9 Maximum Drawdown Under Distinct Heavy-Tailed Return Models
What the Gaussian Analytical Result Cannot Tell You
Status: Corrected synthetic diagnostic; point-in-time empirical rebuild pending
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 non-Gaussian power-law returns, the model class matters. Non-degenerate stable laws permit \(0<\alpha\leq2\); they cannot represent a finite-variance regime with \(\alpha>2\). Finite-variance, infinite-fourth-moment returns require a different family such as Student-t, Pareto-tailed mixtures, or tempered models. For \(1<\alpha<2\), \(T^{1/\alpha}\) grows faster than \(\sqrt{T}\) but remains sublinear in \(T\).
9.2 The Claim
The revised study will not assume that one estimated exponent identifies the drawdown law. It will compare valid model families and use exact, dated 252-session breach labels. The earlier manuscript’s claim that the 715-name panel was free of survivorship bias is withdrawn; the historical panel was survivorship-conditioned. A stronger empirical version requires a point-in-time universe and explicit delisting treatment.
Event attribution can be retained as metadata in a future empirical comparison, but it does not define a drawdown law. Attributed and unattributed paths may differ because their complete return processes differ; that possibility must be tested after preserving attribution uncertainty, sector composition, dependence, and point-in-time membership. No categorical drawdown prediction follows from the withdrawn FAT_STRUCTURAL/FAT_BINARY labels.
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: comparable drawdown calibration across valid infinite-variance and finite-variance regularly varying models, with point-in-time empirical validation. Novelty: potentially high, pending replication.