Model

Unimod is built on a single definition. The liquidity of a market is

where $\vartheta$ is the volume flowing through the liquidity (denominated in the quote asset) and $\sigma^2$ is the variance of the log-price increment over the same period.

The definition reads as a transfer function: volume is what a market receives, price is what it emits, and $\Lambda$ is the gain of that conversion — how much volume the market absorbs per unit of realized log-price variance. A deep market takes in large flow and barely moves; an illiquid one converts small flow into large price motion.

Why this definition

The quantity is agnostic to market structure: it applies to order books and automated market makers alike. And it is where the empirical literature keeps pointing:

• The square-root law. The price impact of a metaorder of volume $Q$ follows $\Delta p \propto \sigma \sqrt{Q/V}$ across venues, epochs and asset classes. Squaring it: $Q / (\Delta p)^2 \propto V/\sigma^2 = \Lambda$ — volume per unit of variance is the invariant quantity.
• The inelastic markets hypothesis. At the macro scale, markets do not absorb flow elastically — liquidity is finite, priced, and slow to regenerate.

Volume and variance are also the invariant coordinates of price processes themselves — every continuous martingale is a Brownian motion run on its variance clock — so a liquidity defined as volume per unit of variance is stated in the coordinates of the price process, not of any particular trading mechanism.

The constant-liquidity model

Fixing $\Lambda$ defines the constant-liquidity regime — the base model. Each market structure has a canonical incarnation of it: for order books, the parabolic order book; for AMMs, precisely what Unimod implements — an AMM whose invariant is not a fixed curve but a dynamic invariant maintaining $\Lambda$ constant as the pool trades.

A useful similitude: the model is the conductance model of electricity, with intensity in the role of volume and tension in the role of variance. Liquidity is a conductance — a reading that becomes literal in the liquidity topology.

Further reading

The full development — the derivation of the model from microstructure (as a scaling limit of order-book dynamics) and from macrostructure, and its resolutions of arbitrage, optimal consumption and optimal production — is in the whitepaper and the underlying research:

• Unimod: a new liquidity primitive — the protocol whitepaper.
• Towards a Theory of Liquidity — the seminal Unimodular research paper (in preparation).
• research.unimodular.xyz — interactive simulations behind the design decisions.