Non-Convex Market Makers: Coherence without Convexity, Friction, and Parallel Composition

Paper: arXiv 2611.00502

Authors: Peter Cotton

Abstract

Cost-function market makers are always assumed convex, and convexity is not what makes them coherent. For a path-independent maker the exact no-arbitrage condition is a chord condition, that every chord slope of the cost lie in the convex hull of payoffs; convexity enters the standard axiomatics through a separate requirement, monotone information incorporation, and dropping it leaves coherent non-convex makers. Rational flow against such a maker trades the convex envelope, so non-convexity costs expressiveness rather than soundness, and a proportional fee of at least the arbitrage depth restores no-arbitrage, which by conjugation is a bid-ask spread with an exact no-trade interval for any cost whatever. Combining fee-bearing makers is an infimal convolution solved by one monotone clearing-price root-find, with sparse participation and a consolidated limit order book as the aggregate, and a deep quadratic co-quoter is Moreau smoothing, which preserves coherence at every depth. Any quoted configuration outside the coherent hull admits a portfolio priced below its worst payoff, so arbitrageurs act as decentralized separation oracles, and coherence is relative to friction and to the cost of finding certificates.

Complexity vs Empirical Score

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

Why this score: The paper presents a highly theoretical and mathematically dense argument for non-convex market makers, challenging a fundamental assumption in the field. While it introduces novel concepts and derivations, it lacks empirical validation or backtesting, focusing purely on theoretical coherence and implications. The mathematical rigor is very high, but the empirical rigor is low.

Research Flowchart

  flowchart TD
    A[Research Goal: Re-evaluate Market Maker Coherence Conditions] --> B{Methodology: Analyze Cost-Function MM Axiomatics & Arbitrage Conditions};
    B --> C[Inputs: Standard MM Theory, Convexity Assumptions, Arbitrage Definitions];
    C --> D{Computational Process: Prove Chord Condition for No-Arbitrage; Analyze Rational Flow & Fee Impact};
    D --> E[Key Finding 1: No-Arbitrage is Chord Condition, not Convexity];
    E --> F[Key Finding 2: Non-Convexity Costs Expressiveness; Fees Restore No-Arbitrage (Bid-Ask Spread)];
    F --> G[Key Finding 3: Parallel Composition via Infimal Convolution (Consolidated LOB), Moreau Smoothing for Co-Quoting];