9 Maximum Drawdown Under Power Laws
What the Gaussian Analytical Result Cannot Tell You
Status: Planned — research literature surveyed, writing not yet begun
9.1 Motivation
Magdon-Ismail and Atiya (2004) derive the analytical distribution of maximum drawdown under Gaussian (Brownian motion) dynamics. The result is elegant: expected drawdown scales as \(\sqrt{T}\) at zero drift, \(\log T\) at positive drift, linearly at negative drift.
For power-law return distributions, no analogous analytical result exists. Vlasiuk (2025) derives single-period drawdown scaling under Lévy-stable dynamics (\(\propto \sigma T^{1/\alpha}\)) but not the full multi-period distributional law.
9.2 The Claim
The distribution of running maximum drawdown as a function of \(\alpha\) is not known. Our breach_35 and breach_50 empirical data — the fraction of stocks in each tail class that breach 35% and 50% peak-to-trough drawdown within a 12-month window — provides an empirical anchor for a simulation study calibrated to our observed \(\hat{\alpha}\) cross-section.
The FAT_STRUCTURAL vs FAT_BINARY sub-classification (Chapter 3) predicts that drawdown distributions within the SUPER_FAT class should differ: FAT_BINARY stocks have discrete jump-driven drawdowns; FAT_STRUCTURAL stocks have continuous power-law drawdowns. The distributional shapes differ even when \(\hat{\alpha}\) is identical.
9.3 Key prior work
- Magdon-Ismail and Atiya (2004) — Gaussian analytical result
- Filimonov and Sornette (2015) — empirical intraday drawdown power laws; dragon-king outliers
- Goldberg and Mahmoud (2017) — CED as coherent risk measure (finite variance)
- Vlasiuk (2025) — single-period scaling under stable dynamics
- Gap: full multi-period distributional law under \(\alpha\)-stable dynamics. Novelty: HIGH.