, not 2 sigma, worst case.
Assume the 30-year event can happen every year.