Federico Carrone
Two wooden-framed Galton boards against a black background: on the left the metal beads rest in the reservoir, on the right they have fallen through rows of pins into narrow columns that rise to a bell-shaped peak in the middle
Series

Leptokurtic

Financial returns have fat tails. Crashes that Gaussian models call impossible happen regularly. This series assembles twenty centuries of financial data, builds a crash-detection toolkit from 17 methods, tests tail hedging with real options data, and then explains the geometric logic tying the whole picture together.

On the image Each bead bounces left or right at every pin, each bounce independent of the last, and the sum of those small coin flips piles up into a bell curve. That independence is the assumption finance borrowed and the one markets break: when shocks cluster and feed on each other, the tails grow far fatter than the board allows. Two Galton boards, before and after a run, in the Matemateca collection of the Institute of Mathematics and Statistics, University of São Paulo, 2017. Exhibit by Estes Objethos Atelier. Photo: Rodrigo Tetsuo Argenton, CC BY-SA 4.0, via Wikimedia Commons.

4 episodes · 163 min in total · In progress

Black Monday should not have happened. Under the Gaussian model finance was built on, a twenty-something-sigma day is closer to never than to once-per-universe. It happened anyway. So did 2008, the 2010 flash crash, March 2020, and a long catalogue of others the textbook curve insists do not exist.

The textbook is wrong. Returns are leptokurtic (fat-tailed), and the rare events the normal distribution wants to round to zero are where most long-run wealth is actually made or destroyed. The series builds the empirical case first, then closes with the geometry that ties it together.

  1. Twenty centuries of data. Forex, gold, silver, debt, and GDP from 240 countries, assembled back to 1 CE. Fat tails are universal. Pegged currencies are the worst place to be. Every currency eventually loses to gold.
  2. Detecting crashes. A Rust and Python toolkit running fifteen methods (LPPLS, DFA, Hill, GSADF, momentum, others) against 96 historical drawdowns, scored with honest precision and recall. The methods that survive transfer to revenue and profit data, which is where this turns practical.
  3. The tail-hedge debate. Spitznagel and AQR are arguing past each other. With seventeen years of real SPY options data, deep out-of-the-money puts beat the index. An externally funded overlay wins by the widest margin. Macro signals are useless for timing.
  4. Jensen’s inequality. Logarithms, Kelly, ergodicity, and tail risk are all the same piece of geometry. Wealth compounds multiplicatively. The log is concave. Variability has a cost you can write down.

Mandelbrot, Taleb, Spitznagel, and Bouchaud are circling the same observation from different sides. The point of the series is to put receipts under it: fat tails are not a footnote on finance, they are most of what finance actually is.

Episodes

  1. Twenty Centuries of Financial Data: What 240 Countries and 2,000 Years Reveal

    Twenty Centuries of Financial Data: What 240 Countries and 2,000 Years Reveal

    · 24 min read

    We assembled forex-centuries, an open dataset of exchange rates, gold, silver, interest rates, commodity prices, GDP, sovereign debt, and more, across 27 sources spanning 1 CE to 2026 and covering 240 countries. Fat tails are universal. Pegged currencies are the most dangerous. Every currency loses against gold.

  2. Detecting Crashes with Fat-Tail Statistics

    Detecting Crashes with Fat-Tail Statistics

    · 39 min read

    We built fatcrash, a Rust+Python toolkit with 15 crash detection methods: LPPLS, DFA, EVT, Hill, Kappa, Hurst, GSADF, momentum/reversal, price velocity, and more. Tested on 96 drawdowns across BTC, SPY, Gold, 23 forex pairs, and equity crises with honest precision/recall/F1 metrics. Plus: which methods transfer to revenue and profit data.

  3. The Tail Hedge Debate: Spitznagel Is Right, AQR Is Answering the Wrong Question

    The Tail Hedge Debate: Spitznagel Is Right, AQR Is Answering the Wrong Question

    · 63 min read

    We tested Spitznagel’s tail hedging strategy and AQR’s critique with 17 years of real SPY options data. In the allocation-reducing framing AQR uses, selling SPY to fund puts, deep OTM puts lose at every budget. In the externally funded overlay Spitznagel actually proposes (100% SPY + put budget on top), the strategy shows a positive raw gap versus plain SPY only when the OTM band is set by strike (not delta), held for longer than the article first published, and tested across rolling windows that include a crash. The edge is regime-conditional: it pays in 6 of 13 rolling 5-year windows (every window that contains a ≥25% SPY drawdown) and drags by 2-3pp/yr in windows that don’t.

  4. Finance Is Geometry, and It All Comes Back to Jensen’s Inequality

    Finance Is Geometry, and It All Comes Back to Jensen’s Inequality

    · 37 min read

    Logarithms, Kelly, ergodicity, and tail risk meet at the same geometry: wealth compounds multiplicatively, the logarithm is concave, and Jensen’s inequality measures the cost of variability.

This series is in progress. More episodes are coming.