How predictable trading can coexist with diffusive prices

A puzzle inside everyday trading

An institutional investor seeking a large position may split the transaction into smaller orders rather than execute everything at once. Those smaller trades share a direction: buying or selling. This creates persistence in the sequence of market orders, which execute immediately against available liquidity.

The paper starts from two empirical regularities discussed in earlier research: long-lasting correlations in order direction and a roughly square-root relationship between a large order’s total size and its price impact. The puzzle is why persistent trading can coexist with diffusive prices, whose mean-squared displacement grows proportionally to elapsed time.

Sato, Fujiwara and Kanazawa propose an exactly solvable mathematical model connecting these regularities. This is a theoretical preprint, with numerical illustrations, rather than a new empirical test of market data.

Building a deliberately simple market

The researchers extend the Lillo–Mike–Farmer model of order splitting. Each simulated trader repeatedly draws a large parent order, called a metaorder, and executes it through unit-sized child orders. Every child order within a metaorder has the same direction. Once execution finishes, the trader independently draws a new size and a new buying or selling direction, with equal probabilities.

Metaorder sizes follow a power-law distribution controlled by an exponent α between one and two. This produces a finite average size but an infinite size variance in the idealized distribution. The order-direction correlation decays with exponent α − 1, slowly enough to constitute long memory.

The added ingredient is an assumed impact function: after executing volume q within a metaorder, its accumulated price contribution is proportional to q raised to an exponent δ. Each additional child order moves the price by the increment in that function.

For linear impact, δ equals one. For square-root impact, δ equals one-half: quadrupling executed volume doubles the accumulated impact rather than quadrupling it. The model assumes this relationship; it does not explain its microscopic origin.

What the exact solution establishes

The authors map each metaorder to a flight in a Lévy walk, a random-motion framework linking movement duration and displacement. Here, execution duration determines displacement through the nonlinear impact function. This mapping makes the model analytically tractable.

The central result is a boundary between two long-time behaviors. When 2δ is smaller than α, mean-squared price displacement grows linearly with time: normal diffusion. When 2δ exceeds α, it grows as time raised to 1 + 2δ − α: superdiffusion. At equality, a logarithmic correction appears.

Square-root impact always falls on the diffusive side for the assumed range 1 < α < 2. Linear impact falls on the superdiffusive side. Other nonlinear impact functions can also produce diffusion, provided they satisfy the same inequality. Square-root impact is therefore sufficient within this model, not uniquely necessary.

The intuition concerns the variance of completed metaorders’ price contributions. Although order sizes have infinite variance, their square-root impacts have finite variance. With independent, randomly signed metaorders and finite mean execution duration, this allows ordinary diffusive scaling to emerge.

Persistent orders, shorter-lived price correlations

The paper also calculates correlations in the continuous-time price velocity, the counterpart of one-step returns in the discrete model. Their long-lag decay exponent is α − 2δ + 1.

Under square-root impact, that becomes α, which exceeds one. Price correlations are therefore short-memory in the technical sense that their long-lag tail is integrable. Order-direction correlations, by contrast, retain long memory.

This does not mean price correlations vanish immediately. Nor does linear growth of mean-squared displacement alone establish complete unpredictability, an exactly Brownian process at every timescale, or an absence of profitable strategies. The result concerns diffusion and correlation scaling under specified assumptions.

What remains outside the model

The baseline uses deterministic impact, independent metaorders and simplified execution. With multiple traders, one is selected randomly to submit each child order. The paper reports that extending to multiple splitters preserves the diffusion criterion; its continuous-time derivation sums independent trader contributions.

The authors also question whether an estimated decay in impact necessarily represents actual fading of an order’s influence. They report that propagator estimation can suggest such decay even for their model, which has no true impact decay. The broader supporting argument is attributed to work listed as in preparation.

Resting periods, post-execution impact decay and connections to volatility clustering are discussed through companion work, not established here as comprehensive explanations of real markets. Order-book dynamics, external shocks and changing market conditions remain outside the baseline. The preprint’s contribution is narrower: a solvable mechanism showing how sufficiently concave impact can reconcile persistent order flow with long-time price diffusion.