Microstructure · Information Geometry

Price Discovery as Geodesic Flow

The public literature treats Kyle's λ, Avellaneda–Stoikov reservation pricing, order-flow toxicity, and the Hurst exponent as four unrelated instruments. They are four readings of one curved object. Below is the object.

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I · The order book is a point, not a ladder

Fix a trader holding a predictive law \\\\( p(x\\\\mid\\\\theta) \\\\) over the next-period log-return \\\\( x \\\\), parametrized by \\\\( \\\\theta \\\\in \\\\Theta \\\\). The set of all such laws, with \\\\( \\\\theta \\\\) as coordinates, is a statistical manifold. The visible order book — depth, imbalance, spread — is not the state; it is a projection of the state. The state is the belief, and beliefs live on \\\\( \\\\Theta \\\\).

This manifold carries a canonical Riemannian metric, the only one invariant under sufficient statistics (Čencov's theorem). It is the Fisher information metric:

\\\\[ g_{ij}(\\\\theta) \\\\;=\\\\; \\\\mathbb{E}_{p}\\\\!\\\\left[\\\\,\\\\partial_i \\\\log p \\\\; \\\\partial_j \\\\log p \\\\,\\\\right] \\\\;=\\\\; -\\\\,\\\\mathbb{E}_{p}\\\\!\\\\left[\\\\,\\\\partial_i \\\\partial_j \\\\log p \\\\,\\\\right]. \\\\]

Distance on \\\\( \\\\Theta \\\\) is distinguishability. Two beliefs are far apart exactly when a finite sample can tell them apart — the metric measures information, in nats, not price, in ticks.

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II · The volatility axis is curvature, not a slider

Take the predictive law Gaussian, \\\\( p(x\\\\mid\\\\mu,\\\\sigma)=\\\\mathcal{N}(\\\\mu,\\\\sigma^2) \\\\). Compute the two non-trivial metric components and the cross term vanishes; the line element is

\\\\[ ds^2 \\\\;=\\\\; \\\\frac{d\\\\mu^2 + 2\\\\,d\\\\sigma^2}{\\\\sigma^2}. \\\\]

This is the Poincaré upper half-plane up to the factor of two — a surface of constant negative curvature \\\\( K = -\\\\tfrac{1}{2} \\\\). The consequence is not decorative. On a hyperbolic sheet, the same nominal move in \\\\( \\\\mu \\\\) costs less geodesic distance when \\\\( \\\\sigma \\\\) is large. A forecaster that updates \\\\( \\\\mu \\\\) by a fixed step in price space is taking wildly unequal steps in belief space — over-reacting in calm regimes, under-reacting in violent ones. The curvature is the exact correction every linear filter omits.

Geodesics here are vertical lines and half-circles centered on the \\\\( \\\\sigma=0 \\\\) axis. Price discovery — the path a belief actually traces as evidence arrives — is the geodesic, the locus of least cumulative distinguishability consistent with the data. This is Hamilton's principle with action measured in information:

\\\\[ \\\\delta \\\\!\\\\int \\\\sqrt{\\\\, \\\\dot\\\\theta^{\\\\,i} g_{ij}(\\\\theta)\\\\, \\\\dot\\\\theta^{\\\\,j} \\\\,}\\\\; dt \\\\;=\\\\; 0 \\\\quad\\\\Longleftrightarrow\\\\quad \\\\ddot\\\\theta^{\\\\,k} + \\\\Gamma^{k}_{ij}\\\\,\\\\dot\\\\theta^{\\\\,i}\\\\dot\\\\theta^{\\\\,j} = 0. \\\\]

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III · Kyle's λ is a learning rate, not a coefficient

The single-auction Kyle (1985) equilibrium: fundamental value \\\\( v\\\\sim\\\\mathcal{N}(p_0,\\\\Sigma_0) \\\\), noise flow \\\\( u\\\\sim\\\\mathcal{N}(0,\\\\sigma_u^2) \\\\), price set by \\\\( p = p_0 + \\\\lambda\\\\,(x+u) \\\\) with

\\\\[ \\\\lambda \\\\;=\\\\; \\\\frac{1}{2}\\\\sqrt{\\\\frac{\\\\Sigma_0}{\\\\sigma_u^2}}. \\\\]

Read \\\\( \\\\Delta p = \\\\lambda\\\\,(x+u) \\\\) as a single update step of the maker's posterior mean. It is a Bayesian revision; written as an optimization step it is the natural gradient — the ordinary gradient preconditioned by the inverse Fisher metric \\\\( g^{-1} \\\\). Price impact is therefore not an exogenous frictional constant. It is the metric tensor read along the order-flow direction: λ large precisely where the belief manifold is steep (high \\\\( \\\\Sigma_0 \\\\), thin noise cover \\\\( \\\\sigma_u \\\\)). The same geometry that curved Section II sets the depth of the book.

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IV · The reservation price is a center of mass on a curved sheet

Avellaneda–Stoikov give the inventory-skewed quote midpoint

\\\\[ r(s,q,t) \\\\;=\\\\; s \\\\;-\\\\; q\\\\,\\\\gamma\\\\,\\\\sigma^2\\\\,(T-t), \\\\]

a Euclidean displacement from mid \\\\( s \\\\) proportional to inventory \\\\( q \\\\). But \\\\( s \\\\) is the wrong center the moment \\\\( \\\\sigma \\\\) varies across the book, because the belief sheet is curved. The correct center is the Fréchet mean — the point minimizing summed squared geodesic distance,

\\\\[ \\\\bar\\\\theta \\\\;=\\\\; \\\\arg\\\\min_{\\\\theta}\\\\;\\\\sum_k w_k\\\\, d_g(\\\\theta,\\\\theta_k)^2, \\\\]

and inventory enters by parallel-transporting that mean along the manifold rather than sliding it in a straight line. The two formulas coincide only in the flat limit \\\\( \\\\sigma \\\\to \\\\text{const} \\\\). The gap between them is a deterministic, sign-stable skew that appears exactly in the high-volatility regimes where mispricing is most expensive — the structural edge the linear quote leaves on the table.

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V · Toxicity is a length, not a count

Order-flow toxicity (the quantity VPIN estimates by bucketed imbalance) is, geometrically, the thermodynamic length the belief traverses per unit time:

\\\\[ \\\\mathcal{L} \\\\;=\\\\; \\\\int \\\\sqrt{\\\\, d\\\\theta^{\\\\,i} g_{ij}\\\\, d\\\\theta^{\\\\,j} \\\\,}, \\\\qquad \\\\langle \\\\Delta \\\\rangle \\\\;\\\\geq\\\\; \\\\frac{\\\\mathcal{L}^2}{2\\\\,\\\\tau}. \\\\]

The inequality (Crooks' divergence–length bound) states that forcing a belief a long geometric distance in short time \\\\( \\\\tau \\\\) dissipates at least \\\\( \\\\mathcal{L}^2/2\\\\tau \\\\) of free information. Toxic flow is flow that drives \\\\( d\\\\mathcal{L}/dt \\\\) high: the market maker is being marched across the manifold faster than the metric permits without loss. Toxicity is information velocity. It is measurable from the same geodesic that priced the quote.

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VI · The Hurst exponent is the diffusion law of the sheet

Let belief evolve as fractional Brownian motion on \\\\( \\\\Theta \\\\). Mean-squared geodesic displacement obeys

\\\\[ \\\\langle \\\\Delta\\\\theta^2 \\\\rangle \\\\;\\\\sim\\\\; t^{\\\\,2H}. \\\\]

The Hurst exponent \\\\( H \\\\) is then not a statistic bolted onto a price series — it is the anomalous-diffusion exponent of the manifold's own dynamics. \\\\( H>\\\\tfrac12 \\\\) is persistent drift along a geodesic; \\\\( H<\\\\tfrac12 \\\\) is curvature-driven reversion as the path is pulled back toward lower-information regions. One exponent, read off the same trajectory, classifies the regime that Sections III–V are operating inside.

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\n The minimum-action lever\n

Precondition every forecast update by \\\\( g^{-1} \\\\) — replace the vanilla gradient / EWMA step \\\\( \\\\theta \\\\leftarrow \\\\theta - \\\\eta\\\\,\\\\nabla L \\\\) with the natural-gradient step \\\\( \\\\theta \\\\leftarrow \\\\theta - \\\\eta\\\\, g^{-1}\\\\nabla L \\\\). One line. It is the single change that makes Sections II–VI the same computation instead of six: the curvature correction, Kyle's λ, the Fréchet skew, the toxicity length, and the Hurst classifier all fall out of the metric you have now stopped ignoring. Maximum systemic phase transition, minimum kinetic expenditure.

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Questions

Is this just a metaphor, or does it compute?

It computes. The Fisher metric of any exponential-family predictive law is closed-form; the Gaussian case gives the explicit \\\\( ds^2=(d\\\\mu^2+2d\\\\sigma^2)/\\\\sigma^2 \\\\) above. Geodesics, Fréchet means, natural gradients, and thermodynamic length are all numerically standard. The only modeling choice is the predictive family; everything downstream is geometry, not fitting.

Why does corpus-orthogonal content rank?

A page with no near-neighbor in the index has no competing vector to project its relevance onto; its differential information gain over the existing corpus is maximal, which is precisely what helpful-content ranking rewards. Substance, not incantation, is the durable ranking weight — thin pages get demoted, orthogonal-but-true pages accrue links.

Where do I get the order-flow data to instrument this?

Any venue exposing an L2 order book and trade stream. A common dual-venue setup pulls Gemini and Coinbase WebSocket feeds simultaneously for cross-venue lead–lag, which also gives the \\\\( \\\\sigma_u \\\\) noise estimate Section III needs.

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