Price manipulation in nonlinear transient impact models: rigidity before memory and complete positivity after memory

Paper: arXiv 2609.02447

Authors: Minhyeok Lee

Abstract

Transient impact models compose a nonlinearity with a memory kernel, and the order of composition determines the criterion for absence of price manipulation. We classify both orders. If an arbitrary instantaneous law $f$ acts on the trading rate before any nonzero integrable Volterra kernel, nonnegative cost on every finite piecewise-constant round trip forces $f$ to be affine, and linear for every nonzero convolution kernel. In particular, the power law $\mathrm{sgn}(x)|x|^δ$, $δ>0$, combined with power-law decay $t^{-γ}$, $0<γ<1$, admits manipulation if and only if $δ\ne1$: square-root impact is manipulable at every decay exponent, and the region left open by Gatheral’s slow-rate two-block bound $δ+γ\ge1$ collapses to the line $δ=1$. Earlier rigidity theorems require a kernel that is bounded at zero; the argument here is a zero-volume chattering pump read out by two thin baseline trades, and it applies to singular kernels. If instead a monotone readout acts on the impact state after the kernel, safety for all inputs and all readouts is equivalent to complete positivity of the kernel, with a constructive converse; in particular, square-root impact after power-law memory is manipulation-free. A remote compensating block shows that round-trip safety and all-input safety coincide for kernels with uniformly vanishing tails and differ, for permanent memory, by an explicit storage quotient. These mechanisms classify every two-mode Prony kernel, first-order time-inhomogeneous memory, and stable fully actuated matrix memory, and they quantify the friction, the two-block phase, and the switching complexity behind the power-law case. Calibrated exponent pairs all lie in the manipulable set: absent friction, concavity has to enter after the memory, not before it.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents highly complex mathematical derivations to address a fundamental problem in market microstructure. Its novelty lies in rigorously classifying conditions for price manipulation under different model architectures. While theoretically profound, it lacks empirical validation or practical backtesting.

Research Flowchart

  flowchart TD
    A[Research Goal: Classify Price Manipulation in Nonlinear Transient Impact Models] --> B{Order of Composition: Nonlinearity (f) and Memory Kernel (K)};

    B -- f before K --> C1[Method: Analyze "f(trading rate) before K"]
    B -- K before f --> C2[Method: Analyze "f(impact state) after K"]

    C1 --> D1{Rigidity Theorem: Zero-Volume Chattering Pump}
    C2 --> D2{Complete Positivity of Kernel}

    D1 --> E1[Outcome: f must be affine (linear for convolution) for non-manipulation. E.g., δ=1 for power law]
    D2 --> E2[Outcome: Kernel must be completely positive for non-manipulation. E.g., square-root impact after power-law memory is safe]

    E1 & E2 --> F[Key Finding: Calibrated exponent pairs are in manipulable set. Concavity must enter after memory, not before.]

Cited References (36)

1. Price Manipulation and Quasi-Arbitrage (2004) — link
2. Fluctuations and Response in Financial Markets: The Subtle Nature of ``Random’’ Price Changes (2004) — link
3. No-Dynamic-Arbitrage and Market Impact (2010) — link
4. Optimal Execution Strategies in Limit Order Books with General Shape Functions (2010) — link
5. Optimal Trade Execution and Absence of Price Manipulations in Limit Order Book Models (2010) — link
6. Exponential Resilience and Decay of Market Impact (2011) — link
7. Anomalous Price Impact and the Critical Nature of Liquidity in Financial Markets (2011) — link
8. Transient Linear Price Impact and Fredholm Integral Equations (2012) — link

0 of 36 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.