Below alpha of 2, distributions have no finite variance at all

Evidence

A value for alpha of 2 corresponds to the normal distribution, with its “nonmonstrous,” well-understood, mathematically tractable properties and rapidly diminishing tails. It is a form of “mild” randomness: the rapidly diminishing tails mean that extreme events are very infrequent. If alpha is less than 2, however, the tails of the distribution are sufficiently “fat” that the statistician’s standard ways of measuring the extent of a distribution’s “spread” (the standard deviation and its square, the variance) no longer exist: the integral expression that defines them does not converge, and so they have no finite value and are in that sense infinite. — Donald MacKenzie, An Engine, Not a Camera, 480

Categorization

• DEFINE: Clarifies the sharp dividing line between normal and Levy distributions: infinite variance below alpha=2.
• QUESTION: This infinite-variance property is what makes Gaussian-based option pricing (Black-Scholes assumes finite variance) strictly inapplicable to alpha<2 regimes.
• XREF: Connects to the Black-Scholes/Merton option pricing model’s reliance on lognormal returns — the core of the derivative-driven financialization story this book tells. — connects to: A finite sample’s variance cannot distinguish a fat-tailed from an infinite-variance distribution — Donald MacKenzie, An Engine, Not a Camera: How Financial…, 499

Broader Connections

• A finite sample’s variance cannot distinguish a fat-tailed from an infinite-variance distribution #expands (0.85) — Donald MacKenzie, An Engine, Not a Camera, 499