QUANTITATIVE RESEARCH / HEAVY-TAILED STOCHASTICS

Risk lives
in the tails.

Every path behind this text is a truncated Lévy flight — α-stable steps, α = 1.7. Watch the histogram: mass keeps landing where the Gaussian (amber, dashed) swears it can't. The empirical distribution (teal) grows tails no bell curve reaches. Ten-sigma days are not outliers. They are the model.

Read the research Why α, not σ

α — tail exponent

empirical F̂ₙ
Gaussian fit

n = 0 paths · α̂_Hill = — · 3σ events: — Gaussian

GENERALIZED CENTRAL LIMIT THEOREM — GNEDENKO & KOLMOGOROV, 1954

Strip away finite variance and the bell curve loses its monopoly. Sums of independent randomness converge to α-stable laws with power-law tails — the Gaussian is merely the α = 2 edge case. Markets live below it.

( Σᵢ Xᵢ − bₙ ) / aₙ ⟶_d S_α(β, σ, μ) , ℙ(|X| > x) ~ x^−α , 0 < α ≤ 2

Mandelbrot measured cotton at α ≈ 1.7 in 1963 — the paper that broke the Gaussian's monopoly. (Modern equity tails run nearer α ≈ 3: still fat, no longer stable.)

The Pareto–Lévy family

ONE PARAMETER, α — FROM CAUCHY CHAOS TO GAUSSIAN ORDER

α = 1.70

DENSITY — f(x) = π⁻¹∫₀^∞ e^{−t^α} cos(xt) dt (no closed form — integrated live)

TAIL, LOG–LOG — ℙ(X > x) ~ C_α x^−α (power law ⇒ straight line, slope −α)

Mandelbrot's 1963 name for the α-stable laws. Drag α down from 2 and watch both panels: the density's peak sharpens while its shoulders fatten, and on the log–log plot the stable tail stays a straight line — a power law — while the Gaussian plunges off a cliff. Two properties make the family inevitable: it is closed under addition (sums of stables are stable — the only laws with that property), and by the generalized CLT it is the complete set of possible limits for sums of i.i.d. randomness. Choosing a return distribution is not a modeling taste; it is choosing α.

Research programs

THREE OPERATORS, ONE OBJECT: THE PRICE PROCESS

dXₜ + dJₜ

∫

Jump-diffusion calculus

Itô is the easy half. We work in semimartingales with jumps — Lévy measures, rough volatility, and the discontinuities your diffusion-only SDE quietly assumes away. Model risk lives in the dropped jump term.

e^{iωt}

𝓕

Spectral analysis

Decomposing return series in frequency space: spectral density of volatility, long-memory signatures, and where the noise floor actually sits. The name of the house, and its method.

x^{−α}

Tail estimation

Hill estimators, extreme value theory, and the honest question every risk model dodges: what is α, really, and how fast does your VaR die when it drifts below 2?