Leptokurtic

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

Twenty Centuries of Financial Data: What 240 Countries and 2,000 Years Reveal Episode 1 – Twenty Centuries of Financial Data: What 240 Countries and 2,000 Years Reveal
23 min read Keywords:  forex, exchange rates, fat tails, gold, silver, interest rates, commodity prices, sovereign debt, volatility, medieval finance, currency crisis, open data

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.

Detecting Crashes with Fat-Tail Statistics Episode 2 – Detecting Crashes with Fat-Tail Statistics
38 min read Keywords:  fat tails, crash detection, LPPLS, extreme value theory, DFA, Hurst, Taleb, Sornette, Bouchaud, bitcoin, finance, rust, revenue, profit, momentum, volatility targeting

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.

The Tail Hedge Debate: Spitznagel Is Right, AQR Is Answering the Wrong Question Episode 3 – The Tail Hedge Debate: Spitznagel Is Right, AQR Is Answering the Wrong Question
52 min read Keywords:  options, tail hedging, Spitznagel, AQR, volatility, SPY, backtesting

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) deep OTM puts beat SPY, with the Sharpe ratio peaking near 0.5%-1.0% annual premium before reversing. Macro signals are useless for timing.

Finance Is Geometry, and It All Comes Back to Jensen’s Inequality Episode 4 – Finance Is Geometry, and It All Comes Back to Jensen’s Inequality
36 min read Keywords:  Jensen's inequality, Kelly criterion, ergodicity economics, tail hedging, logarithms, Spitznagel, Taleb, Peters

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, stay tuned!