🤖 AI Summary
This work derives, from first principles, a uniquely determined market maker quoting rule by imposing eight natural axioms and six environmental assumptions. It establishes that the mid-price is linear in inventory and that the bid-ask spread decomposes into components attributable to inventory risk and adverse selection. The study presents the first axiomatic framework guaranteeing the enforced uniqueness of the quoting rule, reveals that the limit order book encodes a recoverable inventory cost function, and uncovers a sharp phase transition between functional and frozen market states. Combining axiomatic derivation, functional equation analysis, and structural identification, the approach yields three identifiable parameters—each estimable from distinct observable moments—and establishes four meta-theoretic invariants that hold across all structural primitives.
📝 Abstract
This paper axiomatizes the bid-ask market maker's quoting rule. A quoting rule maps the maker's state, namely inventory, belief, variance, trade intensity, and informed-trader fraction, to a bid-ask pair. Eight natural axioms, together with six environmental assumptions on the maker's inventory cost, force a unique three-parameter family: the mid-quote is linear in inventory, and the spread decomposes additively into inventory and adverse-selection components. Each of the three parameters is identified from a distinct moment of the observable quoting rule, with the three identifications mutually decoupled. The eight axioms partition into a four-axiom indispensable core, one structural choice, and three modularity extensions. Two structural corollaries follow: the latent inventory cost function is recoverable from the limit order book, and a sharp phase transition separates a functioning regime from a frozen one. A closing meta-theorem identifies four features invariant across all admissible structural primitives within the axiom system. To our knowledge, this is the first forced-uniqueness axiomatization of the quoting rule.