Axiomatic Market Making

Battle Count: 64

Extremely relevant to quantitative trading. The paper provides a closed-form, empirically identifiable quoting rule for market makers with three interpretable parameters (risk aversion, carrying cost, adverse-selection loss) that can be recovered from observable quoting behavior via three distinct moments. The phase transition theorem provides structural thresholds for market viability (functioning vs. frozen regimes) with explicit conditions on volatility, informed flow, and liquidity. The Recovery Theorem shows the full inventory cost function is observable in the limit order book depth profile. The empirical implementation roadmap (Section 10.6) gives a concrete step-by-step procedure for parameter identification and diagnostic testing. The Bertrand equivalence connects the axiomatic rule to competitive equilibrium. The three-liquidity-regime classification provides actionable regime identification for trading strategies. The paper directly addresses the core problem of how market makers should set bid-ask quotes.

The theoretical framework is mathematically sophisticated, involving convex analysis, Legendre-Fenchel duality, Cauchy functional equations, and axiomatic proof techniques. However, the empirical implementation is relatively straightforward once the framework is understood: three separate regressions identify the three parameters (mid-quote on inventory for gamma_0, spread on inverse liquidity at low pi for kappa_inv, spread on informed-fraction proxy for kappa_adv). The Recovery Theorem requires integrating cost gradients across order book levels. The main practical challenges are: (1) proxying the unobservable informed-trader fraction pi, (2) finding valid instruments for liquidity variation exogenous to other state variables, (3) handling boundary-regime nonlinearity near the phase transition, and (4) obtaining sufficient quote-level data with inventory records. The diagnostic tests for axiom violations are well-specified but require careful econometric implementation.

This is a purely theoretical/mathematical paper with complete formal proofs. All theorems (Full Characterization, Bijection, Irredundancy, Structural Separability, Recovery, Legendre-Fenchel Duality, Phase Transition, Bertrand Equivalence, Universal Structure) are stated with rigorous proofs. The axioms are precisely defined, the state space is fully specified, and the closed-form quoting rule is explicit. No empirical data or computational experiments are required for verification. The logical structure is fully self-contained and verifiable by any reader with the relevant mathematical background.