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arxiv: 2605.01878 · v1 · submitted 2026-05-03 · 🧮 math.PR

Random trade timing and power-law tails in realized prices

Won-Ki Seo This is my paper

Pith reviewed 2026-05-09 16:49 UTC · model grok-4.3

classification 🧮 math.PR
keywords power-law tailsrealized pricesrandom trade timingPareto distributionsMarkov-modulated Lévy processestrading heterogeneitymarket microstructurefat tails
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The pith

Random trade timing can generate power-law tails in realized prices from light-tailed latent processes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that heterogeneity in when trades occur can create Pareto-type tails in observed prices even if the underlying price movements follow a light-tailed process. Two models are analyzed: one where trades happen on a grid with type-specific probabilities, and another where waiting times follow generalized Erlang distributions with different rates. In each case the tail index is fixed by the slowest trading type while faster types only rescale the constant. Sufficient conditions are identified that turn the approximate power-law tails into exact Pareto distributions.

Core claim

We show that random trade timing can generate Pareto-type tails, possibly with a logarithmic correction, in realized prices even when the latent price process is light-tailed. In both models, the tail exponent is determined by the least frequent trading type, while the proportions of faster-trading types affect only the scale constant. We also provide sufficient conditions under which these Pareto-type tails sharpen to exact Paretian tails.

What carries the argument

The random trade time T applied to the Markov-modulated Lévy process X in the realized price P_T = exp(X_T), where heterogeneity across trader types shapes the tail of T through either discrete incidence probabilities or generalized Erlang intertrade times.

If this is right

  • The power-law exponent is controlled exclusively by the least frequent trading type.
  • Faster trading types change only the scale prefactor of the tail.
  • Under stated conditions the approximate tails become exact Pareto distributions.
  • The same qualitative tail generation occurs in both the incidence-grid and generalized-Erlang timing models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The mechanism supplies a microstructure route to fat tails that does not require heavy-tailed innovations or jumps in the latent price process.
  • Empirical checks could compare measured intertrade-time distributions against observed price-tail exponents to test consistency with the slowest-type prediction.
  • Greater dispersion in trading speeds across participants should, all else equal, produce heavier tails under this account.
  • If trading frequency correlates with price movements the independence assumption fails and the tail-generation result need not hold.

Load-bearing premise

Trade timing is independent of the latent price dynamics and the models of heterogeneous trading frequencies accurately capture real behavior.

What would settle it

Empirical data or a simulation in which all traders share the same trading frequency yet realized prices still display power-law tails with an exponent unrelated to any single frequency parameter would contradict the mechanism.

read the original abstract

This paper studies stochastic mechanisms under which light-tailed latent price dynamics yield realized prices with power-law tails. The realized price is modeled as $P_T=e^{X_T}$, where $X$ is a Markov-modulated L\'evy process and $T$ is the random time of the next trade. We consider two trade-timing environments. In the intertrade-incidence model, trades occur on a discrete grid with type-dependent probabilities. In the intertrade-time model, the waiting time to the next trade is generalized Erlang, allowing for heterogeneous arrival rates and transaction-completion delays. We show that random trade timing can generate Pareto-type tails, possibly with a logarithmic correction, in realized prices even when the latent price process is light-tailed. In both models, the tail exponent is determined by the least frequent trading type, while the proportions of faster-trading types affect only the scale constant. We also provide sufficient conditions under which these Pareto-type tails sharpen to exact Paretian tails. These results identify random trade timing and heterogeneity in trading behavior as a general mechanism for generating power-law tails in realized prices.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper studies mechanisms by which light-tailed latent price dynamics, modeled as a Markov-modulated Lévy process X, produce realized prices P_T = exp(X_T) with Pareto-type tails when the trade time T is random. Two explicit timing models are analyzed: an intertrade-incidence model on a discrete grid with type-dependent probabilities, and an intertrade-time model with generalized Erlang waiting times allowing heterogeneous rates and delays. The central claims are that the tail exponent is governed by the least frequent trading type (with faster types affecting only the scale constant) and that sufficient conditions exist under which the tails become exactly Paretian.

Significance. If the derivations hold, the work identifies random trade timing and heterogeneity in trading frequencies as a general, parameter-light mechanism capable of generating power-law tails from light-tailed fundamentals. This is a substantive contribution to mathematical finance and stochastic processes, offering a clean explanation for empirical heavy tails without requiring heavy-tailed latent dynamics. The explicit identification of the slowest trader as the tail driver, together with the sufficient conditions for exact Pareto behavior, provides falsifiable predictions and strengthens the result's applicability.

minor comments (3)
  1. The abstract and introduction would benefit from a brief statement of the precise regularity conditions on the Lévy process X (e.g., moment-generating-function domain) that are used throughout the tail-asymptotics arguments.
  2. In the intertrade-time model, the notation for the generalized Erlang parameters (rates and delays) should be introduced with an explicit table or diagram to avoid ambiguity when the mixture over types is formed.
  3. A short remark on the robustness of the tail exponent to small correlations between trading intensity and the price process X would clarify the scope of the independence assumption without altering the main theorems.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. We appreciate the recognition of the contribution regarding random trade timing as a mechanism for power-law tails from light-tailed processes. Since no specific major comments were raised in the report, we provide no point-by-point responses below and note that we are ready to implement any minor clarifications or improvements as needed.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained stochastic analysis

full rationale

The paper establishes tail asymptotics for the realized price P_T = exp(X_T) where X is a Markov-modulated Lévy process with light tails and T is a random stopping time drawn from one of two explicit heterogeneous trading models (discrete-grid type-dependent probabilities or generalized Erlang intertrade times). The claimed results—that the tail exponent is governed by the slowest trading type while faster types affect only scale, and that sufficient conditions yield exact Pareto tails—follow directly from the mixture structure of the stopping distribution and standard large-deviation or renewal-type arguments applied to the stopped process. No parameter is fitted to data and then re-labeled as a prediction, no result is defined in terms of itself, and no load-bearing step reduces to a self-citation whose content is merely the present claim. The derivation therefore remains independent of its inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The paper's results depend on the specific constructions of the two trade-timing models and standard properties of Lévy processes. No new entities are postulated, but the models introduce several parameters for trading heterogeneity.

free parameters (2)
  • type-dependent trading probabilities
    Parameters in the intertrade-incidence model that control the likelihood of different trade types occurring on the discrete grid.
  • heterogeneous arrival rates and delays
    Parameters in the intertrade-time model defining the generalized Erlang distribution for waiting times.
axioms (2)
  • domain assumption The latent price process X is a Markov-modulated Lévy process with light tails.
    This is the starting point for the realized price P_T = exp(X_T) to be light-tailed without random T.
  • domain assumption Trade timing T is independent of the price process X.
    Assumed in the modeling of random trade times in both environments.

pith-pipeline@v0.9.0 · 5482 in / 1549 out tokens · 60853 ms · 2026-05-09T16:49:31.055801+00:00 · methodology

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Reference graph

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